On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra

On homomorphism and Cartesian products of fuzzy PUP... | 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/14-115" }, "headline": "On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra", "datePublished": "2025-01-20T16:05:38", "dateModified": "2025-01-20T16:05:38", "author": [ { "@type": "Person", "name": "Alachew Amaneh Mechderso" }, { "@type": "Person", "name": "Berhanu Assaye Alaba" }, { "@type": "Person", "name": "Tilahun Mekonnen Munie" } ], "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": " Background In this paper, we explore the Cartesian product of fuzzy PUP ideals and subalgebras within a PUP (pseudo-UP)algebra X. We investigate how fuzzy pseudo-UP ideals are affected under PUP homomorphisms, particularly focusing on the image and inverse image of the fuzzy sets K α and K v , which generalize classical fuzzy sets. These fuzzy sets aim to quantify the level of ambiguity in fuzzy situations. Methods We introduced the concept of the Cartesian product of fuzzy PUP ideals and subalgebras of PUP algebra and analyzed their properties. Specifically, we studied the behavior of fuzzy pseudo-UP ideals under PUP homomorphism, using the fuzzy sets K α and K v to model ambiguity. We also examined the strongest fuzzy PUP relation on a fuzzy PUP subalgebra and ideal of PUP algebra. Results We proved that the Cartesian product of fuzzy PUP ideals forms a fuzzy pseudo-UP ideal. This result was characterized through its level sets, providing a deeper understanding of the interaction between fuzzy PUP ideals and their algebraic structure. Additionally, the strongest fuzzy PUP relation on a fuzzy PUP subalgebra was defined, and several important properties were derived. Conclusions This work extends the concept of fuzzy PUP ideals in the context of PUP algebras and explores how fuzzy homomorphism influence their structure. Our results contribute to a better understanding of fuzzy relations in algebraic settings and open the door for further research into the properties and applications of fuzzy ideals in PUP algebras. " } { "@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/14-115/v1", "name": "On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra" } } ] } Home Browse On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Mechderso AA, Alaba BA and Munie TM. On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra [version 1; peer review: 1 approved with reservations] . F1000Research 2025, 14 :115 ( https://doi.org/10.12688/f1000research.160312.1 ) 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 On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra [version 1; peer review: 1 approved with reservations] Alachew Amaneh Mechderso https://orcid.org/0009-0007-9566-9901 1,2 , Berhanu Assaye Alaba 2 , Tilahun Mekonnen Munie 2 Alachew Amaneh Mechderso https://orcid.org/0009-0007-9566-9901 1,2 , Berhanu Assaye Alaba 2 , Tilahun Mekonnen Munie 2 PUBLISHED 20 Jan 2025 Author details Author details 1 Department of Maths,, Kabridahar University, Kebri Dehar, Somali, Ethiopia 2 Mathematics, Bahir Dar University, Bahir Dar, Amhara, 6000, Ethiopia Alachew Amaneh Mechderso Roles: Conceptualization, Writing – Original Draft Preparation Berhanu Assaye Alaba Roles: Conceptualization, Supervision, Writing – Review & Editing Tilahun Mekonnen Munie Roles: Conceptualization, Supervision, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS Abstract Background In this paper, we explore the Cartesian product of fuzzy PUP ideals and subalgebras within a PUP (pseudo-UP)algebra X. We investigate how fuzzy pseudo-UP ideals are affected under PUP homomorphisms, particularly focusing on the image and inverse image of the fuzzy sets K α and K v , which generalize classical fuzzy sets. These fuzzy sets aim to quantify the level of ambiguity in fuzzy situations. Methods We introduced the concept of the Cartesian product of fuzzy PUP ideals and subalgebras of PUP algebra and analyzed their properties. Specifically, we studied the behavior of fuzzy pseudo-UP ideals under PUP homomorphism, using the fuzzy sets K α and K v to model ambiguity. We also examined the strongest fuzzy PUP relation on a fuzzy PUP subalgebra and