Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra

Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra | F1000Research "use strict";function _typeof(t){return(_typeof="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t})(t)}!function(){var t=function(){var t,e,o=[],n=window,r=n;for(;r;){try{if(r.frames.__tcfapiLocator){t=r;break}}catch(t){}if(r===n.top)break;r=r.parent}t||(!function t(){var e=n.document,o=!!n.frames.__tcfapiLocator;if(!o)if(e.body){var r=e.createElement("iframe");r.style.cssText="display:none",r.name="__tcfapiLocator",e.body.appendChild(r)}else setTimeout(t,5);return!o}(),n.__tcfapi=function(){for(var t=arguments.length,n=new Array(t),r=0;r 3&&2===parseInt(n[1],10)&&"boolean"==typeof n[3]&&(e=n[3],"function"==typeof n[2]&&n[2]("set",!0)):"ping"===n[0]?"function"==typeof n[2]&&n[2]({gdprApplies:e,cmpLoaded:!1,cmpStatus:"stub"}):o.push(n)},n.addEventListener("message",(function(t){var e="string"==typeof t.data,o={};if(e)try{o=JSON.parse(t.data)}catch(t){}else o=t.data;var n="object"===_typeof(o)&&null!==o?o.__tcfapiCall:null;n&&window.__tcfapi(n.command,n.version,(function(o,r){var a={__tcfapiReturn:{returnValue:o,success:r,callId:n.callId}};t&&t.source&&t.source.postMessage&&t.source.postMessage(e?JSON.stringify(a):a,"*")}),n.parameter)}),!1))};"undefined"!=typeof module?module.exports=t:t()}(); dataLayer = dataLayer || []; // Standard GTM initialization - Google Consent Mode handles consent automatically (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl+ '>m_auth=hzk0Vc3qFsQYhCrIoHz68A>m_preview=env-1>m_cookies_win=x';f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-MWFK8L5J'); ;window.NREUM||(NREUM={});NREUM.init={distributed_tracing:{enabled:true},privacy:{cookies_enabled:true},ajax:{deny_list:["bam.nr-data.net"]}}; ;NREUM.loader_config={accountID:"438030",trustKey:"438030",agentID:"772317073",licenseKey:"97f8f67f26",applicationID:"772317073"} ;NREUM.info={beacon:"bam.nr-data.net",errorBeacon:"bam.nr-data.net",licenseKey:"97f8f67f26",applicationID:"772317073",sa:1} ;/*! For license information please see nr-loader-spa-1.236.0.min.js.LICENSE.txt */ (()=>{"use strict";var e,t,r={5763:(e,t,r)=>{r.d(t,{P_:()=>l,Mt:()=>g,C5:()=>s,DL:()=>v,OP:()=>T,lF:()=>D,Yu:()=>y,Dg:()=>h,CX:()=>c,GE:()=>b,sU:()=>_});var n=r(8632),i=r(9567);const o={beacon:n.ce.beacon,errorBeacon:n.ce.errorBeacon,licenseKey:void 0,applicationID:void 0,sa:void 0,queueTime:void 0,applicationTime:void 0,ttGuid:void 0,user:void 0,account:void 0,product:void 0,extra:void 0,jsAttributes:{},userAttributes:void 0,atts:void 0,transactionName:void 0,tNamePlain:void 0},a={};function s(e){if(!e)throw new Error("All info objects require an agent identifier!");if(!a[e])throw new Error("Info for ".concat(e," was never set"));return a[e]}function c(e,t){if(!e)throw new Error("All info objects require an agent identifier!");a[e]=(0,i.D)(t,o),(0,n.Qy)(e,a[e],"info")}var u=r(7056);const d=()=>{const e={blockSelector:"[data-nr-block]",maskInputOptions:{password:!0}};return{allow_bfcache:!0,privacy:{cookies_enabled:!0},ajax:{deny_list:void 0,enabled:!0,harvestTimeSeconds:10},distributed_tracing:{enabled:void 0,exclude_newrelic_header:void 0,cors_use_newrelic_header:void 0,cors_use_tracecontext_headers:void 0,allowed_origins:void 0},session:{domain:void 0,expiresMs:u.oD,inactiveMs:u.Hb},ssl:void 0,obfuscate:void 0,jserrors:{enabled:!0,harvestTimeSeconds:10},metrics:{enabled:!0},page_action:{enabled:!0,harvestTimeSeconds:30},page_view_event:{enabled:!0},page_view_timing:{enabled:!0,harvestTimeSeconds:30,long_task:!1},session_trace:{enabled:!0,harvestTimeSeconds:10},harvest:{tooManyRequestsDelay:60},session_replay:{enabled:!1,harvestTimeSeconds:60,sampleRate:.1,errorSampleRate:.1,maskTextSelector:"*",maskAllInputs:!0,get blockClass(){return"nr-block"},get ignoreClass(){return"nr-ignore"},get maskTextClass(){return"nr-mask"},get blockSelector(){return e.blockSelector},set blockSelector(t){e.blockSelector+=",".concat(t)},get maskInputOptions(){return e.maskInputOptions},set maskInputOptions(t){e.maskInputOptions={...t,password:!0}}},spa:{enabled:!0,harvestTimeSeconds:10}}},f={};function l(e){if(!e)throw new Error("All configuration objects require an agent identifier!");if(!f[e])throw new Error("Configuration for ".concat(e," was never set"));return f[e]}function h(e,t){if(!e)throw new Error("All configuration objects require an agent identifier!");f[e]=(0,i.D)(t,d()),(0,n.Qy)(e,f[e],"config")}function g(e,t){if(!e)throw new Error("All configuration objects require an agent identifier!");var r=l(e);if(r){for(var n=t.split("."),i=0;i {r.d(t,{D:()=>i});var n=r(50);function i(e,t){try{if(!e||"object"!=typeof e)return(0,n.Z)("Setting a Configurable requires an object as input");if(!t||"object"!=typeof t)return(0,n.Z)("Setting a Configurable requires a model to set its initial properties");const r=Object.create(Object.getPrototypeOf(t),Object.getOwnPropertyDescriptors(t)),o=0===Object.keys(r).length?e:r;for(let a in o)if(void 0!==e[a])try{"object"==typeof e[a]&&"object"==typeof t[a]?r[a]=i(e[a],t[a]):r[a]=e[a]}catch(e){(0,n.Z)("An error occurred while setting a property of a Configurable",e)}return r}catch(e){(0,n.Z)("An error occured while setting a Configurable",e)}}},6818:(e,t,r)=>{r.d(t,{Re:()=>i,gF:()=>o,q4:()=>n});const n="1.236.0",i="PROD",o="CDN"},385:(e,t,r)=>{r.d(t,{FN:()=>a,IF:()=>u,Nk:()=>f,Tt:()=>s,_A:()=>o,il:()=>n,pL:()=>c,v6:()=>i,w1:()=>d});const n="undefined"!=typeof window&&!!window.document,i="undefined"!=typeof WorkerGlobalScope&&("undefined"!=typeof self&&self instanceof WorkerGlobalScope&&self.navigator instanceof WorkerNavigator||"undefined"!=typeof globalThis&&globalThis instanceof WorkerGlobalScope&&globalThis.navigator instanceof WorkerNavigator),o=n?window:"undefined"!=typeof WorkerGlobalScope&&("undefined"!=typeof self&&self instanceof WorkerGlobalScope&&self||"undefined"!=typeof globalThis&&globalThis instanceof WorkerGlobalScope&&globalThis),a=""+o?.location,s=/iPad|iPhone|iPod/.test(navigator.userAgent),c=s&&"undefined"==typeof SharedWorker,u=(()=>{const e=navigator.userAgent.match(/Firefox[/\s](\d+\.\d+)/);return Array.isArray(e)&&e.length>=2?+e[1]:0})(),d=Boolean(n&&window.document.documentMode),f=!!navigator.sendBeacon},1117:(e,t,r)=>{r.d(t,{w:()=>o});var n=r(50);const i={agentIdentifier:"",ee:void 0};class o{constructor(e){try{if("object"!=typeof e)return(0,n.Z)("shared context requires an object as input");this.sharedContext={},Object.assign(this.sharedContext,i),Object.entries(e).forEach((e=>{let[t,r]=e;Object.keys(i).includes(t)&&(this.sharedContext[t]=r)}))}catch(e){(0,n.Z)("An error occured while setting SharedContext",e)}}}},8e3:(e,t,r)=>{r.d(t,{L:()=>d,R:()=>c});var n=r(2177),i=r(1284),o=r(4322),a=r(3325);const s={};function c(e,t){const r={staged:!1,priority:a.p[t]||0};u(e),s[e].get(t)||s[e].set(t,r)}function u(e){e&&(s[e]||(s[e]=new Map))}function d(){let e=arguments.length>0&&void 0!==arguments[0]?arguments[0]:"",t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:"feature";if(u(e),!e||!s[e].get(t))return a(t);s[e].get(t).staged=!0;const r=[...s[e]];function a(t){const r=e?n.ee.get(e):n.ee,a=o.X.handlers;if(r.backlog&&a){var s=r.backlog[t],c=a[t];if(c){for(var u=0;s&&u {let[t,r]=e;return r.staged}))&&(r.sort(((e,t)=>e[1].priority-t[1].priority)),r.forEach((e=>{let[t]=e;a(t)})))}function f(e,t){var r=e[1];(0,i.D)(t[r],(function(t,r){var n=e[0];if(r[0]===n){var i=r[1],o=e[3],a=e[2];i.apply(o,a)}}))}},2177:(e,t,r)=>{r.d(t,{c:()=>f,ee:()=>u});var n=r(8632),i=r(2210),o=r(1284),a=r(5763),s="nr@context";let c=(0,n.fP)();var u;function d(){}function f(e){return(0,i.X)(e,s,l)}function l(){return new d}function h(){u.aborted=!0,u.backlog={}}c.ee?u=c.ee:(u=function e(t,r){var n={},c={},f={},g=!1;try{g=16===r.length&&(0,a.OP)(r).isolatedBacklog}catch(e){}var p={on:b,addEventListener:b,removeEventListener:y,emit:v,get:x,listeners:w,context:m,buffer:A,abort:h,aborted:!1,isBuffering:E,debugId:r,backlog:g?{}:t&&"object"==typeof t.backlog?t.backlog:{}};return p;function m(e){return e&&e instanceof d?e:e?(0,i.X)(e,s,l):l()}function v(e,r,n,i,o){if(!1!==o&&(o=!0),!u.aborted||i){t&&o&&t.emit(e,r,n);for(var a=m(n),s=w(e),d=s.length,f=0;fn,p:()=>i});var n=r(2177).ee.get("handle");function i(e,t,r,i,o){o?(o.buffer([e],i),o.emit(e,t,r)):(n.buffer([e],i),n.emit(e,t,r))}},4322:(e,t,r)=>{r.d(t,{X:()=>o});var n=r(5546);o.on=a;var i=o.handlers={};function o(e,t,r,o){a(o||n.E,i,e,t,r)}function a(e,t,r,i,o){o||(o="feature"),e||(e=n.E);var a=t[o]=t[o]||{};(a[r]=a[r]||[]).push([e,i])}},3239:(e,t,r)=>{r.d(t,{bP:()=>s,iz:()=>c,m$:()=>a});var n=r(385);let i=!1,o=!1;try{const e={get passive(){return i=!0,!1},get signal(){return o=!0,!1}};n._A.addEventListener("test",null,e),n._A.removeEventListener("test",null,e)}catch(e){}function a(e,t){return i||o?{capture:!!e,passive:i,signal:t}:!!e}function s(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2],n=arguments.length>3?arguments[3]:void 0;window.addEventListener(e,t,a(r,n))}function c(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2],n=arguments.length>3?arguments[3]:void 0;document.addEventListener(e,t,a(r,n))}},4402:(e,t,r)=>{r.d(t,{Ht:()=>u,M:()=>c,Rl:()=>a,ky:()=>s});var n=r(385);const i="xxxxxxxx-xxxx-4xxx-yxxx-xxxxxxxxxxxx";function o(e,t){return e?15&e[t]:16*Math.random()|0}function a(){const e=n._A?.crypto||n._A?.msCrypto;let t,r=0;return e&&e.getRandomValues&&(t=e.getRandomValues(new Uint8Array(31))),i.split("").map((e=>"x"===e?o(t,++r).toString(16):"y"===e?(3&o()|8).toString(16):e)).join("")}function s(e){const t=n._A?.crypto||n._A?.msCrypto;let r,i=0;t&&t.getRandomValues&&(r=t.getRandomValues(new Uint8Array(31)));const a=[];for(var s=0;s {r.d(t,{Bq:()=>n,Hb:()=>o,oD:()=>i});const n="NRBA",i=144e5,o=18e5},7894:(e,t,r)=>{function n(){return Math.round(performance.now())}r.d(t,{z:()=>n})},7243:(e,t,r)=>{r.d(t,{e:()=>o});var n=r(385),i={};function o(e){if(e in i)return i[e];if(0===(e||"").indexOf("data:"))return{protocol:"data"};let t;var r=n._A?.location,o={};if(n.il)t=document.createElement("a"),t.href=e;else try{t=new URL(e,r.href)}catch(e){return o}o.port=t.port;var a=t.href.split("://");!o.port&&a[1]&&(o.port=a[1].split("/")[0].split("@").pop().split(":")[1]),o.port&&"0"!==o.port||(o.port="https"===a[0]?"443":"80"),o.hostname=t.hostname||r.hostname,o.pathname=t.pathname,o.protocol=a[0],"/"!==o.pathname.charAt(0)&&(o.pathname="/"+o.pathname);var s=!t.protocol||":"===t.protocol||t.protocol===r.protocol,c=t.hostname===r.hostname&&t.port===r.port;return o.sameOrigin=s&&(!t.hostname||c),"/"===o.pathname&&(i[e]=o),o}},50:(e,t,r)=>{function n(e,t){"function"==typeof console.warn&&(console.warn("New Relic: ".concat(e)),t&&console.warn(t))}r.d(t,{Z:()=>n})},2587:(e,t,r)=>{r.d(t,{N:()=>c,T:()=>u});var n=r(2177),i=r(5546),o=r(8e3),a=r(3325);const s={stn:[a.D.sessionTrace],err:[a.D.jserrors,a.D.metrics],ins:[a.D.pageAction],spa:[a.D.spa],sr:[a.D.sessionReplay,a.D.sessionTrace]};function c(e,t){const r=n.ee.get(t);e&&"object"==typeof e&&(Object.entries(e).forEach((e=>{let[t,n]=e;void 0===u[t]&&(s[t]?s[t].forEach((e=>{n?(0,i.p)("feat-"+t,[],void 0,e,r):(0,i.p)("block-"+t,[],void 0,e,r),(0,i.p)("rumresp-"+t,[Boolean(n)],void 0,e,r)})):n&&(0,i.p)("feat-"+t,[],void 0,void 0,r),u[t]=Boolean(n))})),Object.keys(s).forEach((e=>{void 0===u[e]&&(s[e]?.forEach((t=>(0,i.p)("rumresp-"+e,[!1],void 0,t,r))),u[e]=!1)})),(0,o.L)(t,a.D.pageViewEvent))}const u={}},2210:(e,t,r)=>{r.d(t,{X:()=>i});var n=Object.prototype.hasOwnProperty;function i(e,t,r){if(n.call(e,t))return e[t];var i=r();if(Object.defineProperty&&Object.keys)try{return Object.defineProperty(e,t,{value:i,writable:!0,enumerable:!1}),i}catch(e){}return e[t]=i,i}},1284:(e,t,r)=>{r.d(t,{D:()=>n});const n=(e,t)=>Object.entries(e||{}).map((e=>{let[r,n]=e;return t(r,n)}))},4351:(e,t,r)=>{r.d(t,{P:()=>o});var n=r(2177);const i=()=>{const e=new WeakSet;return(t,r)=>{if("object"==typeof r&&null!==r){if(e.has(r))return;e.add(r)}return r}};function o(e){try{return JSON.stringify(e,i())}catch(e){try{n.ee.emit("internal-error",[e])}catch(e){}}}},3960:(e,t,r)=>{r.d(t,{K:()=>a,b:()=>o});var n=r(3239);function i(){return"undefined"==typeof document||"complete"===document.readyState}function o(e,t){if(i())return e();(0,n.bP)("load",e,t)}function a(e){if(i())return e();(0,n.iz)("DOMContentLoaded",e)}},8632:(e,t,r)=>{r.d(t,{EZ:()=>u,Qy:()=>c,ce:()=>o,fP:()=>a,gG:()=>d,mF:()=>s});var n=r(7894),i=r(385);const o={beacon:"bam.nr-data.net",errorBeacon:"bam.nr-data.net"};function a(){return i._A.NREUM||(i._A.NREUM={}),void 0===i._A.newrelic&&(i._A.newrelic=i._A.NREUM),i._A.NREUM}function s(){let e=a();return e.o||(e.o={ST:i._A.setTimeout,SI:i._A.setImmediate,CT:i._A.clearTimeout,XHR:i._A.XMLHttpRequest,REQ:i._A.Request,EV:i._A.Event,PR:i._A.Promise,MO:i._A.MutationObserver,FETCH:i._A.fetch}),e}function c(e,t,r){let i=a();const o=i.initializedAgents||{},s=o[e]||{};return Object.keys(s).length||(s.initializedAt={ms:(0,n.z)(),date:new Date}),i.initializedAgents={...o,[e]:{...s,[r]:t}},i}function u(e,t){a()[e]=t}function d(){return function(){let e=a();const t=e.info||{};e.info={beacon:o.beacon,errorBeacon:o.errorBeacon,...t}}(),function(){let e=a();const t=e.init||{};e.init={...t}}(),s(),function(){let e=a();const t=e.loader_config||{};e.loader_config={...t}}(),a()}},7956:(e,t,r)=>{r.d(t,{N:()=>i});var n=r(3239);function i(e){let t=arguments.length>1&&void 0!==arguments[1]&&arguments[1],r=arguments.length>2?arguments[2]:void 0,i=arguments.length>3?arguments[3]:void 0;return void(0,n.iz)("visibilitychange",(function(){if(t)return void("hidden"==document.visibilityState&&e());e(document.visibilityState)}),r,i)}},1214:(e,t,r)=>{r.d(t,{em:()=>v,u5:()=>N,QU:()=>S,_L:()=>I,Gm:()=>L,Lg:()=>M,gy:()=>U,BV:()=>Q,Kf:()=>ee});var n=r(2177);const i="nr@original";var o=Object.prototype.hasOwnProperty,a=!1;function s(e,t){return e||(e=n.ee),r.inPlace=function(e,t,n,i,o){n||(n="");var a,s,c,u="-"===n.charAt(0);for(c=0;c 2?n-2:0),o=2;o {r(A[T],e,w),r(E[T],e,w)})),r(l._A,"fetch",y),t.on(y+"end",(function(e,r){var n=this;if(r){var i=r.headers.get("content-length");null!==i&&(n.rxSize=i),t.emit(y+"done",[null,r],n)}else t.emit(y+"done",[e],n)})),t}const O={},j=["pushState","replaceState"];function S(e){const t=function(e){return(e||n.ee).get("history")}(e);return!l.il||O[t.debugId]++||(O[t.debugId]=1,s(t).inPlace(window.history,j,"-")),t}var P=r(3239);const C={},R=["appendChild","insertBefore","replaceChild"];function I(e){const t=function(e){return(e||n.ee).get("jsonp")}(e);if(!l.il||C[t.debugId])return t;C[t.debugId]=!0;var r=s(t),i=/[?&](?:callback|cb)=([^&#]+)/,o=/(.*)\.([^.]+)/,a=/^(\w+)(\.|$)(.*)$/;function c(e,t){var r=e.match(a),n=r[1],i=r[3];return i?c(i,t[n]):t[n]}return r.inPlace(Node.prototype,R,"dom-"),t.on("dom-start",(function(e){!function(e){if(!e||"string"!=typeof e.nodeName||"script"!==e.nodeName.toLowerCase())return;if("function"!=typeof e.addEventListener)return;var n=(a=e.src,s=a.match(i),s?s[1]:null);var a,s;if(!n)return;var u=function(e){var t=e.match(o);if(t&&t.length>=3)return{key:t[2],parent:c(t[1],window)};return{key:e,parent:window}}(n);if("function"!=typeof u.parent[u.key])return;var d={};function f(){t.emit("jsonp-end",[],d),e.removeEventListener("load",f,(0,P.m$)(!1)),e.removeEventListener("error",l,(0,P.m$)(!1))}function l(){t.emit("jsonp-error",[],d),t.emit("jsonp-end",[],d),e.removeEventListener("load",f,(0,P.m$)(!1)),e.removeEventListener("error",l,(0,P.m$)(!1))}r.inPlace(u.parent,[u.key],"cb-",d),e.addEventListener("load",f,(0,P.m$)(!1)),e.addEventListener("error",l,(0,P.m$)(!1)),t.emit("new-jsonp",[e.src],d)}(e[0])})),t}var k=r(5763);const H={};function L(e){const t=function(e){return(e||n.ee).get("mutation")}(e);if(!l.il||H[t.debugId])return t;H[t.debugId]=!0;var r=s(t),i=k.Yu.MO;return i&&(window.MutationObserver=function(e){return this instanceof i?new i(r(e,"fn-")):i.apply(this,arguments)},MutationObserver.prototype=i.prototype),t}const z={};function M(e){const t=function(e){return(e||n.ee).get("promise")}(e);if(z[t.debugId])return t;z[t.debugId]=!0;var r=n.c,o=s(t),a=k.Yu.PR;return a&&function(){function e(r){var n=t.context(),i=o(r,"executor-",n,null,!1);const s=Reflect.construct(a,[i],e);return t.context(s).getCtx=function(){return n},s}l._A.Promise=e,Object.defineProperty(e,"name",{value:"Promise"}),e.toString=function(){return a.toString()},Object.setPrototypeOf(e,a),["all","race"].forEach((function(r){const n=a[r];e[r]=function(e){let i=!1;[...e||[]].forEach((e=>{this.resolve(e).then(a("all"===r),a(!1))}));const o=n.apply(this,arguments);return o;function a(e){return function(){t.emit("propagate",[null,!i],o,!1,!1),i=i||!e}}}})),["resolve","reject"].forEach((function(r){const n=a[r];e[r]=function(e){const r=n.apply(this,arguments);return e!==r&&t.emit("propagate",[e,!0],r,!1,!1),r}})),e.prototype=a.prototype;const n=a.prototype.then;a.prototype.then=function(){var e=this,i=r(e);i.promise=e;for(var a=arguments.length,s=new Array(a),c=0;c e())),t};function m(e,t){i.inPlace(t,["onreadystatechange"],"fn-",E)}function b(){var e=this,t=r.context(e);e.readyState>3&&!t.resolved&&(t.resolved=!0,r.emit("xhr-resolved",[],e)),i.inPlace(e,f,"fn-",E)}if(function(e,t){for(var r in e)t[r]=e[r]}(o,p),p.prototype=o.prototype,i.inPlace(p.prototype,J,"-xhr-",E),r.on("send-xhr-start",(function(e,t){m(e,t),function(e){h.push(e),a&&(y?y.then(A):u?u(A):(w=-w,x.data=w))}(t)})),r.on("open-xhr-start",m),a){var y=c&&c.resolve();if(!u&&!c){var w=1,x=document.createTextNode(w);new a(A).observe(x,{characterData:!0})}}else t.on("fn-end",(function(e){e[0]&&e[0].type===d||A()}));function A(){for(var e=0;e {r.d(t,{t:()=>n});const n=r(3325).D.ajax},6660:(e,t,r)=>{r.d(t,{A:()=>i,t:()=>n});const n=r(3325).D.jserrors,i="nr@seenError"},3081:(e,t,r)=>{r.d(t,{gF:()=>o,mY:()=>i,t9:()=>n,vz:()=>s,xS:()=>a});const n=r(3325).D.metrics,i="sm",o="cm",a="storeSupportabilityMetrics",s="storeEventMetrics"},4649:(e,t,r)=>{r.d(t,{t:()=>n});const n=r(3325).D.pageAction},7633:(e,t,r)=>{r.d(t,{Dz:()=>i,OJ:()=>a,qw:()=>o,t9:()=>n});const n=r(3325).D.pageViewEvent,i="firstbyte",o="domcontent",a="windowload"},9251:(e,t,r)=>{r.d(t,{t:()=>n});const n=r(3325).D.pageViewTiming},3614:(e,t,r)=>{r.d(t,{BST_RESOURCE:()=>i,END:()=>s,FEATURE_NAME:()=>n,FN_END:()=>u,FN_START:()=>c,PUSH_STATE:()=>d,RESOURCE:()=>o,START:()=>a});const n=r(3325).D.sessionTrace,i="bstResource",o="resource",a="-start",s="-end",c="fn"+a,u="fn"+s,d="pushState"},7836:(e,t,r)=>{r.d(t,{BODY:()=>A,CB_END:()=>E,CB_START:()=>u,END:()=>x,FEATURE_NAME:()=>i,FETCH:()=>_,FETCH_BODY:()=>v,FETCH_DONE:()=>m,FETCH_START:()=>p,FN_END:()=>c,FN_START:()=>s,INTERACTION:()=>l,INTERACTION_API:()=>d,INTERACTION_EVENTS:()=>o,JSONP_END:()=>b,JSONP_NODE:()=>g,JS_TIME:()=>T,MAX_TIMER_BUDGET:()=>a,REMAINING:()=>f,SPA_NODE:()=>h,START:()=>w,originalSetTimeout:()=>y});var n=r(5763);const i=r(3325).D.spa,o=["click","submit","keypress","keydown","keyup","change"],a=999,s="fn-start",c="fn-end",u="cb-start",d="api-ixn-",f="remaining",l="interaction",h="spaNode",g="jsonpNode",p="fetch-start",m="fetch-done",v="fetch-body-",b="jsonp-end",y=n.Yu.ST,w="-start",x="-end",A="-body",E="cb"+x,T="jsTime",_="fetch"},5938:(e,t,r)=>{r.d(t,{W:()=>o});var n=r(5763),i=r(2177);class o{constructor(e,t,r){this.agentIdentifier=e,this.aggregator=t,this.ee=i.ee.get(e,(0,n.OP)(this.agentIdentifier).isolatedBacklog),this.featureName=r,this.blocked=!1}}},9144:(e,t,r)=>{r.d(t,{j:()=>m});var n=r(3325),i=r(5763),o=r(5546),a=r(2177),s=r(7894),c=r(8e3),u=r(3960),d=r(385),f=r(50),l=r(3081),h=r(8632);function g(){const e=(0,h.gG)();["setErrorHandler","finished","addToTrace","inlineHit","addRelease","addPageAction","setCurrentRouteName","setPageViewName","setCustomAttribute","interaction","noticeError","setUserId"].forEach((t=>{e[t]=function(){for(var r=arguments.length,n=new Array(r),i=0;i 1?r-1:0),i=1;i {e.exposed&&e.api[t]&&o.push(e.api[t](...n))})),o.length>1?o:o[0]}(t,...n)}}))}var p=r(2587);function m(e){let t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:{},m=arguments.length>2?arguments[2]:void 0,v=arguments.length>3?arguments[3]:void 0,{init:b,info:y,loader_config:w,runtime:x={loaderType:m},exposed:A=!0}=t;const E=(0,h.gG)();y||(b=E.init,y=E.info,w=E.loader_config),(0,i.Dg)(e,b||{}),(0,i.GE)(e,w||{}),(0,i.sU)(e,x),y.jsAttributes??={},d.v6&&(y.jsAttributes.isWorker=!0),(0,i.CX)(e,y),g();const T=function(e,t){t||(0,c.R)(e,"api");const h={};var g=a.ee.get(e),p=g.get("tracer"),m="api-",v=m+"ixn-";function b(t,r,n,o){const a=(0,i.C5)(e);return null===r?delete a.jsAttributes[t]:(0,i.CX)(e,{...a,jsAttributes:{...a.jsAttributes,[t]:r}}),x(m,n,!0,o||null===r?"session":void 0)(t,r)}function y(){}["setErrorHandler","finished","addToTrace","inlineHit","addRelease"].forEach((e=>h[e]=x(m,e,!0,"api"))),h.addPageAction=x(m,"addPageAction",!0,n.D.pageAction),h.setCurrentRouteName=x(m,"routeName",!0,n.D.spa),h.setPageViewName=function(t,r){if("string"==typeof t)return"/"!==t.charAt(0)&&(t="/"+t),(0,i.OP)(e).customTransaction=(r||"http://custom.transaction")+t,x(m,"setPageViewName",!0)()},h.setCustomAttribute=function(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2];if("string"==typeof e){if(["string","number"].includes(typeof t)||null===t)return b(e,t,"setCustomAttribute",r);(0,f.Z)("Failed to execute setCustomAttribute.\nNon-null value must be a string or number type, but a type of was provided."))}else(0,f.Z)("Failed to execute setCustomAttribute.\nName must be a string type, but a type of was provided."))},h.setUserId=function(e){if("string"==typeof e||null===e)return b("enduser.id",e,"setUserId",!0);(0,f.Z)("Failed to execute setUserId.\nNon-null value must be a string type, but a type of was provided."))},h.interaction=function(){return(new y).get()};var w=y.prototype={createTracer:function(e,t){var r={},i=this,a="function"==typeof t;return(0,o.p)(v+"tracer",[(0,s.z)(),e,r],i,n.D.spa,g),function(){if(p.emit((a?"":"no-")+"fn-start",[(0,s.z)(),i,a],r),a)try{return t.apply(this,arguments)}catch(e){throw p.emit("fn-err",[arguments,this,"string"==typeof e?new Error(e):e],r),e}finally{p.emit("fn-end",[(0,s.z)()],r)}}}};function x(e,t,r,i){return function(){return(0,o.p)(l.xS,["API/"+t+"/called"],void 0,n.D.metrics,g),i&&(0,o.p)(e+t,[(0,s.z)(),...arguments],r?null:this,i,g),r?void 0:this}}function A(){r.e(439).then(r.bind(r,7438)).then((t=>{let{setAPI:r}=t;r(e),(0,c.L)(e,"api")})).catch((()=>(0,f.Z)("Downloading runtime APIs failed...")))}return["actionText","setName","setAttribute","save","ignore","onEnd","getContext","end","get"].forEach((e=>{w[e]=x(v,e,void 0,n.D.spa)})),h.noticeError=function(e,t){"string"==typeof e&&(e=new Error(e)),(0,o.p)(l.xS,["API/noticeError/called"],void 0,n.D.metrics,g),(0,o.p)("err",[e,(0,s.z)(),!1,t],void 0,n.D.jserrors,g)},d.il?(0,u.b)((()=>A()),!0):A(),h}(e,v);return(0,h.Qy)(e,T,"api"),(0,h.Qy)(e,A,"exposed"),(0,h.EZ)("activatedFeatures",p.T),T}},3325:(e,t,r)=>{r.d(t,{D:()=>n,p:()=>i});const n={ajax:"ajax",jserrors:"jserrors",metrics:"metrics",pageAction:"page_action",pageViewEvent:"page_view_event",pageViewTiming:"page_view_timing",sessionReplay:"session_replay",sessionTrace:"session_trace",spa:"spa"},i={[n.pageViewEvent]:1,[n.pageViewTiming]:2,[n.metrics]:3,[n.jserrors]:4,[n.ajax]:5,[n.sessionTrace]:6,[n.pageAction]:7,[n.spa]:8,[n.sessionReplay]:9}}},n={};function i(e){var t=n[e];if(void 0!==t)return t.exports;var o=n[e]={exports:{}};return r[e](o,o.exports,i),o.exports}i.m=r,i.d=(e,t)=>{for(var r in t)i.o(t,r)&&!i.o(e,r)&&Object.defineProperty(e,r,{enumerable:!0,get:t[r]})},i.f={},i.e=e=>Promise.all(Object.keys(i.f).reduce(((t,r)=>(i.f[r](e,t),t)),[])),i.u=e=>(({78:"page_action-aggregate",147:"metrics-aggregate",242:"session-manager",317:"jserrors-aggregate",348:"page_view_timing-aggregate",412:"lazy-feature-loader",439:"async-api",538:"recorder",590:"session_replay-aggregate",675:"compressor",733:"session_trace-aggregate",786:"page_view_event-aggregate",873:"spa-aggregate",898:"ajax-aggregate"}[e]||e)+"."+{78:"ac76d497",147:"3dc53903",148:"1a20d5fe",242:"2a64278a",317:"49e41428",348:"bd6de33a",412:"2f55ce66",439:"30bd804e",538:"1b18459f",590:"cf0efb30",675:"ae9f91a8",733:"83105561",786:"06482edd",860:"03a8b7a5",873:"e6b09d52",898:"998ef92b"}[e]+"-1.236.0.min.js"),i.o=(e,t)=>Object.prototype.hasOwnProperty.call(e,t),e={},t="NRBA:",i.l=(r,n,o,a)=>{if(e[r])e[r].push(n);else{var s,c;if(void 0!==o)for(var u=document.getElementsByTagName("script"),d=0;d {s.onerror=s.onload=null,clearTimeout(h);var i=e[r];if(delete e[r],s.parentNode&&s.parentNode.removeChild(s),i&&i.forEach((e=>e(n))),t)return t(n)},h=setTimeout(l.bind(null,void 0,{type:"timeout",target:s}),12e4);s.onerror=l.bind(null,s.onerror),s.onload=l.bind(null,s.onload),c&&document.head.appendChild(s)}},i.r=e=>{"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},i.j=364,i.p="https://js-agent.newrelic.com/",(()=>{var e={364:0,953:0};i.f.j=(t,r)=>{var n=i.o(e,t)?e[t]:void 0;if(0!==n)if(n)r.push(n[2]);else{var o=new Promise(((r,i)=>n=e[t]=[r,i]));r.push(n[2]=o);var a=i.p+i.u(t),s=new Error;i.l(a,(r=>{if(i.o(e,t)&&(0!==(n=e[t])&&(e[t]=void 0),n)){var o=r&&("load"===r.type?"missing":r.type),a=r&&r.target&&r.target.src;s.message="Loading chunk "+t+" failed.\n("+o+": "+a+")",s.name="ChunkLoadError",s.type=o,s.request=a,n[1](s)}}),"chunk-"+t,t)}};var t=(t,r)=>{var n,o,[a,s,c]=r,u=0;if(a.some((t=>0!==e[t]))){for(n in s)i.o(s,n)&&(i.m[n]=s[n]);if(c)c(i)}for(t&&t(r);u {i.r(o);var e=i(3325),t=i(5763);const r=Object.values(e.D);function n(e){const n={};return r.forEach((r=>{n[r]=function(e,r){return!1!==(0,t.Mt)(r,"".concat(e,".enabled"))}(r,e)})),n}var a=i(9144);var s=i(5546),c=i(385),u=i(8e3),d=i(5938),f=i(3960),l=i(50);class h extends d.W{constructor(e,t,r){let n=!(arguments.length>3&&void 0!==arguments[3])||arguments[3];super(e,t,r),this.auto=n,this.abortHandler,this.featAggregate,this.onAggregateImported,n&&(0,u.R)(e,r)}importAggregator(){let e=arguments.length>0&&void 0!==arguments[0]?arguments[0]:{};if(this.featAggregate||!this.auto)return;const r=c.il&&!0===(0,t.Mt)(this.agentIdentifier,"privacy.cookies_enabled");let n;this.onAggregateImported=new Promise((e=>{n=e}));const o=async()=>{let t;try{if(r){const{setupAgentSession:e}=await Promise.all([i.e(860),i.e(242)]).then(i.bind(i,3228));t=e(this.agentIdentifier)}}catch(e){(0,l.Z)("A problem occurred when starting up session manager. This page will not start or extend any session.",e)}try{if(!this.shouldImportAgg(this.featureName,t))return void(0,u.L)(this.agentIdentifier,this.featureName);const{lazyFeatureLoader:r}=await i.e(412).then(i.bind(i,8582)),{Aggregate:o}=await r(this.featureName,"aggregate");this.featAggregate=new o(this.agentIdentifier,this.aggregator,e),n(!0)}catch(e){(0,l.Z)("Downloading and initializing ".concat(this.featureName," failed..."),e),this.abortHandler?.(),n(!1)}};c.il?(0,f.b)((()=>o()),!0):o()}shouldImportAgg(r,n){return r!==e.D.sessionReplay||!1!==(0,t.Mt)(this.agentIdentifier,"session_trace.enabled")&&(!!n?.isNew||!!n?.state.sessionReplay)}}var g=i(7633),p=i(7894);class m extends h{static featureName=g.t9;constructor(r,n){let i=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];if(super(r,n,g.t9,i),("undefined"==typeof PerformanceNavigationTiming||c.Tt)&&"undefined"!=typeof PerformanceTiming){const n=(0,t.OP)(r);n[g.Dz]=Math.max(Date.now()-n.offset,0),(0,f.K)((()=>n[g.qw]=Math.max((0,p.z)()-n[g.Dz],0))),(0,f.b)((()=>{const t=(0,p.z)();n[g.OJ]=Math.max(t-n[g.Dz],0),(0,s.p)("timing",["load",t],void 0,e.D.pageViewTiming,this.ee)}))}this.importAggregator()}}var v=i(1117),b=i(1284);class y extends v.w{constructor(e){super(e),this.aggregatedData={}}store(e,t,r,n,i){var o=this.getBucket(e,t,r,i);return o.metrics=function(e,t){t||(t={count:0});return t.count+=1,(0,b.D)(e,(function(e,r){t[e]=w(r,t[e])})),t}(n,o.metrics),o}merge(e,t,r,n,i){var o=this.getBucket(e,t,n,i);if(o.metrics){var a=o.metrics;a.count+=r.count,(0,b.D)(r,(function(e,t){if("count"!==e){var n=a[e],i=r[e];i&&!i.c?a[e]=w(i.t,n):a[e]=function(e,t){if(!t)return e;t.c||(t=x(t.t));return t.min=Math.min(e.min,t.min),t.max=Math.max(e.max,t.max),t.t+=e.t,t.sos+=e.sos,t.c+=e.c,t}(i,a[e])}}))}else o.metrics=r}storeMetric(e,t,r,n){var i=this.getBucket(e,t,r);return i.stats=w(n,i.stats),i}getBucket(e,t,r,n){this.aggregatedData[e]||(this.aggregatedData[e]={});var i=this.aggregatedData[e][t];return i||(i=this.aggregatedData[e][t]={params:r||{}},n&&(i.custom=n)),i}get(e,t){return t?this.aggregatedData[e]&&this.aggregatedData[e][t]:this.aggregatedData[e]}take(e){for(var t={},r="",n=!1,i=0;i t.max&&(t.max=e),e 2&&void 0!==arguments[2])||arguments[2];super(e,r,j.t,n),c.il&&((0,t.OP)(e).initHidden=Boolean("hidden"===document.visibilityState),(0,N.N)((()=>(0,s.p)("docHidden",[(0,p.z)()],void 0,j.t,this.ee)),!0),(0,O.bP)("pagehide",(()=>(0,s.p)("winPagehide",[(0,p.z)()],void 0,j.t,this.ee))),this.importAggregator())}}var P=i(3081);class C extends h{static featureName=P.t9;constructor(e,t){let r=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];super(e,t,P.t9,r),this.importAggregator()}}var R,I=i(2210),k=i(1214),H=i(2177),L={};try{R=localStorage.getItem("__nr_flags").split(","),console&&"function"==typeof console.log&&(L.console=!0,-1!==R.indexOf("dev")&&(L.dev=!0),-1!==R.indexOf("nr_dev")&&(L.nrDev=!0))}catch(e){}function z(e){try{L.console&&z(e)}catch(e){}}L.nrDev&&H.ee.on("internal-error",(function(e){z(e.stack)})),L.dev&&H.ee.on("fn-err",(function(e,t,r){z(r.stack)})),L.dev&&(z("NR AGENT IN DEVELOPMENT MODE"),z("flags: "+(0,b.D)(L,(function(e,t){return e})).join(", ")));var M=i(6660);class B extends h{static featureName=M.t;constructor(r,n){let i=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];super(r,n,M.t,i),this.skipNext=0;try{this.removeOnAbort=new AbortController}catch(e){}const o=this;o.ee.on("fn-start",(function(e,t,r){o.abortHandler&&(o.skipNext+=1)})),o.ee.on("fn-err",(function(t,r,n){o.abortHandler&&!n[M.A]&&((0,I.X)(n,M.A,(function(){return!0})),this.thrown=!0,(0,s.p)("err",[n,(0,p.z)()],void 0,e.D.jserrors,o.ee))})),o.ee.on("fn-end",(function(){o.abortHandler&&!this.thrown&&o.skipNext>0&&(o.skipNext-=1)})),o.ee.on("internal-error",(function(t){(0,s.p)("ierr",[t,(0,p.z)(),!0],void 0,e.D.jserrors,o.ee)})),this.origOnerror=c._A.onerror,c._A.onerror=this.onerrorHandler.bind(this),c._A.addEventListener("unhandledrejection",(t=>{const r=function(e){let t="Unhandled Promise Rejection: ";if(e instanceof Error)try{return e.message=t+e.message,e}catch(t){return e}if(void 0===e)return new Error(t);try{return new Error(t+(0,D.P)(e))}catch(e){return new Error(t)}}(t.reason);(0,s.p)("err",[r,(0,p.z)(),!1,{unhandledPromiseRejection:1}],void 0,e.D.jserrors,this.ee)}),(0,O.m$)(!1,this.removeOnAbort?.signal)),(0,k.gy)(this.ee),(0,k.BV)(this.ee),(0,k.em)(this.ee),(0,t.OP)(r).xhrWrappable&&(0,k.Kf)(this.ee),this.abortHandler=this.#e,this.importAggregator()}#e(){this.removeOnAbort?.abort(),this.abortHandler=void 0}onerrorHandler(t,r,n,i,o){"function"==typeof this.origOnerror&&this.origOnerror(...arguments);try{this.skipNext?this.skipNext-=1:(0,s.p)("err",[o||new F(t,r,n),(0,p.z)()],void 0,e.D.jserrors,this.ee)}catch(t){try{(0,s.p)("ierr",[t,(0,p.z)(),!0],void 0,e.D.jserrors,this.ee)}catch(e){}}return!1}}function F(e,t,r){this.message=e||"Uncaught error with no additional information",this.sourceURL=t,this.line=r}let U=1;const q="nr@id";function G(e){const t=typeof e;return!e||"object"!==t&&"function"!==t?