ideal of PUP algebra. Results We proved that the Cartesian product of fuzzy PUP ideals forms a fuzzy pseudo-UP ideal. This result was characterized through its level sets, providing a deeper understanding of the interaction between fuzzy PUP ideals and their algebraic structure. Additionally, the strongest fuzzy PUP relation on a fuzzy PUP subalgebra was defined, and several important properties were derived. Conclusions This work extends the concept of fuzzy PUP ideals in the context of PUP algebras and explores how fuzzy homomorphism influence their structure. Our results contribute to a better understanding of fuzzy relations in algebraic settings and open the door for further research into the properties and applications of fuzzy ideals in PUP algebras. READ ALL READ LESS Keywords Keywords: level set, fuzzy PUP ideal and subalgebra, homomorphism, cartesian product and strongest fuzzy relation. 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: © 2025 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 and Munie TM. On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra [version 1; peer review: 1 approved with reservations] . F1000Research 2025, 14 :115 ( https://doi.org/10.12688/f1000research.160312.1 ) First published: 20 Jan 2025, 14 :115 ( https://doi.org/10.12688/f1000research.160312.1 ) Latest published: 20 Jan 2025, 14 :115 ( https://doi.org/10.12688/f1000research.160312.1 ) Introduction The concepts of set theory and set hypothesis are fascinating in mathematics. However, the core principle of the fundamental set hypothesis, which states that an element either belongs to a set or does not, can make it challenging to represent much of human communication. In a traditional (crisp) set, we have a distinct understanding of whether an element is part of the set or not. Fuzzy sets enable components to be modestly in a set. Each element is assigned a degree of membership in a set, with values ranging from 0 to 1. If only the extreme values of 0 and 1 were allowed, it would be equivalent to traditional (crisp) sets. This theory is being applied across various fields, including medical diagnosis, vulnerability assessment of gas pipeline networks, travel time estimation, and neural network models. Zadeh 1 introduced the concept of fuzzy sets in 1965. The primary motivation for studying fuzzy sets is their enhanced ability to handle uncertainty and vagueness in physical problems, surpassing the capabilities of classic fuzzy set theory. This is particularly useful in areas such as logic programming, decision-making, financial services, psychological assessments, medical diagnosis, career planning, and artificial intelligence. Rosenfeld 2 explored fuzzy subgroups within fuzzy sets and their fundamental implications in 1971. In 2018 Akram 3 the strongest fuzzy relations on fuzzy ideals of Lie were studied, leading to several interesting findings. In 2020, Romano 4 introduced a new class of algebras known as pseudo-UP algebras, extending the concept of UP-algebras. He also explored pseudo-UP filters and PUP ideals within pseudo-UP algebras in. 5 Additionally, Romano introduced the concept of homomorphisms between pseudo-UP algebras in. 6 Choudhury et al. 7 introduced the concept of fuzzy homomorphism and its related implications for fuzzy subgroups in 1988 and in 2012 Mostafa et al. 8 introduced fuzzy Cartesian product and fuzzy strongest relation on KU algebra. In our previous paper, 9 we introduced the concept of a fuzzy PUP subalgebra and fuzzy PUP ideal in PUP algebra and examined some of its properties. In this article, we continue from the previous discussions, we present and examine the concepts of Cartesian product of fuzzy PUP ideal and fuzzy PUP subalgebra within PUP Algebra. The key contributions of this paper are as follows: 1) The concept of κ α and κ ν have been presented in pseudo-UP algebra. 2) The behavior of κ α and κ ν fuzzy PUP ideal homomorphic image of this specific homomorphism has been examined. 3) The strongest fuzzy PUP relations on a fuzzy set within a PUP algebra are explored, demonstrating that a fuzzy PUP relation on a fuzzy set in a PUP algebra is a fuzzy PUP ideal and subalgebra if and only if the corresponding fuzzy set in the PUP algebra is itself a fuzzy pseudo-UP ideal and subalgebra. 