-1:e===c._A?0:(0,I.X)(e,q,(function(){return U++}))}function V(e){if("string"==typeof e&&e.length)return e.length;if("object"==typeof e){if("undefined"!=typeof ArrayBuffer&&e instanceof ArrayBuffer&&e.byteLength)return e.byteLength;if("undefined"!=typeof Blob&&e instanceof Blob&&e.size)return e.size;if(!("undefined"!=typeof FormData&&e instanceof FormData))try{return(0,D.P)(e).length}catch(e){return}}}var X=i(7243);class W{constructor(e){this.agentIdentifier=e,this.generateTracePayload=this.generateTracePayload.bind(this),this.shouldGenerateTrace=this.shouldGenerateTrace.bind(this)}generateTracePayload(e){if(!this.shouldGenerateTrace(e))return null;var r=(0,t.DL)(this.agentIdentifier);if(!r)return null;var n=(r.accountID||"").toString()||null,i=(r.agentID||"").toString()||null,o=(r.trustKey||"").toString()||null;if(!n||!i)return null;var a=(0,_.M)(),s=(0,_.Ht)(),c=Date.now(),u={spanId:a,traceId:s,timestamp:c};return(e.sameOrigin||this.isAllowedOrigin(e)&&this.useTraceContextHeadersForCors())&&(u.traceContextParentHeader=this.generateTraceContextParentHeader(a,s),u.traceContextStateHeader=this.generateTraceContextStateHeader(a,c,n,i,o)),(e.sameOrigin&&!this.excludeNewrelicHeader()||!e.sameOrigin&&this.isAllowedOrigin(e)&&this.useNewrelicHeaderForCors())&&(u.newrelicHeader=this.generateTraceHeader(a,s,c,n,i,o)),u}generateTraceContextParentHeader(e,t){return"00-"+t+"-"+e+"-01"}generateTraceContextStateHeader(e,t,r,n,i){return i+"@nr=0-1-"+r+"-"+n+"-"+e+"----"+t}generateTraceHeader(e,t,r,n,i,o){if(!("function"==typeof c._A?.btoa))return null;var a={v:[0,1],d:{ty:"Browser",ac:n,ap:i,id:e,tr:t,ti:r}};return o&&n!==o&&(a.d.tk=o),btoa((0,D.P)(a))}shouldGenerateTrace(e){return this.isDtEnabled()&&this.isAllowedOrigin(e)}isAllowedOrigin(e){var r=!1,n={};if((0,t.Mt)(this.agentIdentifier,"distributed_tracing")&&(n=(0,t.P_)(this.agentIdentifier).distributed_tracing),e.sameOrigin)r=!0;else if(n.allowed_origins instanceof Array)for(var i=0;i 2&&void 0!==arguments[2])||arguments[2];super(r,n,Z.t,i),(0,t.OP)(r).xhrWrappable&&(this.dt=new W(r),this.handler=(e,t,r,n)=>(0,s.p)(e,t,r,n,this.ee),(0,k.u5)(this.ee),(0,k.Kf)(this.ee),function(r,n,i,o){function a(e){var t=this;t.totalCbs=0,t.called=0,t.cbTime=0,t.end=E,t.ended=!1,t.xhrGuids={},t.lastSize=null,t.loadCaptureCalled=!1,t.params=this.params||{},t.metrics=this.metrics||{},e.addEventListener("load",(function(r){_(t,e)}),(0,O.m$)(!1)),c.IF||e.addEventListener("progress",(function(e){t.lastSize=e.loaded}),(0,O.m$)(!1))}function s(e){this.params={method:e[0]},T(this,e[1]),this.metrics={}}function u(e,n){var i=(0,t.DL)(r);i.xpid&&this.sameOrigin&&n.setRequestHeader("X-NewRelic-ID",i.xpid);var a=o.generateTracePayload(this.parsedOrigin);if(a){var s=!1;a.newrelicHeader&&(n.setRequestHeader("newrelic",a.newrelicHeader),s=!0),a.traceContextParentHeader&&(n.setRequestHeader("traceparent",a.traceContextParentHeader),a.traceContextStateHeader&&n.setRequestHeader("tracestate",a.traceContextStateHeader),s=!0),s&&(this.dt=a)}}function d(e,t){var r=this.metrics,i=e[0],o=this;if(r&&i){var a=V(i);a&&(r.txSize=a)}this.startTime=(0,p.z)(),this.listener=function(e){try{"abort"!==e.type||o.loadCaptureCalled||(o.params.aborted=!0),("load"!==e.type||o.called===o.totalCbs&&(o.onloadCalled||"function"!=typeof t.onload)&&"function"==typeof o.end)&&o.end(t)}catch(e){try{n.emit("internal-error",[e])}catch(e){}}};for(var s=0;s 1?e[1]=i:e.push(i)}else e[0]&&e[0].headers&&s(e[0].headers,n)&&(this.dt=n);function s(e,t){var r=!1;return t.newrelicHeader&&(e.set("newrelic",t.newrelicHeader),r=!0),t.traceContextParentHeader&&(e.set("traceparent",t.traceContextParentHeader),t.traceContextStateHeader&&e.set("tracestate",t.traceContextStateHeader),r=!0),r}}function x(e,t){this.params={},this.metrics={},this.startTime=(0,p.z)(),this.dt=t,e.length>=1&&(this.target=e[0]),e.length>=2&&(this.opts=e[1]);var r,n=this.opts||{},i=this.target;"string"==typeof i?r=i:"object"==typeof i&&i instanceof Y?r=i.url:c._A?.URL&&"object"==typeof i&&i instanceof URL&&(r=i.href),T(this,r);var o=(""+(i&&i instanceof Y&&i.method||n.method||"GET")).toUpperCase();this.params.method=o,this.txSize=V(n.body)||0}function A(t,r){var n;this.endTime=(0,p.z)(),this.params||(this.params={}),this.params.status=r?r.status:0,"string"==typeof this.rxSize&&this.rxSize.length>0&&(n=+this.rxSize);var o={txSize:this.txSize,rxSize:n,duration:(0,p.z)()-this.startTime};i("xhr",[this.params,o,this.startTime,this.endTime,"fetch"],this,e.D.ajax)}function E(t){var r=this.params,n=this.metrics;if(!this.ended){this.ended=!0;for(var o=0;o 2&&void 0!==arguments[2])||arguments[2];super(e,t,we.t,r),this.importAggregator()}}new class{constructor(e){let t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:(0,_.ky)(16);c._A?(this.agentIdentifier=t,this.sharedAggregator=new y({agentIdentifier:this.agentIdentifier}),this.features={},this.desiredFeatures=new Set(e.features||[]),this.desiredFeatures.add(m),Object.assign(this,(0,a.j)(this.agentIdentifier,e,e.loaderType||"agent")),this.start()):(0,l.Z)("Failed to initial the agent. Could not determine the runtime environment.")}get config(){return{info:(0,t.C5)(this.agentIdentifier),init:(0,t.P_)(this.agentIdentifier),loader_config:(0,t.DL)(this.agentIdentifier),runtime:(0,t.OP)(this.agentIdentifier)}}start(){const t="features";try{const r=n(this.agentIdentifier),i=[...this.desiredFeatures];i.sort(((t,r)=>e.p[t.featureName]-e.p[r.featureName])),i.forEach((t=>{if(r[t.featureName]||t.featureName===e.D.pageViewEvent){const n=function(t){switch(t){case e.D.ajax:return[e.D.jserrors];case e.D.sessionTrace:return[e.D.ajax,e.D.pageViewEvent];case e.D.sessionReplay:return[e.D.sessionTrace];case e.D.pageViewTiming:return[e.D.pageViewEvent];default:return[]}}(t.featureName);n.every((e=>r[e]))||(0,l.Z)("".concat(t.featureName," is enabled but one or more dependent features has been disabled (").concat((0,D.P)(n),"). This may cause unintended consequences or missing data...")),this.features[t.featureName]=new t(this.agentIdentifier,this.sharedAggregator)}})),(0,T.Qy)(this.agentIdentifier,this.features,t)}catch(e){(0,l.Z)("Failed to initialize all enabled instrument classes (agent aborted) -",e);for(const e in this.features)this.features[e].abortHandler?.();const r=(0,T.fP)();return delete r.initializedAgents[this.agentIdentifier]?.api,delete r.initializedAgents[this.agentIdentifier]?.[t],delete this.sharedAggregator,r.ee?.abort(),delete r.ee?.get(this.agentIdentifier),!1}}}({features:[J,m,S,class extends h{static featureName=oe;constructor(t,r){if(super(t,r,oe,!(arguments.length>2&&void 0!==arguments[2])||arguments[2]),!c.il)return;const n=this.ee;let i;(0,k.QU)(n),this.eventsEE=(0,k.em)(n),this.eventsEE.on(se,(function(e,t){this.bstStart=(0,p.z)()})),this.eventsEE.on(ae,(function(t,r){(0,s.p)("bst",[t[0],r,this.bstStart,(0,p.z)()],void 0,e.D.sessionTrace,n)})),n.on(ce+ne,(function(e){this.time=(0,p.z)(),this.startPath=location.pathname+location.hash})),n.on(ce+ie,(function(t){(0,s.p)("bstHist",[location.pathname+location.hash,this.startPath,this.time],void 0,e.D.sessionTrace,n)}));try{i=new PerformanceObserver((t=>{const r=t.getEntries();(0,s.p)(te,[r],void 0,e.D.sessionTrace,n)})),i.observe({type:re,buffered:!0})}catch(e){}this.importAggregator({resourceObserver:i})}},C,xe,B,class extends h{static featureName=de;constructor(e,r){if(super(e,r,de,!(arguments.length>2&&void 0!==arguments[2])||arguments[2]),!c.il)return;if(!(0,t.OP)(e).xhrWrappable)return;try{this.removeOnAbort=new AbortController}catch(e){}let n,i=0;const o=this.ee.get("tracer"),a=(0,k._L)(this.ee),s=(0,k.Lg)(this.ee),u=(0,k.BV)(this.ee),d=(0,k.Kf)(this.ee),f=this.ee.get("events"),l=(0,k.u5)(this.ee),h=(0,k.QU)(this.ee),g=(0,k.Gm)(this.ee);function m(e,t){h.emit("newURL",[""+window.location,t])}function v(){i++,n=window.location.hash,this[ve]=(0,p.z)()}function b(){i--,window.location.hash!==n&&m(0,!0);var e=(0,p.z)();this[pe]=~~this[pe]+e-this[ve],this[ye]=e}function y(e,t){e.on(t,(function(){this[t]=(0,p.z)()}))}this.ee.on(ve,v),s.on(be,v),a.on(be,v),this.ee.on(ye,b),s.on(ge,b),a.on(ge,b),this.ee.buffer([ve,ye,"xhr-resolved"],this.featureName),f.buffer([ve],this.featureName),u.buffer(["setTimeout"+le,"clearTimeout"+fe,ve],this.featureName),d.buffer([ve,"new-xhr","send-xhr"+fe],this.featureName),l.buffer([me+fe,me+"-done",me+he+fe,me+he+le],this.featureName),h.buffer(["newURL"],this.featureName),g.buffer([ve],this.featureName),s.buffer(["propagate",be,ge,"executor-err","resolve"+fe],this.featureName),o.buffer([ve,"no-"+ve],this.featureName),a.buffer(["new-jsonp","cb-start","jsonp-error","jsonp-end"],this.featureName),y(l,me+fe),y(l,me+"-done"),y(a,"new-jsonp"),y(a,"jsonp-end"),y(a,"cb-start"),h.on("pushState-end",m),h.on("replaceState-end",m),window.addEventListener("hashchange",m,(0,O.m$)(!0,this.removeOnAbort?.signal)),window.addEventListener("load",m,(0,O.m$)(!0,this.removeOnAbort?.signal)),window.addEventListener("popstate",(function(){m(0,i>1)}),(0,O.m$)(!0,this.removeOnAbort?.signal)),this.abortHandler=this.#e,this.importAggregator()}#e(){this.removeOnAbort?.abort(),this.abortHandler=void 0}}],loaderType:"spa"})})(),window.NRBA=o})(); window.jQuery || document.write(' ') CKEDITOR_BASEPATH='https://f1000research.com/js/vendor/ckeditor/' window.reactTheme = 'research'; window.MathJax = { CommonHTML: { linebreaks: { automatic: true } }, 'HTML-CSS': { linebreaks: { automatic: true } }, SVG: { linebreaks: { automatic: true } }, AuthorInit: function() { MathJax.Hub.Register.MessageHook('End Process', function () { let timeout = false; // holder for timeout id const delay = 250; // delay after event is "complete" to run callback const reflowMath = function() { const dispFormulas = document.querySelectorAll('.disp-formula.panel'); if (!dispFormulas) { return; } for (const dispFormula of dispFormulas) { const child = dispFormula.querySelector('.MathJax_Preview').nextSibling.firstChild; const isMultiline = MathJax.Hub.getAllJax(dispFormula)[0].root.isMultiline; if (dispFormula.offsetWidth < child.offsetWidth || isMultiline) { MathJax.Hub.Queue(['Rerender', MathJax.Hub, dispFormula]); } } }; window.addEventListener('resize', function() { clearTimeout(timeout); // clear the timeout timeout = setTimeout(reflowMath, delay); // start timing for event "completion" }); }); }, }; if (window.location.hash == '#_=_'){ window.location = window.location.href.split('#')[0] } !function(f,b,e,v,n,t,s){if(f.fbq)return;n=f.fbq=function() {n.callMethod? n.callMethod.apply(n,arguments):n.queue.push(arguments)} ;if(!f._fbq)f._fbq=n; n.push=n;n.loaded=!0;n.version='2.0';n.queue=[];t=b.createElement(e);t.async=!0; t.src=v;s=b.getElementsByTagName(e)[0];s.parentNode.insertBefore(t,s)}(window, document,'script','https://connect.facebook.net/en_US/fbevents.js'); fbq('init', '1641728616063202'); fbq('track', "PixelInitialized", {}); (function(h,o,t,j,a,r){ h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)}; h._hjSettings={hjid:2318163,hjsv:6}; a=o.getElementsByTagName('head')[0]; r=o.createElement('script');r.async=1; r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv; a.appendChild(r); })(window,document,'https://static.hotjar.com/c/hotjar-','.js?sv='); search file_upload Submit your research search menu close search Browse Gateways & Collections How to Publish Submit your Research My Submissions Article Guidelines Article Guidelines (New Versions) Open Data, Software and Code Guidelines Open Data and Accessible Source Materials Guidelines (HSS) Open Data, Software and Code Guidelines (PSE) Prepublication Checks Production Process Posters and Slides Guidelines Document Guidelines Article Processing Charges Peer Review Finding Article Reviewers About How it Works For Reviewers Our Advisors Policies Glossary FAQs For Developers Newsroom Contact My Research Submissions Content and Tracking Alerts My Details Sign In file_upload Submit your research { "@context": "https://schema.org", "@type": "ScholarlyArticle", "mainEntityOfPage": { "@type": "WebPage", "@id": "https://f1000research.com/articles/13-1386" }, "headline": "Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra", "datePublished": "2024-11-18T16:55:31", "dateModified": "2024-11-18T16:55:31", "author": [ { "@type": "Person", "name": "Alachew Amaneh Mechderso" }, { "@type": "Person", "name": "Berhanu Assaye Alaba" }, { "@type": "Person", "name": "Tilahun Mekonnen Munie" }, { "@type": "Person", "name": "Teferi Getachew Alemayhu" } ], "publisher": { "@type": "Organization", "name": "F1000Research", "logo": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 480, "width": 60 } }, "image": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 1200, "width": 150 }, "description": "In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. MSC: 03G25, 06D30." } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/13-1386/v1", "name": "Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra" } } ] } Home Browse Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Mechderso AA, Alaba BA, Munie TM and Alemayhu TG. Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.12688/f1000research.157173.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] Alachew Amaneh Mechderso https://orcid.org/0009-0007-9566-9901 1 , Berhanu Assaye Alaba 1 , Tilahun Mekonnen Munie 1 , Teferi Getachew Alemayhu 2 Alachew Amaneh Mechderso https://orcid.org/0009-0007-9566-9901 1 , Berhanu Assaye Alaba 1 , Tilahun Mekonnen Munie 1 , Teferi Getachew Alemayhu 2 PUBLISHED 18 Nov 2024 Author details Author details 1 Department of Mathematics, Bahir Dar University College of Science, Bahir Dar, Amhara, 6000, Ethiopia 2 Department of Mathematics, Debre Berhan University, Debre Birhan, Amhara, Ethiopia Alachew Amaneh Mechderso Roles: Conceptualization, Writing – Original Draft Preparation, Writing – Review & Editing Berhanu Assaye Alaba Roles: Conceptualization Tilahun Mekonnen Munie Roles: Conceptualization Teferi Getachew Alemayhu Roles: Conceptualization OPEN PEER REVIEW DETAILS REVIEWER STATUS Abstract In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. MSC: 03G25, 06D30. READ ALL READ LESS Keywords Pseudo-UP algebra, pseudo-UP ideal, Fuzzy pseudo-UP ideal, bipolar Fuzzy pseudo- UP ideal Corresponding Author(s) Alachew Amaneh Mechderso ( [email protected] ) Close Corresponding author: Alachew Amaneh Mechderso Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2024 Mechderso AA et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: Mechderso AA, Alaba BA, Munie TM and Alemayhu TG. Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.12688/f1000research.157173.1 ) First published: 18 Nov 2024, 13 :1386 ( https://doi.org/10.12688/f1000research.157173.1 ) Latest published: 18 Nov 2024, 13 :1386 ( https://doi.org/10.12688/f1000research.157173.1 ) Introduction Iseki 1 introduced BCK algebras. It is known that the class of BCK algebras is a proper subclass of the class of BCI-algebras. In 2020, Romano 2 introduced the concept of pseudo-UP ideals and pseudo-UP filters and derived basic properties. In 1965, Zadeh 3 popularized the concept of fuzzy sets. Since then, the concepts of fuzzy sets have been extensively used in many branches of mathematics. The fuzzifications of algebraic structures were initiated by Rosenfeld, 4 and he introduced the notion of fuzzy subgroups. In 2023, Mechderso 5 investigated the concept of fuzzy pseudo-UP ideals of pseudo-UP algebra. Wechler 6 studied how fuzzy algebraic structures plaid vital roles in mathematics with wide applications in many other branches such as computer sciences, theoretical physics, information sciences, control engineering, topological spaces and coding theory. For a fuzzy set, its degree of membership expresses its degree of containment of elements in it. Sometimes, the degree of membership also means the degree of satisfaction of elements to some property or constraint corresponding to a fuzzy set. 7 Keeping in view this notion, the membership degree 0 is in general assigned to those elements of the set that do not satisfy some property. In the usual study of fuzzy set representation, the elements with membership degree 0 are usually regarded as having the same characteristic. Keeping in view these facts, Lee 8 proposed an extension of fuzzy sets named bipolar fuzzy sets. Lee 9 used the notion of bipolar fuzzy set and worked on bipolar valued fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI algebra. In recent times, Alaba 10 studied the notion of intuitionist fuzzy PMS ideals under homomorphism and Cartesian product and investigated several related properties. A wide variety of human decision-making is based on double-sided or bipolar judgmental thinking on a positive side and a negative side, for instance, cooperation and competition, friendship and hostility, common interests and conflict interests, effect and side effect, likelihood and unlikelihood, feed forward and feedback. The notion of bipolar fuzzy sets (YinYang bipolar fuzzy sets) was introduced by Zhang 11 as a generaztion of fuzzy set. Consider a bipolar fuzzy set, as follows: Frog’s prey = {(mosquito, 1, 0), (dragon fly, 0.4, 0), (turtle, 0, 0), (snake, 0, −1)}. We can see that membership degree 0 and non-membership degree 0 of turtle mean that frog never hunts turtle and turtle never hunts frog. While membership degree 0 and non-membership degree –1 of snake mean that frog never hunts snake but snake always hunts frog. Here, the implicit counter-property is “predator of frog,” which created the difference between fuzzy set and bipolar fuzzy set of frog’s prey. Motivated by this, we introduced the notion of a bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra and proved some results. We present several results related to the bipolar fuzzy pseudo-UP ideal in pseudo-UP algebras. We investigate the properties of the homomorphic image and preimage of bipolar fuzzy pseudo-UP ideals and demonstrate that both the homomorphic image and preimage of a bipolar fuzzy pseudo-UP ideal are themselves bipolar fuzzy pseudo-UP ideals. Furthermore, the Cartesian product of bipolar fuzzy pseudo-UP ideals of pseudo-UP algebras is introduced, and several properties are investigated. Preliminaries Definition 2.1. [Ref. 12 ] A pseudo-UP algebra is algebra ( X , ·, * , 0) of type (2, 2, 0) which satisfies the following axioms: for any x , y , and z ∈ X 1) ( y ⋅ z ) ⋅ ( ( x ⋅ y ) ∗ ( x ⋅ z ) ) = 0 and ( y ∗ z ) ∗ ( ( x ∗ y ) ⋅ ( x ∗ z ) ) = 0 , 2) x ⋅ y = 0 = y ⋅ x and x ∗ y = 0 = y ∗ x ⇒ x = y , 3) ( y ⋅ 0 ) ∗ x = x and ( y ∗ 0 ) ⋅ x = x , 4) x ≤ y if and only if x · y = 0 and x ≤ y if and only if x * y = 0. Proposition 2.2. [Ref. 12 ] In a pseudo-UP algebra X the following holds: for any x , y ∈ X 1) x ≤ y ⋅ x and 2) x ≤ y ∗ x . Definition 2.3. [Ref. 2 ] A pseudo-UP ideal of X is a nonempty subset J of a pseudo-UP algebra X That has the following: for each x , y , z ∈ X. 1) 0 ∈ J , 2) x ⋅ ( y ∗ z ) ∈ J and y ∈ J ⇒ x ⋅ z ∈ J and 3) x ∗ ( y ⋅ z ) ∈ J and y ∈ J ⇒ x ∗ z ∈ J . Lemma 2.4. [Ref. 2 ] In a pseudo-UP algebra X the following holds, for each x ∈ X , 1) x · 0 = 0 and x * 0 = 0, 2) 0 · x = x and 0 * x = x and 3) x · x = 0 and x * x = 0. Definition 2.5. [Ref. 5 ] A fuzzy subset λ of a pseudo-UP algebras of X is called fuzzy pseudo-UP ideal of X if and only if it fulfill the following axioms: for any x, y, z ∈ X 1) λ ( 0 ) ≥ λ ( x ) , 2) λ ( x ⋅ y ) ≥ min { λ ( x ⋅ ( y ∗ z ) ) , λ ( y ) } and 3) λ ( x ∗ y ) ≥ min { λ ( x ∗ ( y ⋅ z ) ) , λ ( y ) } . Definition 2.6. [Ref. 8 ] Let X be the universe of discourse. A bipolar fuzzy set λ in X is an object having the form λ = { ( x , λ − ( x ) , λ + ( x ) ) : x ∈ X } Where λ − : X → [−1, 0] and λ + : X → [0, 1] are mappings. Form this paper we use the symbol λ = ( X ; λ − , λ + ) for the bipolar valued fuzzy set λ = {( x , λ − ( x ), λ + ( x )): x ∈ X }, and use the notion of bipolar fuzzy sets instead of the notion of bipolar valued fuzzy sets. Definition 2.7. [Ref. 13 ] A bipolar fuzzy set λ = ( X ; λ + , λ − ) in a set X with the positive membership λ + : X → [0, 1] and negative membership λ − : X → [−1, 0] is indicated to have Sup-Inf property, if for any subset T of X , there exists x 0 ∈ T such that λ + ( x 0 ) = sup t ∈ T λ + ( t ) and λ − ( x 0 ) = inf t ∈ T λ − ( t ) Lemma 2.8. Let λ = ( X , λ + , λ − ) be bipolar fuzzy set in X. Then the following statements hold for any x, y ∈ X. 1) 1 − max { λ + ( x ) , λ + ( y ) } = min { 1 − λ + ( x ) , 1 − λ + ( y ) } , 2) 1 − min { λ + ( x ) , λ + ( y ) } = max { 1 − λ + ( x ) , 1 − λ + ( y ) } , 3) − 1 − max { λ − ( x ) , λ − ( y ) } = min { − 1 − λ − ( x ) , − 1 − λ − ( y ) } and 4) − 1 − min { λ − ( x ) , λ − ( y ) } = max { − 1 − λ − ( x ) , − 1 − λ − ( y ) } . Definition 2.9. [Ref. 14 ] Let f : X → Y be a homomorphism from a set X onto a set Y and let λ = ( X ; λ − , λ + ) be a bipolar fuzzy set of X and σ = ( Y ; σ − , σ + ) be two bipolar fuzzy set of Y , then the homomorphic image f ( λ ) is f ( λ ) = ( f ( λ − ), f ( λ + )) defined as for all y ∈ Y. And f ( λ − ) ( x ) = inf λ − ( x ) / x ∈ f − 1 ( y ) if f − 1 ( y ) ≠ Ø f ( λ + ) ( x ) = sup λ + ( x ) / x ∈ f − 1 ( y ) if f − 1 ( y ) ≠ Ø . The pre-image f −1 ( σ ) of σ under f is a bipolar set defined as f −1 ( σ − ) ( x ) = σ − (f(x)) and f −1 ( σ + ) ( x )) = σ + ( f ( x )), for all x ∈ X. Definition 2.10. [Ref. 15 ] Let λ and σ are any two bipolar fuzz set of X and Y respectively. The Cartesian product of λ and σ is defined as λ × σ = ( X × Y , λ + × σ + , λ − × σ − ) with λ + × σ + ( x , y ) = min { λ + ( x ) , σ + ( y ) } and λ − × σ − ( x , y ) = max { λ − ( x ) , σ − ( y ) } where λ + × σ + : X × Y → [ 0 , 1 ] and λ − × σ − : X × Y → [ − 1 , 0 ] , ∀ ( x , y ) ∈ X × Y . Bipolar fuzzy pseudo-UP ideal In this section, we introduce the concept of a bipolar fuzzy pseudo-UP ideal in pseudo-UP algebras and examine several important properties associated with bipolar fuzzy pseudo-UP ideals. Definition 3.1. A bipolar fuzzy set λ = ( X , λ + , λ − ) in X is called bipolar fuzzy pseudo-UP ideal of X if it satisfies the following conditions for all x, y, z ∈ X. i) λ + (0) ≥ λ + ( x ) and λ − (0) ≤ λ − ( x ), ii) λ + ( x · z ) ≥ min { λ + ( x · ( y * z )), λ + ( y )}, iii) λ + ( x * z ) ≥ min { λ + ( x * ( y · z )), λ + ( y )}, iv) λ − ( x · z ) ≤ max { λ − ( x · ( y * z )), λ − ( y )}, v) λ − ( x * z ) ≤ max { λ − ( x * ( y · z )), λ − ( y )}. Example 3.2. Let X = {0, a , b , c } be a set with binary operations “·” and “*” defined by the following cayley Table 1 Clearly, ( X , ·, *, 0) is pseudo-UP algebra. We define a bipolar fuzzy pseudo-UP ideal of X as follows. X 0 a b c λ + 0.8 0.6 0.5 0.4 λ − -0.7 -0.5 -0.4 -0.4 Then, λ is a bipolar fuzzy pseudo-UP ideal of X. Theorem 3.3. Let λ = ( X , λ + , λ − ) be a bipolar fuzzy pseudo-UP ideal of a pseudo-UP algebra X with y ≤ z , for any y , z ∈ X , then λ + ( y ) ≥ λ + ( z ) and λ − ( y ) ≤ λ − ( z ), that is λ + is order reversing and λ − is order preserving. Proof. Let λ = ( X , λ + , λ − ) be a bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra X such that y ≤ z , for all y, z ∈ X. Then by the binary relation ≤ define in X , we have z · y = 0 and z * y = 0. Now , λ + ( y ) = λ + ( 0 ⋅ y ) , by Lemma 2.4 ≥ min { λ + ( 0 ⋅ ( z ∗ y ) ) , λ + ( z ) } , ( by Definition 3.1 ) = min { λ + ( z ∗ y ) , λ + ( z ) } , ( by Lemma 2.4 ) = min { λ + ( 0 ) , λ + ( z ) } , ( since y ≤ z ⇒ z ∗ y = 0 ) = λ + ( z ) . ⇒ λ + ( y ) ≥ λ + ( z ) . λ + ( y ) = λ + ( 0 ∗ y ) , by Lemma 2.4 ≥ min { λ + ( 0 ∗ ( z ⋅ y ) ) , λ + ( z ) } , ( by Definition 3.1 ) = min { λ + ( z ⋅ y ) , λ + ( z ) } , ( by Lemma 2.4 ) = min { λ + ( 0 ) , λ + ( z ) } , ( since y ≤ z ⇒ z ⋅ y = 0 ) = λ + ( z ) . ⇒ λ + ( y ) ≥ λ + ( z ) , and λ − ( y ) = λ − ( 0 ⋅ y ) , ( by Lemma 2.4 ) ≤ max { λ − ( 0 ⋅ ( z ∗ y ) ) , λ − ( z ) } , ( by Definition 3.1 ) = max { λ − ( z ∗ y ) , λ − ( z ) } , ( by Lemma 2.4 ) = max { λ − ( 0 ) , λ − ( z ) } , ( since y ≤ z ⇒ z ∗ y = 0 ) = λ − ( z ) . ⇒ λ − ( y ) ≤ λ − ( z ) . λ − ( y ) = λ − ( 0 ∗ y ) , ( by Lemma 2.4 ) ≤ max { λ − ( 0 ∗ ( z ⋅ y ) ) , λ − ( z ) } , ( by Definition 3.1 ) = max { λ − ( z ⋅ y ) , λ − ( z ) } , ( by Lemma 2.4 ) = max { λ − ( 0 ) , λ − ( z ) } , ( since y ≤ z ⇒ z ⋅ y = 0 ) = λ − ( z ) . ⇒ λ − ( y ) ≤ λ − ( z ) . Hence, λ + ( y ) ≥ λ + ( z ) and λ − ( y ) ≤ λ − ( z ). □ Theorem 3.4. Every bipolar fuzzy pseudo-UP ideal λ = ( X , λ + , λ − ) of X is a bipolar fuzzy pseudo- UP subalgebra of X. Proof. Suppose λ is a fuzzy pseudo-UP ideal of X. Then, λ + ( x ⋅ y ) ≥ min { λ + ( x ⋅ ( y ∗ y ) ) , λ + ( y ) } , ( by Definition 3.1 ) = min { λ + ( x ⋅ 0 ) , λ + ( y ) } , ( by Lemma 2.4 ) = min { λ + ( 0 ) , λ + ( y ) } , ( by Lemma 2.4 ) ≥ min { λ + ( x ) , λ + ( y ) } , ( since λ + ( 0 ) ≥ λ + ( x ) ) ⇒ λ + ( x ⋅ y ) ≥ min { λ + ( x ) , λ + ( y ) } . λ + ( x ∗ y ) ≥ min { λ + ( x ∗ ( y ⋅ y ) ) , λ + ( y ) } , ( by Definition 3.1 ) = min { λ + ( x ∗ 0 ) , λ + ( y ) } , ( by Proposition 2.2 ) = min { λ + ( 0 ) , λ + ( y ) } , ( by Lemma 2.4 ) ≥ min { λ + ( x ) , λ + ( y ) } ( since λ + ( 0 ) ≥ λ + ( x ) ) . ⇒ λ + ( x ∗ y ) ≥ min { λ + ( x ) , λ + ( y ) } , and λ − ( x ⋅ y ) ≤ max { λ − ( x ⋅ ( y ∗ y ) ) , λ − ( y ) ) } , ( by Definition 3.1 ) = max { λ − ( x ⋅ 0 ) , λ − ( y ) } , ( by Lemma 2.4 ) = max { λ − ( 0 ) , λ − ( y ) } , ( by Lemma 2.4 ) ≤ max { λ − ( x ) , λ − ( y ) } . ⇒ λ − ( x ⋅ y ) ≤ max { λ − ( x ) , λ − ( y ) } . Finally , λ − ( x ∗ y ) ≤ max { λ − ( x ∗ ( y ≥ y ) ) , λ − ( y ) } , ( by Definition 3.1 ) = max { λ − ( x ∗ 0 ) , λ − ( y ) } , ( by Proposition 2.2 ) = max { λ − ( 0 ) , λ − ( y ) } , ( by Lemma 2.4 ) ≤ max { λ − ( x ) , λ − ( y ) } . ⇒ λ − ( x ∗ y ) ≤ max { λ − ( x ) , λ − ( y ) } . Thus, λ = ( X , λ + , λ − ) is a bipolar fuzzy pseudo-UP subalgebra of pseudo-UP algebra X. □ Remark 3.5. The converse is may not be true. Example 3.6. Let X = {0, 1, 2, and 3} be a set with a binary operations “·” and “*” defined by the following cayley Table 2 . Then, ( X , ·, *, 0) is a pseudo-UP algebra. We define a bipolar fuzzy set λ = ( X ; λ + , λ − ) in X as follows: X 0 1 2 3 λ + 0.8 0.4 0.2 0.1 λ − -0.9 -0.5 -0.3 -0.2 Then λ = ( X ; λ + , λ − ) is a bipolar fuzzy pseudo-UP subalgebra of X , but it is not a bipolar fuzzy pseudo-UP ideal of X. Indeed, λ + (0 · 2) ≥ min { λ + (0 · (1 * 2)), λ + (1)} = λ + (2) ≥ min { λ + (0 · 1), λ + (1)} implies that 0.2 ≥ 0.4 which is contradict to Definition 3.1 . And similarly, λ − (0 · 2) ≤ max { λ − (0 · (1 * 2)), λ − (1)} = λ − (2) ≤ max { λ − (0 · 1), λ − (1)} implies that −0.3 ≤ −0.5 which is contradict to Definition 3.1 . Theorem 3.7. Let λ = ( X ; λ + , λ − ) be a bipolar fuzzy pseudo-UP ideal of a pseudo-UP algebra X. If the inequality x ≤ y · z and x ≤ y * z holds in X for all x , y , z ∈ X , the λ + ( z ) ≥ min { λ + (x), λ + ( y )} and λ − ( z ) ≤ max { λ − ( x ), λ − ( y )}. Proof. Let x , y , z ∈ X such that x ≤ y * z and x ≤ y · z. Then by the binary relation “≤” defined in X , we have x · ( y * z ) = 0 and x * ( y · z ) = 0. By Definition 3.1 , we have (3.1) λ + ( x ⋅ z ) ≥ min { λ + ( x ⋅ ( y ∗ z ) ) , λ + ( y ) } And by Theorem 3.4 , we have (3.2) λ + ( z ) = λ + ( 0 ∗ z ) ≥ min { λ + ( 0 ∗ ( x ⋅ z ) ) , λ + ( x ) } = min { λ + ( x ⋅ z ) , λ + ( x ) } By (3.1) and Definition 3.1 (3.3) λ + ( x ⋅ z ) ≥ min { λ + ( x ∗ ( y ⋅ z ) ) , λ + ( y ) } = min { λ + ( 0 ) , λ + ( y ) } By (3.2) and (3.3), we have Similarly, λ + ( z ) ≥ min { λ + ( x ⋅ z ) , λ + ( x ) } ≥ min { λ + ( y ) , λ + ( x ) } = min { λ + ( x ) , λ + ( y ) } . (3.4) λ − ( x ⋅ z ) ≤ max { λ − ( x ⋅ ( y ∗ z ) ) , λ − ( y ) } And by Theorem 3.4 , we have (3.5) λ − ( z ) = λ − ( 0 ∗ z ) ≤ max { λ − ( 0 ∗ ( x ⋅ z ) ) , λ − ( x ) } = max { λ − ( x ⋅ z ) , λ − ( x ) } By (3.4) and Definition 3.1 (3.6) λ − ( x ⋅ z ) ≤ max { λ − ( x ∗ ( y ⋅ z ) ) , λ − ( y ) } = max { λ − ( 0 ) , λ − ( y ) } By (3.5) and (3.6), we have λ − ( z ) ≤ max { λ − ( x ⋅ z ) , λ − ( x ) } ≤ max { λ − ( y ) , λ − ( x ) } = max { λ − ( x ) , λ − ( y ) } . Hence, λ + ( z ) ≥ min { λ + ( x ) , λ + ( y ) } and λ − ( z ) ≤ max { λ − ( x ) , λ − ( y ) } . □ Theorem 3.8. The intersection of any two bipolar fuzzy pseudo-UP ideals of a pseudo-UP algebra X is also a bipolar fuzzy pseudo-UP ideal. Proof. Let λ = ( X ; λ + , λ − ) and η = ( X ; η + , η − ) be two bipolar fuzzy pseudo-UP ideals of X. Then we claim that λ ∩ η is a bipolar fuzzy pseudo-UP ideal of X. Let x , y , z ∈ X. Then λ + ∩ η + ( 0 ) = min { λ + ( 0 ) , η + ( 0 ) } ≥ min { λ + ( x ) , η + ( x ) } = λ + ∩ η + ( x ) . And λ − ∩ η − ( 0 ) = max { λ − ( 0 ) , η − ( 0 ) } ≤ max { λ − ( x ) , η − ( x ) } = λ − ∩ η − ( x ) . Also , λ + ∩ η + ( x ⋅ z ) = min { λ + ( x ⋅ z ) , η + ( x ⋅ z ) } ≥ min { min { λ + ( x ⋅ ( y ∗ z ) ) , λ + ( y ) } , min { η + ( x ⋅ ( y ∗ z ) ) , η + ( y ) } } = min { min { λ + ( x ⋅ ( y ∗ z ) ) , η + ( x ⋅ ( y ∗ z ) ) } , min { λ + ( y ) , η + ( y ) } } = min { λ + ∩ η + ( x ⋅ ( y ∗ z ) ) , λ + ∩ η + ( y ) } , and λ + ∩ η + ( x ∗ z ) = min { λ + ( x ∗ z ) , η + ( x ∗ z ) } ≥ min { min { λ + ( x ∗ ( y ⋅ z ) ) , λ + ( y ) } , min { η + ( x ∗ ( y ⋅ z ) ) , η + ( y ) } } = min { min { λ + ( x ∗ ( y ⋅ z ) ) , η + ( x ∗ ( y ⋅ z ) ) } , min { λ + ( y ) , η + ( y ) } } = min { λ + ∩ η + ( x ∗ ( y ⋅ z ) ) , λ + ∩ η + ( y ) } . Additionally , λ − ∩ η − ( x ⋅ z ) = max { λ − ( x ⋅ z ) , η − ( x ⋅ z ) } ≤ max { max { λ − ( x ⋅ ( y ∗ z ) ) , λ − ( y ) } , max { η − ( x ⋅ ( y ∗ z ) ) , η − ( y ) } } = max { max { λ − ( x ⋅ ( y ∗ z ) ) , η − ( x ⋅ ( y ∗ z ) ) } , max { λ − ( y ) , η − ( y ) } } = max { λ − ∩ η − ( x ⋅ ( y ∗ z ) ) , λ − ∩ η − ( y ) } . And λ − ∩ η − ( x ∗ z ) = max { λ − ( x ∗ z ) , η − ( x ∗ z ) } ≤ max { max { λ − ( x ∗ ( y ⋅ z ) ) , λ − ( y ) } , max { η − ( x ∗ ( y ⋅ z ) ) , η − ( y ) } } = max { max { λ − ( x ∗ ( y ⋅ z ) ) , η − ( x ∗ ( y ⋅ z ) ) } , max { λ − ( y ) , η − ( y ) } } = max { λ − ∩ η − ( x ∗ ( y ⋅ z ) ) , λ − ∩ η − ( y ) } . Hence, λ ∩ η is a bipolar fuzzy pseudo-UP ideal of X. □ Corollary 3.9. The intersection of any set of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra X is also a bipolar fuzzy pseudo-UP ideal. Remark 3.10. The union of two bipolar fuzzy pseudo-UP ideals may not be bipolar fuzzy pseudo-UP ideal. Example 3.11. Let X = {0, 1, 2, and 3} be a set with a binary operation “·” and “*” defined by the following cayley Table 3 . Clearly, ( X , ·, *, 0) is a pseudo-UP algebra. We define a fuzzy set λ + : X → [0, 1] as follows, λ + (0) = 1, λ + (1) = 0.6, λ + (2) = 0.4, λ + (3) = 0.3 and we define a fuzzy set λ − : X → [−1, 0] as follows λ − (0) = −1, λ − (1) = −0.5, λ − (2) = −0.6, λ − (3) = −0.4 and a fuzzy set η + : X → [0, 1] define as follows, η + (0) = 1, η + (1) = 0.4, η + (2) = 0.5, η + (3) = 0.3. and η − : X → [−1, 0] define as follows η − (0) = −0.9, η − (1) = −0.5, η − (2) = −0.3, η − (3) = −0.2. Now, ( λ + ∪ η + )(1 · 3) = max { λ + (1 · 3), η + (1 · 3)} = max { λ + (3), η + (3)} = max{0.3, 0.3} = 0.3. Hence, ( λ + ∪ η + ) (1 · 3) = 0.3=.....................................(*). ( λ + ∪ η + ) ( 1 ⋅ 3 ) = max { λ + ( 1 ⋅ 3 ) , η + ( 1 ⋅ 3 ) } ≥ max { min { λ + ( 1 ⋅ ( 2 ∗ 3 ) ) , λ + ( 2 ) } , min { η + ( 1 ⋅ ( 2 ∗ 3 ) ) , η + ( 2 ) } } = max { min { λ + ( 1 ⋅ 0 ) , η + ( 1 ⋅ 0 ) } , min { λ + ( 2 ) , η + ( 2 ) } } = max { min { λ + ( 0 ) , η + ( 0 ) } , min { λ + ( 2 ) , η + ( 2 ) } } = max { min { 1 , 1 } , min { 0.4 , 0.5 } } = max { 1 , 0.5 } = 1 . From (*) we get 0.3 ≥ 1 which is contradict to Definition 3.1 . And ( λ − ∪ η − ) ( 1 ⋅ 3 ) = min { λ − ( 1 ⋅ 3 ) , η − ( 1 ⋅ 3 ) } = min { λ − ( 3 ) , η − ( 3 ) } = min { − 0.4 , − 0.2 } = − 0.4 ( ∗ ∗ ) . ( λ − ∪ η − ) ( 1 ⋅ 3 ) = min { λ − ( 1 ⋅ 3 ) , η − ( 1 ⋅ 3 ) } ≤ min { max { λ − ( 1 ⋅ ( 2 ∗ 3 ) ) , λ − ( 2 ) } , max { η − ( 1 ⋅ ( 2 ∗ 3 ) ) , η − ( 2 ) } } = min { max { λ − ( 1 ⋅ 0 ) , η − ( 1 ⋅ 0 ) } , max { λ − ( 2 ) , η − ( 2 ) } } = min { max { λ − ( 0 ) , η − ( 0 ) } , max { λ − ( 2 ) , η − ( 2 ) } } = min { max { − 1 , − 0.9 } , max { − 0.6 , − 0.3 } } = min { − 0.9 , − 0.3 } = − 0.9 . From (**), we get −0.4 ≤ −0.9 which is contradict to Definition 3.1 . This shows that the union of any two bipolar fuzzy pseudo-UP ideal of X may not be a bipolar fuzzy pseudo-UP ideal of X. Theorem 3.12. A bipolar fuzzy set λ = ( X , λ + , λ − ) in X is a bipolar fuzzy pseudo-UP ideal of X If and only if the fuzzy subset λ + and ( λ −) c are fuzzy pseudo-UP ideals of X. Proof. Let λ = ( X , λ + , λ − ) be a bipolar fuzzy pseudo-UP algebra of X. We need to show that the fuzzy subset λ + and ( λ −) c are fuzzy pseudo-UP ideals of X. Clearly, λ + is a fuzzy pseudo-UP ideal of X follows from the fact that ( X , λ + , λ − ) is a bipolar fuzzy pseudo-UP ideal of X. Now, it remains to show that ( λ −) c is a fuzzy pseudo-UP ideal of X. Let x , y , z ∈ X , then we have ( λ −) c (0) = −1 − λ − (0) ≥ −1 − λ − ( x ) = ( λ − ) c ( x ). Next , ( λ − ) c ( x ⋅ z ) = − 1 − λ − ( x ⋅ z ) ≥ − 1 − max { λ − ( x ⋅ ( y ∗ z ) ) , λ − ( y ) } = min { − 1 − λ − ( x ⋅ ( y ∗ z ) ) , − 1 − λ − ( y ) } ( by Lemma ) 2.8 ( 3 ) = min { ( λ − ) c ( x ⋅ ( y ∗ z ) ) , ( λ − ) c ( y ) } . Moreover , ( λ − ) c ( x ∗ z ) = − 1 − λ − ( x ∗ z ) ≥ − 1 − max { λ − ( x ∗ ( y ⋅ z ) ) , λ − ( y ) } = min { − 1 − λ − ( x ∗ ( y ⋅ z ) ) , − 1 − λ − ( y ) } ( by Lemma ) 2.8 ( 3 ) = min { ( λ − ) c ( x ∗ ( y ⋅ z ) ) , ( λ − ) c ( y ) } . Hence, ( λ −) c is a fuzzy pseudo-UP ideal of X. Conversely, assume that λ + and ( λ −) c is fuzzy pseudo-UP ideal of X. Then for every x , y , z ∈ X , we get λ + (0) ≥ λ + ( x ) and ( λ −) c (0) ≥ ( λ −) c ( x ), by Definition 3.1 Now, ( λ −) c (0) ≥ ( λ −) c ( x ) implies that −1 − λ − (0) ≥ −1 − λ − ( x ), then λ − (0) ≤ λ − ( x ). Now it is suffices to show that λ − ( x ~ z ) ≤ max { λ − ( x ·( y * z )), λ − ( y )} and λ − ( x * z ) ≤ max { λ − ( x *( y · z )), λ − ( y )}, for all x , y , z ∈ X. − 1 − λ − ( x ⋅ z ) = ( λ − ) c ( x ⋅ z ) ≥ min { ( λ − ) c ( x ⋅ ( y ∗ z ) ) , ( λ − ) c ( y ) } = min { − 1 − λ − ( x ⋅ ( y ∗ z ) ) , − 1 − λ − ( y ) } = − 1 − max { λ − ( x ⋅ ( y ∗ z ) ) , λ − ( y ) } , ( by Lemma ) 2.8 ⇒ λ − ( x ⋅ z ) ≤ max { λ − ( x ⋅ ( y ∗ z ) ) , λ − ( y ) } . Finally , − 1 − λ − ( x ∗ z ) = ( λ − ) c ( x ∗ z ) ≥ min { ( λ − ) c x ∗ ( y ⋅ z ) , ( λ − ) c ( y ) } = min { − 1 − λ − ( x ∗ ( y ⋅ z ) ) , − 1 − λ − ( y ) } = − 1 − max { λ − ( x ∗ ( y ⋅ z ) ) , λ − ( y ) } , ⇒ λ − ( x ∗ z ) ≤ max { λ − ( x ∗ ( y ⋅ z ) ) , λ − ( y ) } . ( by Lemma ) 2.8 Hence, λ = ( X , λ + , λ − ) is a bipolar fuzzy pseudo-UP ideal of X. □ Corollary 3.13. If λ + is a fuzzy pseudo-UP ideal of X , then λ = ( X , λ + , (( λ + ) c ) is a bipolar fuzzy pseudo-UP ideal of X. Proof. Suppose λ + is a fuzzy pseudo-UP ideal of X. Then we need to show that λ = ( X , λ + , ( λ −) c ) is a bipolar fuzzy pseudo-UP ideal of X. Since λ + is a fuzzy pseudo-UP ideal of X , it follows that λ + (0) ≥ λ + ( x ), λ + ( x · z ) ≥ min { λ + ( x · ( y * z )), λ + ( y )} and λ + ( x * z ) ≥ min { λ + ( x *( y · z )), λ + ( y )}, for all x , y , z ∈ X. Then it is enough to show that ( λ + ) c (0) ≥ ( λ + ) c ( x ), ( λ + ) c ( x · z ) ≥ min{( λ + ) c ( x · ( y * z )), ( λ + ) c ( y )} and ( λ + ) c ( x * z ) ≥ min{( λ + ) c ( x * ( y · z )), ( λ + ) c ( y )}. Now , ( λ + ) c ( 0 ) = 1 − λ + ( 0 ) ≤ 1 − λ + ( x ) = ( λ + ) c ( x ) ⇒ ( λ + ) c ( 0 ) ≤ ( λ + ) c ( x ) and ( λ + ) c ( x ⋅ z ) = 1 − λ + ( x ⋅ z ) ≤ 1 − min { λ + ( x ⋅ ( y ∗ z ) ) , λ + ( y ) } = max { 1 − λ + ( x ⋅ ( y ∗ z ) ) , 1 − λ + ( y ) } = max { ( λ + ) c ( x ⋅ ( y ∗ z ) ) , ( λ + ) c ( y ) } . ⇒ ( λ + ) c ( x ⋅ z ) ≤ max { ( λ + ) c ( x ⋅ ( y ∗ z ) ) , ( λ + ) c ( y ) } . Finally ( λ + ) c ( x ∗ z ) = 1 − λ + ( x ∗ z ) ≤ 1 − min { λ + ( x ∗ ( y . z ) ) , λ + ( y ) } = max { 1 − λ + ( x ∗ ( y . z ) ) , 1 − λ + ( y ) } = max { ( λ + ) c ( x ∗ ( y . z ) ) , ( λ + ) c ( y ) } . ⇒ ( λ + ) c ( x · z ) ≤ max { ( λ + ) c ( x ∗ ( y . z ) ) , ( λ + ) c ( y ) } . Therefore, λ = ( X , λ + , ( λ + ) c ) is a bipolar fuzzy pseudo-UP ideal of X. □ Corollary 3.14. If ( λ −) c is a fuzzy pseudo-UP ideal of X , then λ = ( X , ( λ − ) c ), λ − is a bipolar fuzzy pseudo-UP ideal of X. Proof. Similar to Corollary 3.13 . Using those, corollary the following is obtained. Theorem 3.15. A bipolar fuzzy set λ = ( X , λ + , λ − ) of X is a bipolar fuzzy pseudo-UP ideal of X if and only if λ = ( X , λ + , ( λ + ) c ) and ♦ λ = ( X , ( λ − ) c , λ − ) are bipolar fuzzy pseudo-UP ideals of X. Proof. Suppose λ = ( X , λ + , λ − ) is bipolar fuzzy pseudo-UP ideal of X , then for any x , y , z ∈ X , we have λ + (0) ≥ λ + ( x ), λ + ( x · z ) ≥ min { λ + ( x ·( y * z )), λ + ( y )} and λ + ( x * z ) ≥ min { λ + ( x *( y · z )), λ + ( y )}. Next we have to show that ( λ + ) c satisfies the conditions such that ( λ + ) c (0) ≤ ( λ + ) c ( x ), ( λ + ) c ( x · z ) ≤ max{( λ + ) c ( x · ( y * z )), ( λ + ) c ( y )} and ( λ + ) c ( x * z ) ≤ max{( λ + ) c ( x * ( y · z )), ( λ + ) c ( y )}. Now , ( λ + ) c ( 0 ) = 1 − λ + ( 0 ) ≤ 1 − λ + ( x ) = ( λ + ) c ( x ) ⇒ ( λ + ) c ( 0 ) ≤ ( λ + ) c ( x ) . Next , ( λ + ) c ( x · z ) = 1 − λ + ( x ⋅ z ) ≤ 1 − min { λ + ( x ⋅ ( y ∗ z ) ) , λ + ( y ) } = max { 1 − λ + ( x ⋅ ( y ∗ z ) ) , 1 − λ + ( y ) } = max { ( λ + ) c ( x ⋅ ( y ∗ z ) ) , ( λ + ) c ( y ) } . ⇒ ( λ + ) c ( x ⋅ z ) ≤ max { ( λ + ) c ( x ⋅ ( y ∗ z ) ) , ( λ + ) c ( y ) } . And also ( λ + ) c ( x ∗ z ) = 1 − λ + ( x ∗ z ) ≤ 1 − min { λ + ( x ∗ ( y . z ) ) , λ + ( y ) } = max { 1 − λ + ( x ∗ ( y . z ) ) , 1 − λ + ( y ) } = max { ( λ + ) c ( x ∗ ( y . z ) ) , ( λ + ) c ( y ) } . ⇒ ( λ + ) c ( x ∗ z ) ≤ max { ( λ + ) c ( x ∗ ( y . z ) ) , ( λ + ) c ( y ) } . Hence, λ is a bipolar fuzzy pseudo-UP ideal of X. Also, for any x , y , z ∈ X , λ − (0) ≤ λ − ( x ), λ − ( x · z ) ≤ max { λ − ( x · ( y * z )), λ − ( y )} and λ − ( x * z ) ≥ max { λ − ( x * ( y · z )), λ − ( y )}. Now we have to show that ( λ + ) c (0) ≥ ( λ + ) c ( x ), ( λ + ) c ( x · z ) ≥ min {( λ + ) c ( x · ( y * z )), ( λ + ) c ( y )} and ( λ + ) c ( x * z ) ≤ min {( λ + ) c ( x * ( y · z )), ( λ + ) c ( y )}. So for any x , y , z ∈ X , we have ( λ - ) c (0) = −1 − λ − (0) ≥ −1 − λ − ( x ) = (λ - ) c ( x ) ⇒ ( λ − ) c (0) ≥ ( λ − ) c ( x ). Next , ( λ − ) c ( ( x ⋅ z ) = − 1 − λ − ( x ⋅ z ) ≥ − 1 − max { λ − ( x ⋅ ( y ∗ z ) ) , λ − ( y ) } = min { − 1 − λ − ( x ⋅ ( y ∗ z ) ) , − 1 − λ − ( y ) } = min { ( ( λ − ) c ( x ⋅ ( y ∗ z ) ) , ( λ − ) c ( y ) } . ⇒ ( λ − ) c ( x ⋅ z ) ≥ min { ( λ − ) c ( x ⋅ ( y ∗ z ) ) , ( λ − ) c ( y ) } . Finally , ( λ − ) c ( x ∗ z ) = − 1 − λ − ( x ∗ z ) ≥ − 1 − max { λ − ( x ∗ ( y ⋅ z ) ) , λ − ( y ) } = min { − 1 − λ − ( x ∗ ( y ⋅ z ) ) , − 1 − λ − ( y ) } = min { ( λ − ) c ( x ∗ ( y ⋅ z ) ) , ( λ − ) c ( y ) } . ⇒ ( λ − ) c ( x ∗ z ) ≥ min { ( λ − ) c ( x ∗ ( y ⋅ z ) ) , ( λ − ) c ( y ) } . Hence, ♦ λ is a bipolar fuzzy pseudo-UP ideal of X. The proof of the converse follows from Definition 3.1 .□ Theorem 3.16. If λ = ( X , λ + , λ − ) is a bipolar fuzzy pseudo-UP ideal of X , then λ c is also a bipolar fuzzy pseudo-UP ideal of X. Proof. Let λ = ( X , λ + , λ − ) be a bipolar fuzzy pseudo-UP ideal of X. Then λ c = ( X , ( λ + ) c ( λ − ) c ), where ( λ + ) c (x) = 1 − λ + ( x ) and ( λ − ) c ( x ) = −1 − λ − ( x ). Now, for any x , y , z ∈ X , we have ( λ − ) c (0) = −1− λ − (0) ≥ −1− λ − ( x ) = ( λ − ( λ − ) c ( x ) ⇒ ( λ − ) c (0) ≥ ( λ − ) c ( x ) and ( λ + ) c 0) = 1− λ + (0) ≤ 1 − λ + (x) = ( λ + ) c (x) ⇒ ( λ + ) c (0) ≤ ( λ + ) c (x). Next , ( λ − ) c ( x ⋅ z ) = − 1 − λ − ( x ⋅ z ) ≥ − 1 − max { λ − ( x ⋅ ( y ∗ z ) ) , λ − ( y ) } = min { − 1 − λ − ( x ⋅ ( y ∗ z ) ) , − 1 − λ − ( y ) } = min { ( λ − ) c ( x ⋅ ( y ∗ z ) ) , ( λ − ) c ( y ) } . ⇒ ( λ − ) c ( x ⋅ z ) ≥ min { ( λ − ) c ( x ⋅ ( y ∗ z ) ) , ( λ − ) c ( y ) } . Finally , ( λ − ) c ( x ∗ z ) = − 1 − λ − ( x ∗ z ) ≥ − 1 − max { λ − ( x ∗ ( y ⋅ z ) ) , λ − ( y ) } = min { − 1 − λ − ( x ∗ ( y ⋅ z ) ) , − 1 − λ − ( y ) } = min { ( λ − ) c ( x ∗ ( y ⋅ z ) ) , ( λ − ) c ( y ) } . ⇒ ( λ − ) c ( x ∗ z ) ≥ min { ( λ − ) c ( x ∗ ( y ⋅ z ) ) , ( λ − ) c ( y ) } . ( λ − ) c ( x ⋅ z ) = 1 − λ + ( x ⋅ z ) ≤ 1 − min { λ + ( x ⋅ ( y ∗ z ) ) , λ + ( y ) } = max { 1 − λ + ( x ⋅ ( y ∗ z ) ) , 1 − λ + ( y ) } = max { ( λ − ) c ( x ⋅ ( y ∗ z ) ) , ( λ − ) c ( y ) } . ⇒ ( λ + ) c ( x ⋅ z ) ≤ max { ( λ + ) c ( x ⋅ ( y ∗ z ) ) , ( λ + ) c ( y ) } . And also , ( λ + ) c ( x ∗ z ) = 1 − λ + ( x ∗ z ) ≤ 1 − min { λ + ( x ∗ ( y ⋅ z ) ) , λ + ( y ) } = max { 1 − λ + ( x ∗ ( y ⋅ z ) ) , 1 − λ + ( y ) } = max { ( λ + ) c ( x ∗ ( y ⋅ z ) ) , ( λ + ) c ( y ) } . ⇒ ( λ + ) c ( x ∗ z ) ≤ max { ( λ + ) c ( x ∗ ( y ⋅ z ) ) , ( λ + ) c ( y ) } . Therefore, λ c is a bipolar fuzzy pseudo-UP ideal of X. □ Table 1. Bipolar fuzzy Pseudo-UP ideal. · 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 a 0 0 * 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 a b 0 Table 2. Bipolar fuzzy Pseudo-UP subalgebras is not bipolar fuzzy Pseudo-UP ideal. · 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 3 0 0 0 0 * 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 3 0 0 2 0 Table 3. The union of bipolar fuzzy Pseudo-UP ideals is not bipolar fuzzy Pseudo-UP ideal. · 0 1 2 3 0 0 1 2 3 1 0 0 3 3 2 0 1 0 0 3 0 1 3 0 * 0 1 2 3 0 0 1 2 3 1 0 0 3 3 2 0 1 0 0 3 0 2 3 0 Homomorphism on bipolar fuzzy pseudo-UP ideal In this section, we explore bipolar fuzzy pseudo-UP ideals in the context of homomorphism. We examine the homomorphic images and inverse images of bipolar fuzzy pseudo-UP ideals in pseudo-UP algebras, and present some key findings. Theorem 4.1. Let f : X → Y is an epimorphism of pseudo-UP algebras. If λ = ( X , λ + , λ − ) is a bipolar fuzzy pseudo-UP ideal of X with sup-inf property, then the image f ( λ ) is a bipolar pseudo-UP ideal of Y. Proof. Assume that λ = ( X , λ + , λ − ) is a bipolar fuzzy pseudo-UP ideal of X with sup-inf property. Let x , y , z ∈ Y with a ∈ f −1 ( x ), b ∈ f −1 ( y ) and c ∈ f −1 ( z ). λ + ( a ) = sup t ∈ f − 1 ( x ) λ + ( t ) , λ + ( b ) = sup t ∈ f − 1 ( y ) λ + ( t ) and λ + ( c ) = sup t ∈ f − 1 ( z ) λ + ( t ) λ − ( a ) = inf t ∈ f − 1 ( x ) λ − ( t ) , λ − ( b ) = inf t ∈ f − 1 ( y ) λ − ( t ) and λ − ( c ) = inf t ∈ f − 1 ( z ) λ − ( t ) Then by Definition 2.7 , we have ( f ) ( λ + ) ( 0 ) = sup t ∈ f − 1 ( 0 ) λ + ( t ) ≥ λ + ( 0 ) ≥ λ + ( a ) = sup t ∈ f − 1 ( x ) λ + ( t ) = ( f ) ( λ + ) ( x ) ( f ) ( λ − ) ( 0 ) = inf t ∈ f − 1 ( 0 ) λ − ( t ) ≥ λ − ( 0 ) ≥ λ − ( a ) = inf t ∈ f − 1 ( x ) λ − ( t ) = ( f ) ( λ − ) ( x ) ( f ) ( λ + ) ( x ⋅ z ) = sup t ∈ f − 1 ( x ⋅ z ) λ + ( t ) = λ + ( a ⋅ c ) ≥ min { λ + ( a ⋅ ( b ∗ c ) ) , λ + ( b ) } = min { sup t ∈ f − 1 ( x ⋅ ( y ∗ z ) ) λ + ( t ) , sup t ∈ f − 1 ( y ) λ + ( t ) } = min { ( f ) ( λ + ) ( x ⋅ ( y ∗ z ) ) , ( f ) ( λ + ) ( y ) } . ⇒ ( f ) ( λ + ) ( x ⋅ z ) ≥ min { ( f ) ( λ + ) ( x ⋅ ( y ∗ z ) ) , ( f ) ( λ + ) ( y ) } . And also , ( f ) ( λ + ) ( x ∗ z ) = sup t ∈ f − 1 ( x ∗ z ) λ + ( t ) = λ + ( a ∗ c ) ≥ min { λ + ( a ∗ ( b ⋅ c ) ) , λ + ( b ) } = min { sup t ∈ f − 1 ( x ∗ ( y ⋅ z ) ) λ + ( t ) , sup t ∈ f − 1 ( y ) λ + ( t ) } = min { ( f ) ( λ + ) ( x ∗ ( y ⋅ z ) ) , ( f ) ( λ + ) ( y ) } . ⇒ ( f ) ( λ + ) ( x ∗ z ) ≥ min { ( f ) ( λ + ) ( x ∗ ( y ⋅ z ) ) , ( f ) ( λ + ) ( y ) } . ( f ) ( λ − ) ( x ⋅ z ) = inf t ∈ f − 1 ( x ⋅ z ) λ − ( t ) = λ − ( a ⋅ c ) ≤ max { λ − ( a ⋅ ( b ∗ c ) ) , λ − ( b ) } = max { inf t ∈ f − 1 ( x ⋅ ( y ∗ z ) ) λ − ( t ) , inf t ∈ f − 1 ( y ) λ − ( t ) } = max { ( f ) ( λ − ) ( x ⋅ ( y ∗ z ) ) , ( f ) ( λ − ) ( y ) } . ⇒ ( f ) ( λ − ) ( x ⋅ z ) ≤ max { ( f ) ( λ − ) ( x ⋅ ( y ∗ z ) ) , ( f ) ( λ − ) ( y ) } . Finally, ( f ) ( λ − ) ( x ∗ z ) ≤ max { ( f ) ( λ − ) ( x ∗ ( y ⋅ z ) ) , ( f ) ( λ − ) ( y ) } . Therefore, f ( λ ) is a bipolar fuzzy pseudo-UP ideal of Y. □ Theorem 4.2. Let f be a homomorphism on pseudo-UP algebra X onto Y and η is a bipolar fuzzy pseudo-UP ideal of Y. Then f −1 ( η ) is a bipolar fuzzy pseudo-UP ideal of X. Proof. Let f be a homomorphism of pseudo-UP algebra. Assume that η is a bipolar pseudo- UP ideal of Y and let x ∈ X. Then f −1 ( η + )(0) = η + ( f (0)) ≥ η + ( f ( x )) = f −1 ( η + )( x ) and f −1 ( η − )(0) = η − ( f (0)) ≤ η − ( f ( x )) = f −1 ( η − )( x ). Let x , y , z ∈ X. Then f − 1 ( η + ) ( x ⋅ z ) = η + ( f ( x ⋅ z ) ) = η + ( f ( x ) ⋅ f ( z ) ) ≥ min { η + ( f ( x ) ⋅ ( f ( y ) ∗ f ( z ) ) , η + ( f ( y ) ) } = min { η + ( f ( x ⋅ ( y ∗ z ) ) , η + ( f ( y ) ) } = min { f − 1 ( η + ( x ⋅ ( y ∗ z ) ) ) , f − 1 ( η + ( y ) ) } . ⇒ f − 1 ( η + ) ( x ⋅ z ) ≥ min { f − 1 ( η + ( x ⋅ ( y ∗ z ) ) ) , f − 1 ( η + ( y ) ) } . And f − 1 ( η + ) ( x ∗ z ) = η + ( f ( x ∗ z ) ) = η + ( f ( x ) ∗ f ( z ) ) ≥ min { η + ( f ( x ) ∗ ( f ( y ) ⋅ f ( z ) ) , η + ( f ( y ) ) } = min { η + ( f ( x ∗ ( y ⋅ z ) ) , η + ( f ( y ) ) } = min { f − 1 ( η + ( x ∗ ( y ⋅ z ) ) ) , f − 1 ( η + ( y ) ) } . And f − 1 ( η − ) ( x ⋅ z ) = η − ( f ( x ⋅ z ) ) = η − ( f ( x ) ⋅ f ( z ) ) ≤ max { η − ( f ( x ) ⋅ ( f ( y ) ∗ f ( z ) ) , η − ( f ( y ) ) } = max { η − ( f ( x ⋅ ( y ∗ z ) ) , η − ( f ( y ) ) } = max { f − 1 ( η − ( x ⋅ ( y ∗ z ) ) ) , f − 1 ( η − ( y ) ) } . Again , f − 1 ( η − ) ( x ∗ z ) ≤ max { f − 1 ( η − ( x ∗ ( y ⋅ z ) ) ) , f − 1 ( η − ( y ) ) } . Hence, f −1 ( η ) is a bipolar fuzzy pseudo-UP ideal of X. The next theorem shows that the converse of the above Theorem 4.2 is also true if f is an epimorphism of pseudo-UP algebra. □ Theorem 4.3. Let f: X → Y be an epimorphism of pseudo-UP algebras. Let η be a bipolar fuzzy subset of Y. If f −1 ( η ) is a bipolar fuzzy pseudo-UP ideal of X , then η is a bipolar fuzzy pseudo-UP ideal of Y. Proof. Assume that f is an epimorphism of pseudo-UP algebra and f −1 ( η ) is a bipolar fuzzy pseudo- UP ideal of X. We need to show that η is a bipolar fuzzy pseudo-UP ideal of Y. Since f is an epimorphism of pseudo-UP algebra for any x ∈ Y , there exist a ∈ X such that f ( a ) = x. Then η + ( 0 ) = η + ( f ( 0 ) ) = f − 1 ( η + ) ( 0 ) ) ≥ f − 1 ( η + ) ( a ) = η + ( f ( a ) ) = η + ( x ) . η − ( 0 ) = η − ( f ( 0 ) ) = f − 1 ( η − ) ( 0 ) ) ≤ f − 1 ( η − ) ( a ) = η − ( f ( a ) ) = η − ( x ) . Let x , y , z ∈ Y. Then f ( a ) = x , f ( b ) = y and f ( c ) = z , for all a , b , c ∈ X. Thus η + ( x ⋅ z ) = η + ( f ( x ) ⋅ f ( z ) ) = η + ( f ( a ⋅ c ) ) = f − 1 ( η + ) ( a ⋅ c ) ≥ min { f − 1 ( η + ) ( a ⋅ ( b ∗ c ) ) , f − 1 ( η + ) ( b ) } = min { η + ( f ( a ⋅ ( b ∗ c ) ) , η + ( f ( b ) ) } = min { η + ( f ( a ) ⋅ ( f ( b ) ∗ f ( c ) ) ) , η + ( f ( b ) ) } = min { η + ( x ⋅ ( y ∗ z ) ) , η + ( y ) } . η + ( x ∗ z ) = η + ( f ( x ) ⋅ f ( z ) ) = η + ( f ( a ∗ c ) ) = f − 1 ( η + ) ( a ∗ c ) ≥ min { f − 1 ( η + ) ( a ∗ ( b ⋅ c ) ) , f − 1 ( η + ) ( b ) } = min { η + ( f ( a ∗ ( b ⋅ c ) ) , η + ( f ( b ) ) } = min { η + ( f ( a ) ∗ ( f ( b ) ⋅ f ( c ) ) ) , η + ( f ( b ) ) } = min { η + ( x ∗ ( y ⋅ z ) ) , η + ( y ) } . And η − ( x ⋅ z ) = η − ( f ( x ) ⋅ f ( z ) ) = η − ( f ( a ⋅ c ) ) = f − 1 ( η − ) ( a ⋅ c ) ≤ max { f − 1 ( η − ) ( a ⋅ ( b ∗ c ) ) , f − 1 ( η − ) ( b ) } = max { η − ( f ( a ⋅ ( b ∗ c ) ) , η − ( f ( b ) ) } = max { η − ( f ( a ) ⋅ ( f ( b ) ∗ f ( c ) ) ) , η − ( f ( b ) ) } = max { η − ( x ⋅ ( y ∗ z ) ) , η − ( y ) } . Similarly, η − ( x ⋅ z ) ≤ max { η − ( x ⋅ ( y ∗ z ) ) , η − ( y ) } . Therefore, η is a bipolar fuzzy pseudo-UP ideal of Y. □ Cartesian products of bipolar fuzzy pseudo-UP ideal In this section, we discuss the Cartesian product of two bipolar fuzzy pseudo-UP ideal is also again a bipolar fuzzy pseudo-UP ideal and some other results are also investigated. Theorem 5.1. Let λ = ( X , λ + , λ − ) and σ = ( Y , σ + , σ − ) be any two bipolar fuzzy pseudo-UP ideals of X and Y respectively. Then λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Proof. Assume that λ and σ be any two bipolar fuzzy pseudo-UP ideals of X and Y respectively. Take ( x , y ) ∈ X × Y. Then λ + × σ + ( 0 , 0 ) = min { λ + ( 0 ) , σ + ( 0 ) } ≥ min { λ + ( x ) , σ + ( y ) } = λ + × σ + ( x , y ) . And λ − × σ − ( 0 , 0 ) = max { λ − ( 0 ) , σ − ( 0 ) } ≤ max { λ − ( x ) , σ − ( y ) } = λ − × σ − ( x , y ) . Let ( x 1 , x 2 ), ( y 1 , y 2 ), ( z 1 , z 2 ) ∈ X × Y. Put x = ( x 1 , x 2 ), y = ( y 1 , y 2 ), z = ( z 1 , z 2 ). Then, λ + × σ + ( x ⋅ z ) = λ + × σ + ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) = λ + × σ + ( x 1 ⋅ z 1 , x 2 ⋅ z 2 ) = min { λ + ( x 1 ⋅ z 1 ) , σ + ( x 2 , z 2 ) } ≥ min { min { λ + ( x 1 ⋅ ( y 1 ∗ z 1 ) ) , λ + ( y 1 ) } , min { σ + ( x 2 ⋅ ( y 2 ∗ z 2 ) ) , σ + ( y 2 ) } } = min { min { λ + ( x 1 ⋅ ( y 1 ∗ z 1 ) ) , σ + ( x 2 ⋅ ( y 2 ∗ z 2 ) ) } , min { λ + ( y 1 ) , σ + ( y 2 ) } } = min { λ + × σ + ( ( x 1 ⋅ ( y 1 ∗ z 1 ) ) , ( x 2 ⋅ ( y 2 ∗ z 2 ) ) , λ + × σ + ( y 1 , y 2 ) } = min { λ + × σ + ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( ( z 1 , z 2 ) ) ) , λ + × σ + ( y 1 , y 2 ) } = min { λ + × σ + ( x ⋅ ( y ∗ z ) ) , λ + × σ + ( y ) } . And, λ + × σ + ( x ∗ z ) ≥ min { λ + × σ + ( x ⋅ ( y ∗ z ) ) , λ + × σ + ( y ) } . And, λ − × σ − ( x ⋅ z ) = λ − × σ − ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) = λ − × σ − ( x 1 ⋅ z 1 , x 2 ⋅ z 2 ) = max { λ − ( x 1 ⋅ z 1 ) , σ − ( x 2 , z 2 ) } ≤ max { max { λ − ( x 1 ⋅ ( y 1 ∗ z 1 ) ) , λ − ( y 1 ) } , max { σ − ( x 2 ⋅ ( y 2 ∗ z 2 ) ) , σ − ( y 2 ) } } = max { max { λ − ( x 1 ⋅ ( y 1 ∗ z 1 ) ) , σ − ( x 2 ⋅ ( y 2 ∗ z 2 ) ) } , max { λ − ( y 1 ) , σ − ( y 2 ) } } = max { λ − × σ − ( ( x 1 ⋅ ( y 1 ∗ z 1 ) ) , ( x 2 ⋅ ( y 2 ∗ z 2 ) ) , λ − × σ − ( y 1 , y 2 ) } = max { λ − × σ − ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( ( z 1 , z 2 ) ) ) , λ − × σ − ( y 1 , y 2 ) } = max { λ − × σ − ( x ⋅ ( y ∗ z ) ) , λ − × σ − ( y ) } . Additionally, λ − × σ − ( x ∗ z ) ≤ max { λ − × σ − ( x ⋅ ( y ∗ z ) ) , λ − × σ − ( y ) } . Hence, λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. □ Lemma 5.2. Let λ and σ be two bipolar fuzzy subsets of X and Y. If λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Then the followings are true. 1) λ + (0) ≥ σ + ( y ) and σ + (0) ≥ λ + ( x ), 2) λ − (0) ≤ σ − ( y ) and σ − (0) ≤ λ − ( x ). Proof. 1) Assume σ + ( y ) ≥ λ + (0) and λ + ( x ) ≥ σ + (0), ∀ x ∈ X , y ∈ Y Then , λ + × σ + ( x , y ) = min { λ + ( x ) , σ + ( y ) } ≥ min { λ + ( 0 ) , σ + ( 0 ) } = λ + × σ + ( 0 , 0 ) which is a contradiction . 2) Similarly, λ − ( x ) ≤ σ − ( 0 ) and σ − ( y ) ≤ λ − ( y ) , ∀ x ∈ X , y ∈ Y Then , λ − × σ − ( x , y ) = max { λ − ( x ) , σ − ( y ) } ≤ max { λ − ( 0 ) , σ − ( 0 ) } = λ − × σ − ( 0 , 0 ) which is a contradiction . Thus proving the result. □ Theorem 5.3. If λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y for any two bipolar fuzzy subsets λ and σ of X and Y respectively, then either λ is a bipolar fuzzy pseudo-UP ideal of X or σ is a bipolar fuzzy pseudo-UP ideal of Y. Proof. Let λ and σ be any two bipolar fuzzy subsets of pseudo-UP algebra X and Y respectively such that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Then λ + × σ + (0, 0) ≥ λ + × σ + ( x , y ). Now by Lemma 5.2 , let λ + (0) ≥ λ + ( x ) or σ + (0) ≥ σ + ( y ) and λ − (0) ≤ λ − ( x ) or σ − (0) ≤ σ − ( y ). Let ( x 1 , x 2 ), ( y 1 , y 2 ), ( z 1 , z 2 ) ∈ X × Y. Put x = ( x 1 , x 2 ), y = ( y 1 , y 2 ), z = ( z 1 , z 2 ). Then λ + × σ + ( x ⋅ z ) ≥ min { λ + × σ + ( x ⋅ ( y ∗ z ) ) , λ + × σ + ( y ) } = min { λ + × σ + ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) , λ + × σ + ( y 1 , y 2 ) } = min { λ + × σ + ( ( x 1 ⋅ ( y 1 ∗ z 1 ) , ( x 2 ⋅ ( y 2 ∗ z 2 ) ) σ + × σ + ( y 1 , y 2 ) } = min { min { λ + ( x 1 ∗ ( y 1 ∗ z 1 ) ) , σ + ( x 2 ⋅ ( y 2 ∗ z 2 ) ) } , min { λ + ( y 1 ) , σ + ( y 2 ) } } λ + × σ + ( x 1 ⋅ z 1 , x 2 ⋅ z 2 ) ≥ min { min { λ + ( x 1 ∗ ( y 1 ∗ z 1 ) ) , σ + ( x 2 ⋅ ( y 2 ∗ z 2 ) ) } , min { λ + ( y 1 ) , σ + ( y 2 ) } } = min { min { λ + ( x 1 ∗ ( y 1 ∗ z 1 ) , λ + ( y 1 ) } , min { σ + ( x 2 ⋅ ( y 2 ∗ z 2 ) ) , σ + ( y 2 ) } } . ⇒ either λ + ( x 1 · z 1 ) ≥ min { λ + ( x 1 · ( y 1 * z 1 )), λ + ( y 1 )} or σ + ( x 2 · z 2 ) ≥ min { σ + ( x 2 · ( y 2 * z 2 )), σ + ( y 2 )}. And either λ + ( x 1 * z 1 ) ≥ min { λ + ( x 1 *( y 1 · z 1 )), λ + ( y 1 )} or σ + ( x 2 * z 2 ) ≥ min { σ + ( x 2 *( y 2 · z 2 )), σ + ( y 2 )}. Similarly we can show that either λ − ( x 1 · z 1 ) ≤ max { λ − ( x 1 · ( y 1 * z 1 )), λ − ( y 1 )} or σ − ( x 2 · z 2 ) ≤ max { σ − ( x 2 · ( y 2 * z 2 )), σ − ( y 2 )} and λ − ( x 1 * z 1 ) ≤ max { λ − ( x 1 * ( y 1 · z 1 )), λ − ( y 1 )} or σ − ( x 2 * z 2 ) ≤ max { σ − ( x 2 * ( y 2 · z 2 )), σ − ( y 2 )}. Hence, either λ is a bipolar fuzzy pseudo-UP ideal of X or σ is a bipolar fuzzy pseudo-UP ideal of Y. □ Theorem 5.4. Let λ and σ be any bipolar fuzzy subsets of X and Y respectively. Then λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y if and only If λ + × σ + and ( λ − × σ − ) c are fuzzy pseudo-UP ideals of X × Y. Proof. Let λ and σ be any two bipolar fuzzy subsets of X and Y respectively such that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Now we need to show that λ + × σ + and ( λ − × σ − ) c are fuzzy pseudo-UP ideals of X × Y. Clearly, λ + × σ + is a fuzzy pseudo-UP ideal of X × Y. So it remains to show that ( λ − × σ − ) c is a fuzzy pseudo-UP ideal of X × Y. Now, for every ( x 1 , x 2 ), ( y 1 , y 2 ), ( z 1 , z 2 ) ∈ X × Y. ( λ − × σ − ) c ( 0 , 0 ) = − 1 − λ − × σ − ( 0 , 0 ) ≥ − 1 − λ − × σ − ( x 1 , x 2 ) = ( λ − × σ − ) c ( x 1 , x 2 ) . ( λ − × σ − ) c ( ( x 1 , x 2 ) ⋅ ( ( z 1 , z 2 ) ) = − 1 − λ − × σ − ( ( x 1 , x 2 ) ⋅ ( ( z 1 , z 2 ) ) ≥ − 1 − max { λ − × σ − ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , λ − × σ − ( y 1 , y 2 ) } = min { − 1 − λ − × σ − ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , − 1 − λ − × σ − ( y 1 , y 2 ) } = min { ( λ − × σ − ) c ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , ( λ − × σ − ) c ( y 1 , y 2 ) } . ( λ − × σ − ) c ( ( x 1 , x 2 ) ∗ ( ( z 1 , z 2 ) ) = − 1 − λ − × σ − ( ( x 1 , x 2 ) ∗ ( ( z 1 , z 2 ) ) ≥ − 1 − max { λ − × σ − ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( z 1 , z 2 ) ) ) , λ − × σ − ( y 1 , y 2 ) } = min { − 1 − λ − × σ − ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( z 1 , z 2 ) ) ) , − 1 − λ − × σ − ( y 1 , y 2 ) } = min { ( λ − × σ − ) c ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( z 1 , z 2 ) ) ) , ( λ − × σ − ) c ( y 1 , y 2 ) } . Hence, ( λ − × σ − ) c is a fuzzy pseudo-UP ideal of X × Y. Conversely, assume that λ + × σ + and ( λ − × σ − ) c are fuzzy pseudo-UP ideals of X × Y. We need to show that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Let ( x 1 , x 2 ), ( y 1 , y 2 ), ( z 1 , z 2 ) ∈ X × Y. Then λ + × σ + ( 0 , 0 ) ≥ λ + × σ + ( x 1 , x 2 ) and λ − × σ − ( 0 , 0 ) ≤ λ − × σ − ( x 1 , x 2 ) . Now, we have ( λ − × σ − ) c ( 0 , 0 ) ≥ ( λ − × σ − ) c ( x 1 , x 2 ) ⇒ − 1 − λ − × σ − ( 0 , 0 ) ≥ − 1 − λ − × σ − ( x 1 , x 2 ) ⇒ ( λ − × σ − ) ( 0 , 0 ) ≤ ( λ − × σ − ) ( x 1 , x 2 ) λ + × σ + ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) ≥ min { λ + × σ + ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , λ + × σ + ( y 1 , y 2 ) } . And, λ + × σ + ( ( x 1 , x 2 ) ∗ ( z 1 , z 2 ) ) ≥ min { λ + × σ + ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( z 1 , z 2 ) ) ) , λ + × σ + ( y 1 , y 2 ) } . − 1 − λ − × σ − ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) = ( λ − × σ − ) c ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) ≥ min { ( λ − × σ − ) c ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , ( λ − × σ − ) c ( y 1 , y 2 ) } = min { − 1 − λ − × σ − ( ( x 1 , x 2 ) · ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , − 1 − λ − × σ − ( y 1 , y 2 ) } = − 1 − max { λ − × σ − ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , λ − × σ − ( y 1 , y 2 ) } ⇒ λ − × σ − ( ( x 1 , x 2 ) · ( z 1 , z 2 ) ) ≤ max { λ − × σ − ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , λ − × σ − ( y 1 , y 2 ) } . Similarly, λ − × σ − ( ( x 1 , x 2 ) ∗ ( z 1 , z 2 ) ) ≤ max { λ − × σ − ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( z 1 , z 2 ) ) ) , λ − × σ − ( y 1 , y 2 ) } . Hence, λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. □ Theorem 5.5. Let λ and σ be any two bipolar fuzzy subsets of X and Y , respectively. Then λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y if and only if □ ( λ × σ ) = ( X × Y , λ + × σ + , ( λ + × σ + ) c ) and ♦ ( λ × σ ) = ( X × Y , ( λ − × σ − ) c , ( λ − × σ − ) are bipolar fuzzy pseudo-UP ideals of X × Y. Proof. Assume that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. Now we need to show that ( λ × σ ) and ♦ ( λ × σ ) are bipolar fuzzy pseudo-UP ideal of X × Y. Then for any ( x 1 , x 2 ), ( y 1 , y 2 ), ( z 1 , z 2 ) ∈ X × Y , we have λ + × σ + (0, 0) ≥ λ + × σ + ( x 1 , x 2 ), λ + × σ + (( x 1 , x 2 ) · (( z 1 , z 2 )) ≥ min { λ + × σ + (( x 1 , x 2 ) · (( y 1 , y 2 )*( z 1 , z 2 ))), λ + × σ + ( y 1 , y 2 )} and λ + × σ + (( x 1 , x 2 )*(( z 1 , z 2 )) ≥ min { λ + × σ + (( x 1 , x 2 )*(( y 1 , y 2 )· ( z 1 , z 2 ))), λ + × σ + ( y 1 , y 2 )}. Next, let ( x 1 , x 2 ), ( y 1 , y 2 ), ( z 1 , z 2 ) ∈ X × Y. we have ( λ + × σ + ) c (0, 0) = 1 − λ + × σ + (0, 0) ≤1 − λ + × σ + ( x 1 , x 2 ) = ( λ + × σ + ) c ( x 1 , x 2 ). ( λ + × σ + ) c ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) = 1 − λ + × σ + ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) ≤ 1 − min { λ + × σ + ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( x 1 , z 2 ) ) ) , λ + × σ + ( y 1 , y 2 ) } = max { 1 − λ + × σ + ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( x 1 , z 2 ) ) ) , 1 − λ + × σ + ( y 1 , y 2 ) } = max { ( λ + × σ + ) c ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( x 1 , z 2 ) ) ) , ( λ + × σ + ) c ( y 1 , y 2 ) } . ( λ + × σ + ) c ( ( x 1 , x 2 ) ∗ ( z 1 , z 2 ) ) = 1 − λ + × σ + ( ( x 1 , x 2 ) ∗ ( z 1 , z 2 ) ) ≤ 1 − min { λ + × σ + ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( x 1 , z 2 ) ) ) , λ + × σ + ( y 1 , y 2 ) } = max { 1 − λ + × σ + ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( x 1 , z 2 ) ) ) , 1 − λ + × σ + ( y 1 , y 2 ) } = max { ( λ + × σ + ) c ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( x 1 , z 2 ) ) ) , ( λ + × σ + ) c ( y 1 , y 2 ) } . Hence, ( λ × σ ) is a bipolar fuzzy pseudo-UP ideal of X × Y. And, ( λ − × σ − ) c ( 0 , 0 ) = − 1 − λ − × σ − ( 0 , 0 ) ≥ − 1 − λ − × σ − ( x 1 , x 2 ) = ( λ − × σ − ) c ( x 1 , x 2 ) . ( λ − × σ − ) c ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) = − 1 − λ − × σ − ( ( x 1 , x 2 ) ⋅ ( z 1 , z 2 ) ) ≥ − 1 − max { λ − × σ − ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , λ − × σ − ( y 1 , y 2 ) } = min { − 1 − λ − × σ − ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , − 1 − λ − × σ − ( y 1 , y 2 ) } = min { ( λ − × σ − ) c ( ( x 1 , x 2 ) ⋅ ( ( y 1 , y 2 ) ∗ ( z 1 , z 2 ) ) ) , ( λ − × σ − ) c ( y 1 , y 2 ) } . ( λ − × σ − ) c ( ( x 1 , x 2 ) ∗ ( z 1 , z 2 ) ) = − 1 − λ − × σ − ( ( x 1 , x 2 ) ∗ ( z 1 , z 2 ) ) ≥ − 1 − max { λ − × σ − ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( z 1 , z 2 ) ) ) , λ − × σ − ( y 1 , y 2 ) } = min { − 1 − λ − × σ − ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( z 1 , z 2 ) ) ) , − 1 − λ − × σ − ( y 1 , y 2 ) } = min { ( λ − × σ − ) c ( ( x 1 , x 2 ) ∗ ( ( y 1 , y 2 ) ⋅ ( z 1 , z 2 ) ) ) , ( λ − × σ − ) c ( y 1 , y 2 ) } . Hence, ♦ ( λ × σ ) is a bipolar fuzzy pseudo-UP ideal of X × Y. Conversely, assume that γ( λ × σ ) and ♦ ( λ × σ ) are bipolar fuzzy pseudo-UP ideals of X × Y. It is clear that λ × σ is a bipolar fuzzy pseudo-UP ideal of X × Y. □ Conclusion In this paper, we discussed the bipolar fuzzy pseudo-UP ideal of a pseudo-UP algebra and investigated several related properties. We proved that the intersection of two bipolar fuzzy pseudo-UP ideals is bipolar fuzzy pseudo-UP ideal. The union of two bipolar fuzzy pseudo-UP ideals is not always a bipolar fuzzy pseudo-UP ideal. We studied the homomorphic image and the inverse image of the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra and obtained some interesting results. We proved the Cartesian product of any two bipolar fuzzy pseudo-UP ideals in pseudo-UP algebra and investigated related results. We hope that the findings of this study will add new dimensions to the structures of bipolar fuzzy pseudo-UP ideals based on bipolar fuzzy sets and will serve as a foundation for further study into the structures of fuzzy pseudo-UP ideals by using the concept of bipolar fuzzy subset of pseudo-UP algebra. Author contributions All the authors are contributed equally in this manuscript and also both authors read and approved the final manuscript. Ethics and consent Ethics and consent were not required. Data availability No data are associated with this article. References 1. Iséki K: On axiom systems of propositional calculi. XV. Proc. Jpn. Acad. 1966; 42 (3): 217–220. Publisher Full Text 2. Romano DA: Pseudo-UP ideals and pseudo-UP filters in pseudo- UP algebras. Math. Sci. Appl. E-Notes. 2020; 8 (1): 155–158. Publisher Full Text 3. Zadeh LA: Fuzzy sets. Inf. Control. 1965; 8 (3): 338–353. Publisher Full Text 4. Rosenfeld A: Fuzzy groups. J. Math. Anal. Appl. 1971; 35 (3): 512–517. Publisher Full Text 5. Mechdesro AA, Alaba BA, Munie TM, et al. : Fuzzy pseudo-UP ideal of pseudo-UP algebra. Res. Math. 2023; 10 (1): 2234217. Publisher Full Text 6. Wechler W: The concept of fuzziness in automata and language theory.1978. (No Title). 7. Zimmermann HJ: Fuzzy set theory and its applications. Springer Science & Business Media; 2011. 8. Lee KM: Bipolar-valued fuzzy sets and their operations. Proc. Int. Conf. on Intelligent Technologies. Bangkok, Thailan: 2000; pp. 307–312. 9. Lee KJ: Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras. Bull. Malaysian Math. Sci. Soc. 2009; 32 (3): 361–373. 10. Alaba BA, Derseh BL, Wondifraw YG: On Intuitionistic Fuzzy PMS-Ideals of a PMS-Algebra under Homomorphism and Cartesian product. Int. J. Comput. Intell. Syst. 2023; 16 (1): 68. Publisher Full Text 11. Zhang WR: Bipolar fuzzy sets and relations. A computational framework for cognitive modeling and multiagent decision analysis. NAFIPS/IFIS/NASA’94. Proceedings of the First International Joint Conference of the North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intelligent. 1994; pp. 305–309. 12. Romano DA: Pseudo-UP algebras. An introduction. Bull. Int. Math. Virtual Inst. 2020; 10 (2): 349–355. 13. Lele C, Wu C, Weke P, et al. : Fuzzy ideals and weak ideals in BCK-algebras.2001. 14. Jun YB: Fuzzy d-ideals of d-algebras. J. Fuzzy Math. 2000; 8 (1): 123–130. 15. Sabarinathan S, Kumar DC, Muralikrishna P: Bipolar valued fuzzy α-ideal of BF-algebra. Circuits and Systems. 2016; 07 (10): 3054–3062. Publisher Full Text Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 18 Nov 2024 ADD YOUR COMMENT Comment Author details Author details 1 Department of Mathematics, Bahir Dar University College of Science, Bahir Dar, Amhara, 6000, Ethiopia 2 Department of Mathematics, Debre Berhan University, Debre Birhan, Amhara, Ethiopia Alachew Amaneh Mechderso Roles: Conceptualization, Writing – Original Draft Preparation, Writing – Review & Editing Berhanu Assaye Alaba Roles: Conceptualization Tilahun Mekonnen Munie Roles: Conceptualization Teferi Getachew Alemayhu Roles: Conceptualization Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (1) version 1 Published: 18 Nov 2024, 13:1386 https://doi.org/10.12688/f1000research.157173.1 Copyright © 2024 Mechderso AA et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article Mechderso AA, Alaba BA, Munie TM and Alemayhu TG. Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.12688/f1000research.157173.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 18 Nov 2024 Views 0 Cite How to cite this report: Balamurugan M. Reviewer Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344908 ) The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344908 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 26 Dec 2024 M Balamurugan , Vel Tech, Chennai, Tamil Nadu, India Approved VIEWS 0 https://doi.org/10.5256/f1000research.172589.r344908 It is quite interesting to review this article entitled “Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra” In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of ... Continue reading READ ALL It is quite interesting to review this article entitled “Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra” In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. However, the paper has been minor revisions in present form but need the following corrections: In page 8, 0.3= ………………(*), remove = In page 9, theorem 3.12 and corollary 3.14, ( l - ) c instead of ( l -) c and l = (X, ( l - ) c , l - ) instead of l = (X, ( l - ) c ), l - In page 9, corollary 313, l = (X, l + , ( l + ) c ) instead of l = (X, l + , (( l + ) c ) . Throughout the paper check it. In page 14, proof of the condition 2) σ - (y) ≤ l - (0) instead of σ - (y) ≤ l - (y). In references, the recent references to add in the related work. [Ref 1[ and [Ref 2]. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? No Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes References 1. Mursaleen M, Balamurugan M, Loganathan K, Nisar K: ()-Bipolar Fuzzy-Ideals of BCK/BCI-Algebras. Journal of Function Spaces . 2021; 2021 : 1-8 Publisher Full Text 2. Balamurugan M, Alessa N, Loganathan K, Kumar M: Bipolar Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras and Its Applications in Decision-Making. Mathematics . 2023; 11 (21). Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fuzzy Algebra I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Balamurugan M. Reviewer Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344908 ) The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344908 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Ur Rehman U. Reviewer Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344905 ) The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344905 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 24 Dec 2024 Ubaid Ur Rehman , University of Management and Technology,, Lahore, Punjab, Pakistan Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.172589.r344905 I have read the manuscript thoroughly and decided that the idea is new but the paper in recent shape cannot be accepted. I am adding some specific comments below. So, my decision regarding the current version of the manuscript is ... Continue reading READ ALL I have read the manuscript thoroughly and decided that the idea is new but the paper in recent shape cannot be accepted. I am adding some specific comments below. So, my decision regarding the current version of the manuscript is a major revision. Comments: The abstract should be improved. The present abstract is not up to the mark. The motivation of the proposed theory need a lot of improvement. The authors must discuss that what is the requirement of this theory to be developed. The authors interpreted the concept of bipolar valued algebraic structure, so they must discuss some recent work on fuzzy groups such as Bipolar Complex Fuzzy Subgroups [ref 1], Bipolar Complex Fuzzy Semigroups [ref 2], Analysis of Γ-Semigroups Based on Bipolar Complex Fuzzy Sets [ref 3] , T-Bipolar Soft Groups and Their Fundamental Laws [ref 4], Bipolar complex fuzzy submodules [ref 5] in the introduction. There must be suitable punctuation mark after every equation. The theoretical work is okay but what is the application of the proposed work? The authors must add an application of the bipolar valued fuzzy groups to show the practicality of their work. Improve the language of the whole manuscript with the help of native English speakers, and the structure of each section to ensure the readability of the manuscript. The conclusion is written like a contribution, but there is no deep analysis of obtained knowledge, no discussion, or wider application. How the authors can expand their work to other mathematical frameworks such bipolar complex fuzzy set devised by Tahir Mahmood? The author should discuss it in the future direction by citing 2 to 3 articles of bipolar complex fuzzy set for better future work. Add some merit and limitations of the proposed work. Is the work clearly and accurately presented and does it cite the current literature? No Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? I cannot comment. A qualified statistician is required. Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Partly References 1. Yang X, Mahmood T, ur Rehman U: Bipolar Complex Fuzzy Subgroups. Mathematics . 2022; 10 (16). Publisher Full Text 2. Rehman U, Mahmood T, Naeem M: Bipolar complex fuzzy semigroups. AIMS Mathematics . 2023; 8 (2): 3997-4021 Publisher Full Text 3. Mahmood T, ur Rehman U, Albaity M: Analysis of $$Gamma $$-semigroups based on bipolar complex fuzzy sets. Computational and Applied Mathematics . 2023; 42 (6). Publisher Full Text 4. Mahmood T, Hussain K, Ahmmad J, Shahab S, et al.: T-Bipolar soft groups and their fundamental laws. Journal of Intelligent & Fuzzy Systems . 2024; 46 (4): 9479-9490 Publisher Full Text 5. Alsuraiheed T, ur Rehman U, Khan M, Mahmood T: Bipolar complex fuzzy submodules. Physica Scripta . 2024; 99 (6). Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: fuzzy algebra, decision making, algebraic structures I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Ur Rehman U. Reviewer Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344905 ) The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344905 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Asaad B. Reviewer Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344899 ) The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344899 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 18 Dec 2024 Baravan Asaad , University of Zakho, Zakho, Kurdistan, Iraq; Cihan University-Duhok, Duhok, Iraq Approved VIEWS 0 https://doi.org/10.5256/f1000research.172589.r344899 The manuscript presented a new concept of pseudo-UP algebras called bipolar fuzzy pseudo-UP ideal which is combine between bipolar fuzzy sets and pseudo-UP ideal. Some theoretical results are presented via this new concept of pseudo-UP algebras. I ... Continue reading READ ALL The manuscript presented a new concept of pseudo-UP algebras called bipolar fuzzy pseudo-UP ideal which is combine between bipolar fuzzy sets and pseudo-UP ideal. Some theoretical results are presented via this new concept of pseudo-UP algebras. I have only the following points: - What is the comparison between the existing concept and the previous concept? Therefore, I recommend to accept this manuscript for indexing. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Topology, Fuzzy Set Theory, Soft Set Theory, Decision Making. I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Asaad B. Reviewer Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344899 ) The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344899 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 18 Nov 2024 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 3 Version 1 18 Nov 24 read read read Baravan Asaad , University of Zakho, Zakho, Iraq; Cihan University-Duhok, Duhok, Iraq Ubaid Ur Rehman , University of Management and Technology,, Lahore, Pakistan M Balamurugan , Vel Tech, Chennai, India Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2024 Balamurugan M. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 26 Dec 2024 | for Version 1 M Balamurugan , Vel Tech, Chennai, Tamil Nadu, India 0 Views copyright © 2024 Balamurugan M. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions It is quite interesting to review this article entitled “Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra” In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. However, the paper has been minor revisions in present form but need the following corrections: In page 8, 0.3= ………………(*), remove = In page 9, theorem 3.12 and corollary 3.14, ( l - ) c instead of ( l -) c and l = (X, ( l - ) c , l - ) instead of l = (X, ( l - ) c ), l - In page 9, corollary 313, l = (X, l + , ( l + ) c ) instead of l = (X, l + , (( l + ) c ) . Throughout the paper check it. In page 14, proof of the condition 2) σ - (y) ≤ l - (0) instead of σ - (y) ≤ l - (y). In references, the recent references to add in the related work. [Ref 1[ and [Ref 2]. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? No Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes References 1. Mursaleen M, Balamurugan M, Loganathan K, Nisar K: ()-Bipolar Fuzzy-Ideals of BCK/BCI-Algebras. Journal of Function Spaces . 2021; 2021 : 1-8 Publisher Full Text 2. Balamurugan M, Alessa N, Loganathan K, Kumar M: Bipolar Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras and Its Applications in Decision-Making. Mathematics . 2023; 11 (21). Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Fuzzy Algebra I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Balamurugan M. Peer Review Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344908) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344908 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2024 Ur Rehman U. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 24 Dec 2024 | for Version 1 Ubaid Ur Rehman , University of Management and Technology,, Lahore, Punjab, Pakistan 0 Views copyright © 2024 Ur Rehman U. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions I have read the manuscript thoroughly and decided that the idea is new but the paper in recent shape cannot be accepted. I am adding some specific comments below. So, my decision regarding the current version of the manuscript is a major revision. Comments: The abstract should be improved. The present abstract is not up to the mark. The motivation of the proposed theory need a lot of improvement. The authors must discuss that what is the requirement of this theory to be developed. The authors interpreted the concept of bipolar valued algebraic structure, so they must discuss some recent work on fuzzy groups such as Bipolar Complex Fuzzy Subgroups [ref 1], Bipolar Complex Fuzzy Semigroups [ref 2], Analysis of Γ-Semigroups Based on Bipolar Complex Fuzzy Sets [ref 3] , T-Bipolar Soft Groups and Their Fundamental Laws [ref 4], Bipolar complex fuzzy submodules [ref 5] in the introduction. There must be suitable punctuation mark after every equation. The theoretical work is okay but what is the application of the proposed work? The authors must add an application of the bipolar valued fuzzy groups to show the practicality of their work. Improve the language of the whole manuscript with the help of native English speakers, and the structure of each section to ensure the readability of the manuscript. The conclusion is written like a contribution, but there is no deep analysis of obtained knowledge, no discussion, or wider application. How the authors can expand their work to other mathematical frameworks such bipolar complex fuzzy set devised by Tahir Mahmood? The author should discuss it in the future direction by citing 2 to 3 articles of bipolar complex fuzzy set for better future work. Add some merit and limitations of the proposed work. Is the work clearly and accurately presented and does it cite the current literature? No Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? I cannot comment. A qualified statistician is required. Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Partly References 1. Yang X, Mahmood T, ur Rehman U: Bipolar Complex Fuzzy Subgroups. Mathematics . 2022; 10 (16). Publisher Full Text 2. Rehman U, Mahmood T, Naeem M: Bipolar complex fuzzy semigroups. AIMS Mathematics . 2023; 8 (2): 3997-4021 Publisher Full Text 3. Mahmood T, ur Rehman U, Albaity M: Analysis of $$Gamma $$-semigroups based on bipolar complex fuzzy sets. Computational and Applied Mathematics . 2023; 42 (6). Publisher Full Text 4. Mahmood T, Hussain K, Ahmmad J, Shahab S, et al.: T-Bipolar soft groups and their fundamental laws. Journal of Intelligent & Fuzzy Systems . 2024; 46 (4): 9479-9490 Publisher Full Text 5. Alsuraiheed T, ur Rehman U, Khan M, Mahmood T: Bipolar complex fuzzy submodules. Physica Scripta . 2024; 99 (6). Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise fuzzy algebra, decision making, algebraic structures I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (0) Ur Rehman U. Peer Review Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344905) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344905 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2024 Asaad B. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 18 Dec 2024 | for Version 1 Baravan Asaad , University of Zakho, Zakho, Kurdistan, Iraq; Cihan University-Duhok, Duhok, Iraq 0 Views copyright © 2024 Asaad B. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The manuscript presented a new concept of pseudo-UP algebras called bipolar fuzzy pseudo-UP ideal which is combine between bipolar fuzzy sets and pseudo-UP ideal. Some theoretical results are presented via this new concept of pseudo-UP algebras. I have only the following points: - What is the comparison between the existing concept and the previous concept? Therefore, I recommend to accept this manuscript for indexing. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Topology, Fuzzy Set Theory, Soft Set Theory, Decision Making. I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Asaad B. Peer Review Report For: Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2024, 13 :1386 ( https://doi.org/10.5256/f1000research.172589.r344899) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/13-1386/v1#referee-response-344899 Alongside their report, reviewers assign a status to the article: Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions Adjust parameters to alter display View on desktop for interactive features Includes Interactive Elements View on desktop for interactive features Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. Consider the following examples, but note that this is not an exhaustive list: Examples of 'Non-Financial Competing Interests' Within the past 4 years, you have held joint grants, published or collaborated with any of the authors of the selected paper. You have a close personal relationship (e.g. parent, spouse, sibling, or domestic partner) with any of the authors. You are a close professional associate of any of the authors (e.g. scientific mentor, recent student). You work at the same institute as any of the authors. You hope/expect to benefit (e.g. favour or employment) as a result of your submission. You are an Editor for the journal in which the article is published. Examples of 'Financial Competing Interests' You expect to receive, or in the past 4 years have received, any of the following from any commercial organisation that may gain financially from your submission: a salary, fees, funding, reimbursements. You expect to receive, or in the past 4 years have received, shared grant support or other funding with any of the authors. You hold, or are currently applying for, any patents or significant stocks/shares relating to the subject matter of the paper you are commenting on. Stay Updated Sign up for content alerts and receive a weekly or monthly email with all newly published articles Register with F1000Research Already registered? Sign in Not now, thanks close PLEASE NOTE If you are an AUTHOR of this article, please check that you signed in with the account associated with this article otherwise we cannot automatically identify your role as an author and your comment will be labelled as a “User Comment”. If you are a REVIEWER of this article, please check that you have signed in with the account associated with this article and then go to your account to submit your report, please do not post your review here. If you do not have access to your original account, please contact us . All commenters must hold a formal affiliation as per our Policies . The information that you give us will be displayed next to your comment. User comments must be in English, comprehensible and relevant to the article under discussion. We reserve the right to remove any comments that we consider to be inappropriate, offensive or otherwise in breach of the User Comment Terms and Conditions . Commenters must not use a comment for personal attacks. When criticisms of the article are based on unpublished data, the data should be made available. I accept the User Comment Terms and Conditions Please confirm that you accept the User Comment Terms and Conditions. Affiliation ✕ refresh Please enter your institution. Note: To add your institution or organisation, start typing the name and then select the correct name from the list. Where applicable, the name will appear in both the original language and in English. Do not paste in the name. If the name does not appear in the drop-down list, we will display the information you have entered. ✕ refresh Country/Region * USA UK Canada China France Germany Afghanistan Aland Islands Albania Algeria American Samoa Andorra Angola Anguilla Antarctica Antigua and Barbuda Argentina Armenia Aruba Australia Austria Azerbaijan Bahamas Bahrain Bangladesh Barbados Belarus Belgium Belize Benin Bermuda Bhutan Bolivia Bosnia and Herzegovina Botswana Bouvet Island Brazil British Indian Ocean Territory British Virgin Islands Brunei Bulgaria Burkina Faso Burundi Cambodia Cameroon Canada Cape Verde Cayman Islands Central African Republic Chad Chile China Christmas Island Cocos (Keeling) Islands Colombia Comoros Congo Cook Islands Costa Rica Cote d'Ivoire Croatia Cuba Cyprus Czech Republic Democratic Republic of the Congo Denmark Djibouti Dominica Dominican Republic Ecuador Egypt El Salvador Equatorial Guinea Eritrea Estonia Ethiopia Falkland Islands Faroe Islands Federated States of Micronesia Fiji Finland France French Guiana French Polynesia French Southern Territories Gabon Georgia Germany Ghana Gibraltar Greece Greenland Grenada Guadeloupe Guam Guatemala Guernsey Guinea Guinea-Bissau Guyana Haiti Heard Island and Mcdonald Islands Holy See (Vatican City State) Honduras Hong Kong Hungary Iceland India Indonesia Iran Iraq Ireland Israel Italy Jamaica Japan Jersey Jordan Kazakhstan Kenya Kiribati Kosovo (Serbia and Montenegro) Kuwait Kyrgyzstan Lao People's Democratic Republic Latvia Lebanon Lesotho Liberia Libya Liechtenstein Lithuania Luxembourg Macao Madagascar Malawi Malaysia Maldives Mali Malta Marshall Islands Martinique Mauritania Mauritius Mayotte Mexico Minor Outlying Islands of the United States Moldova Monaco Mongolia Montenegro Montserrat Morocco Mozambique Myanmar Namibia Nauru Nepal Netherlands Antilles New Caledonia New Zealand Nicaragua Niger Nigeria Niue Norfolk Island North Korea North Macedonia Northern Mariana Islands Norway Oman Pakistan Palau Palestinian Territory Panama Papua New Guinea Paraguay Peru Philippines Pitcairn Poland Portugal Puerto Rico Qatar Reunion Romania Russian Federation Rwanda Saint Helena Saint Kitts and Nevis Saint Lucia Saint Pierre and Miquelon Saint Vincent and the Grenadines Samoa San Marino Sao Tome and Principe Saudi Arabia Senegal Serbia Seychelles Sierra Leone Singapore Slovakia Slovenia Solomon Islands Somalia South Africa South Georgia and the South Sandwich Is South Korea South Sudan Spain Sri Lanka Sudan Suriname Svalbard and Jan Mayen Swaziland Sweden Switzerland Syria Taiwan Tajikistan Tanzania Thailand The Gambia The Netherlands Timor-Leste Togo Tokelau Tonga Trinidad and Tobago Tunisia Turkey Turkmenistan Turks and Caicos Islands Tuvalu UK USA Uganda Ukraine United Arab Emirates United States Virgin Islands Uruguay Uzbekistan Vanuatu Venezuela Vietnam Wallis and Futuna West Bank and Gaza Strip Western Sahara Yemen Zambia Zimbabwe Please select your country/region. You must enter a comment. Competing Interests Please disclose any competing interests that might be construed to influence your judgment of the article's or peer review report's validity or importance. Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. Consider the following examples, but note that this is not an exhaustive list: Examples of 'Non-Financial Competing Interests' Within the past 4 years, you have held joint grants, published or collaborated with any of the authors of the selected paper. You have a close personal relationship (e.g. parent, spouse, sibling, or domestic partner) with any of the authors. You are a close professional associate of any of the authors (e.g. scientific mentor, recent student). You work at the same institute as any of the authors. You hope/expect to benefit (e.g. favour or employment) as a result of your submission. You are an Editor for the journal in which the article is published. Examples of 'Financial Competing Interests' You expect to receive, or in the past 4 years have received, any of the following from any commercial organisation that may gain financially from your submission: a salary, fees, funding, reimbursements. You expect to receive, or in the past 4 years have received, shared grant support or other funding with any of the authors. You hold, or are currently applying for, any patents or significant stocks/shares relating to the subject matter of the paper you are commenting on. Please state your competing interests The comment has been saved. An error has occurred. Please try again. Cancel Post var lTitle = "Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP...".replace("'", ''); var linkedInUrl = "http://www.linkedin.com/shareArticle?url=https://f1000research.com/articles/13-1386/v1" + "&title=" + encodeURIComponent(lTitle) + "&summary=" + encodeURIComponent('Read the article by '); var deliciousUrl = "https://del.icio.us/post?url=https://f1000research.com/articles/13-1386/v1&title=" + encodeURIComponent(lTitle); var redditUrl = "http://reddit.com/submit?url=https://f1000research.com/articles/13-1386/v1" + "&title=" + encodeURIComponent(lTitle); linkedInUrl += encodeURIComponent('Mechderso AA et al.'); var offsetTop = /chrome/i.test( navigator.userAgent ) ? 4 : -10; var addthis_config = { ui_offset_top: offsetTop, services_compact : "facebook,twitter,www.linkedin.com,www.mendeley.com,reddit.com", services_expanded : "facebook,twitter,www.linkedin.com,www.mendeley.com,reddit.com", services_custom : [ { name: "LinkedIn", url: linkedInUrl, icon:"/img/icon/at_linkedin.svg" }, { name: "Mendeley", url: "http://www.mendeley.com/import/?url=https://f1000research.com/articles/13-1386/v1/mendeley", icon:"/img/icon/at_mendeley.svg" }, { name: "Reddit", url: redditUrl, icon:"/img/icon/at_reddit.svg" }, ] }; var addthis_share = { url: "https://f1000research.com/articles/13-1386", templates : { twitter : "Bipolar Fuzzy Pseudo-UP Ideal Of Pseudo-UP Algebra. Mechderso AA et al., published by " + "@F1000Research" + ", https://f1000research.com/articles/13-1386/v1" } }; if (typeof(addthis) != "undefined"){ addthis.addEventListener('addthis.ready', checkCount); addthis.addEventListener('addthis.menu.share', checkCount); } $(".f1r-shares-twitter").attr("href", "https://twitter.com/intent/tweet?text=" + addthis_share.templates.twitter); $(".f1r-shares-facebook").attr("href", "https://www.facebook.com/sharer/sharer.php?u=" + addthis_share.url); $(".f1r-shares-linkedin").attr("href", addthis_config.services_custom[0].url); $(".f1r-shares-reddit").attr("href", addthis_config.services_custom[2].url); $(".f1r-shares-mendelay").attr("href", addthis_config.services_custom[1].url); function checkCount(){ setTimeout(function(){ $(".addthis_button_expanded").each(function(){ var count = $(this).text(); if (count !== "" && count != "0") $(this).removeClass("is-hidden"); else $(this).addClass("is-hidden"); }); }, 1000); } close How to cite this report {{reportCitation}} Cancel Copy Citation Details $(function(){R.ui.buttonDropdowns('.dropdown-for-downloads');}); $(function(){R.ui.toolbarDropdowns('.toolbar-dropdown-for-downloads');}); $.get("/articles/acj/157173/172589") new F1000.Clipboard(); new F1000.ThesaurusTermsDisplay("articles", "article", "172589"); $(document).ready(function() { $( "#frame1" ).on('load', function() { var mydiv = $(this).contents().find("div"); var h = mydiv.height(); console.log(h) }); var tooltipLivingFigure = jQuery(".interactive-living-figure-label .icon-more-info"), titleLivingFigure = tooltipLivingFigure.attr("title"); tooltipLivingFigure.simpletip({ fixed: true, position: ["-115", "30"], baseClass: 'small-tooltip', content:titleLivingFigure + " " }); tooltipLivingFigure.removeAttr("title"); $("body").on("click", ".cite-living-figure", function(e) { e.preventDefault(); var ref = $(this).attr("data-ref"); $(this).closest(".living-figure-list-container").find("#" + ref).fadeIn(200); }); $("body").on("click", ".close-cite-living-figure", function(e) { e.preventDefault(); $(this).closest(".popup-window-wrapper").fadeOut(200); }); $(document).on("mouseup", function(e) { var metricsContainer = $(".article-metrics-popover-wrapper"); if (!metricsContainer.is(e.target) && metricsContainer.has(e.target).length === 0) { $(".article-metrics-close-button").click(); } }); var articleId = $('#articleId').val(); if($("#main-article-count-box").attachArticleMetrics) { $("#main-article-count-box").attachArticleMetrics(articleId, { articleMetricsView: true }); } }); var figshareWidget = $(".new_figshare_widget"); if (figshareWidget.length > 0) { window.figshare.load("f1000", function(Widget) { // Select a tag/tags defined in your page. In this tag we will place the widget. _.map(figshareWidget, function(el){ var widget = new Widget({ articleId: $(el).attr("figshare_articleId") //height:300 // this is the height of the viewer part. [Default: 550] }); widget.initialize(); // initialize the widget widget.mount(el); // mount it in a tag that's on your page // this will save the widget on the global scope for later use from // your JS scripts. This line is optional. //window.widget = widget; }); }); } close Error Close Add Reset F1000.MICROSERVICES.AFFILIATION = ''; $(document).ready(function () { $('.js-affiliations-form').each((index, form) => { new AffiliationForm({ formId: form.id, institutionErrorSelector: '.comment-enter-institution', departmentErrorSelector: '.comment-enter-department', placeSelector: '.js-add-comment-place', stateSelector: '.js-add-comment-state', zipCodeSelector: '.js-add-comment-zipcode', countrySelector: '.js-add-comment-country', countryErrorSelector: '.comment-enter-country', }); }); }); $(document).ready(function () { var reportIds = { "344901": 0, "344900": 0, "344903": 0, "344902": 0, "344899": 18, "344908": 8, "344905": 15, "344904": 0, "344907": 0, "344906": 0, }; $(".referee-response-container,.js-referee-report").each(function(index, el) { var reportId = $(el).attr("data-reportid"), reportCount = reportIds[reportId] || 0; $(el).find(".comments-count-container,.js-referee-report-views").html(reportCount); }); var uuidInput = $("#article_uuid"), oldUUId = uuidInput.val(), newUUId = "11378bd6-c320-4837-b865-83560ba68a46"; uuidInput.val(newUUId); $("a[href*='article_uuid=']").each(function(index, el) { var newHref = $(el).attr("href").replace(oldUUId, newUUId); $(el).attr("href", newHref); }); }); An innovative open access publishing platform offering rapid publication and open peer review, whilst supporting data deposition and sharing. Browse Gateways Collections How it Works Contact For Developers Cookie Notice Privacy Notice RSS Submit Your Research Follow us © 2012-2026 F1000 Research Ltd. ISSN 2046-1402 | Legal | Partner of Research4Life • CrossRef • ORCID • FAIRSharing R.templateTests.simpleTemplate = R.template(' $text $text $text $text $text '); R.templateTests.runTests(); var F1000platform = new F1000.Platform({ name: "f1000research", displayName: "F1000Research", hostName: "f1000research.com", id: "1", editorialEmail: "[email protected]", infoEmail: "[email protected]", usePmcStats: true }); $(function(){R.ui.dropdowns('.dropdown-for-authors, .dropdown-for-about, .dropdown-for-myresearch');}); // $(function(){R.ui.dropdowns('.dropdown-for-referees');}); $(document).ready(function () { if ($(".cookie-warning").is(":visible")) { $(".sticky").css("margin-bottom", "35px"); $(".devices").addClass("devices-and-cookie-warning"); } $(".cookie-warning .close-button").click(function (e) { $(".devices").removeClass("devices-and-cookie-warning"); $(".sticky").css("margin-bottom", "0"); }); $("#tweeter-feed .tweet-message").each(function (i, message) { var self = $(message); self.html(linkify(self.html())); }); $(".partner").on("mouseenter mouseleave", function() { $(this).find(".gray-scale, .colour").toggleClass("is-hidden"); }); }); Sign In Remember me Forgotten your password? Sign In Cancel Email or password not correct. Please try again Please wait... $(function(){ // Note: All the setup needs to run against a name attribute and *not* the id due the clonish // nature of facebox... $("a[id=googleSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("GOOGLE"); $("form[id=oAuthForm]").submit(); }); $("a[id=facebookSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("FACEBOOK"); $("form[id=oAuthForm]").submit(); }); $("a[id=orcidSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("ORCID"); $("form[id=oAuthForm]").submit(); }); }); If you've forgotten your password, please enter your email address below and we'll send you instructions on how to reset your password. The email address should be the one you originally registered with F1000. Email address not valid, please try again You registered with F1000 via Google, so we cannot reset your password. To sign in, please click here . If you still need help with your Google account password, please click here . You registered with F1000 via Facebook, so we cannot reset your password. To sign in, please click here . If you still need help with your Facebook account password, please click here . Code not correct, please try again Reset password Cancel Email us for further assistance. Server error, please try again. If your email address is registered with us, we will email you instructions to reset your password. If you think you should have received this email but it has not arrived, please check your spam filters and/or contact for further assistance. Please wait... Register $(document).ready(function () { signIn.createSignInAsRow($("#sign-in-form-gfb-popup")); $(".target-field").each(function () { var uris = $(this).val().split("/"); if (uris.pop() === "login") { $(this).val(uris.toString().replace(",","/")); } }); });