4) We investigate the concept of fuzzy PUP ideals and subalgebras in the context of the Cartesian product and analyze several associated properties. 5) We explore some idea of level set of fuzzy PUP ideal and subalgebra of PUP algebra. Method In this section, we review some fundamental definitions and results that are utilized in the study presented in this paper. Definition 2.1. 5 Let J be a non-empty subset of PUP algebra X . It is termed a PUP ideal of X , if it satisfies the following conditions, for all x, y, z ∈ X. i) 0 ∈ J, ii) If x · (y ٨ z), y ∈ J , then x · z ∈ J, iii) If x ٨ (y · z), y ∈ J , then x ٨ z ∈ J, for all x, y, z ∈ X. Lemma 2.2. 4 In a pseudo-UP algebra X the following holds, for each x ∈ X , 1) x · 0 = 0 and x ٨ 0 = 0 , 2) 0 · x = x and 0 ٨ x = x , and 3) x · x = 0 and x ٨ x = 0 . Definition 2.3. 9 A fuzzy subset κ is called fuzzy PUP subalgebra of X , if it satisfies the following axioms: i) κ(x · y) ≥ min{κ(x), κ(y)} , for each x, y ∈ X, ii) κ(x ٨ y) ≥ min{κ(x), κ(y)} , for every x, y ∈ X. Definition 2.4. 9 A mapping κ: X → [0, 1] is a fuzzy PUP ideal of X , if i) κ(0) ≥ κ(x), ii) κ(x · z) ≥ min{κ(x · (y ٨ z)), κ(y)} , iii) κ(x ٨ z) ≥ min{κ(x ٨ (y · z)), κ(y)} , for any x, y, z ∈ X . Definition 2.5. 8 Let κ and η be fuzzy subset of a set X and Y , respectively, the Cartesian product of κ and η is define by (κ × η)(x, y) = min{κ(x), η(y)} , for (x, y) ∈ X × Y . Definition 2.6. 8 If κ is a fuzzy subset of X , the strongest fuzzy relation on X , that is a fuzzy relation on κ is ξ defined by ξ(x, y) = min{κ(x), κ(y)} , for each x, y ∈ X . Theorem 2.7. 9 Let f: X → Y be a homomorphism of a PUP algebra. If κ be a fuzzy PUP ideal of Y . Then f − 1 (κ) is a fuzzy PUP ideal of X Theorem 2.8. 9 Let (X, · , ٨, 0) and (Y, · , ٨, 0) be a PUP algebra. f: X → Y is surjective and κ is a fuzzy PUP ideals of X . Then f(κ) is a fuzzy pseudo-UP ideals of Y , provided that the sup property holds. Remark 2.9. Let X and Y be pseudo-UP algebras, we define on X × Y by, For every (x, y), (u, v) ∈ X × Y , (x, y) · (u, v) = (x · u, y · v) , (x, y) ٨ (u, v) = (x٨, u, y ٨ v) , then clearly, (X × Y, · , ٨, (0, 0)) is a pseudo-UP algebra. Main Results In this section, we examine homomorphism and Cartesian product of fuzzy PUP ideal and fuzzy PUP subalgebra within PUP algebras. Theorem 3.1. If κ is a fuzzy PUP ideal of a PUP algebra X and define a fuzzy subset κ α (x) = min{α, κ(x)} , for all x ∈ X and α ∈ [0, 1] is a fuzzy PUP ideal of X . Proof. Let κ be a fuzzy PUP ideal of X and α ∈ [0, 1]. Then we need to show that κ α is a fuzzy pseudo-UP ideal of X. Since κ is a fuzzy PUP ideal, we have κ(0) ≥ κ(x) ⇒ κ α (0) = min{α, κ(0)} ≥ min{α, κ(x)} = κ α (x), for all α ∈ [0, 1]. Also, κ is a fuzzy PUP ideal of gives the κ(x · z) ≥ min{κ(x · (y ٨ z)), κ(y)} , for all x, y, z ∈ X. κ α ( x ⋅ z ) = min { α , κ ( x ⋅ z ) } ≥ min { α , min { κ ( x ⋅ ( y ٨ z ) ) , κ ( y ) } } = min { min { α , κ ( x ⋅ ( y ٨ z ) ) , min { α , κ ( y ) } } = min { κ α ( x ⋅ ( y ٨ z ) ) , κ α ( y ) } . And, we have κ(x ٨ z) ≥ min{κ(x ٨ (y · z)), κ(y)} , for all x, y, z ∈ X. κ α ( x ٨ z ) = min { α , κ ( x ٨ z ) } ≥ min { α , min { κ ( x ٨ ( y ⋅ z ) ) , κ ( y ) } } = min { min { α , κ ( x ٨ ( y ⋅ z ) ) , min { α , κ ( y ) } } = min { κ α ( x ٨ ( y ⋅ z ) ) , κ α ( y ) } . This shows that, κ α is a fuzzy PUP ideal of X. Since this is true for all α ∈ [0, 1], κ α is a fuzzy PUP ideal of X. □ Remark 3.2. The converse of above theorem need not be necessarily true. This fact is shown by the following example. Example 3.3. Let X = {0, a, b, c} be a set with binary operations “ · ” and “٨” defined by the following cayley table: Table 1. · 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 a 0 0 Table 2. ٨ 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 a b 0 See 4 (X , ⋅ , ∗ , 0 ) is PUP algebra . Define the fuzzy set κ ( x ) = { 0.7 , if x = 0 0.8 , if x = a 0.5 , if x = b 0.4 , if x = c . Since κ(0) = 0.7 < 0.8 = κ(a) . It follows that κ is not a fuzzy PUP ideal of a PUP algebra X since it does not meet the requirements of the Definition 2.4 (i). If α = 0.4 , then κ α (x) = 0.4 , for all x ∈ X . Accordingly, using routine calculations, we determine κ α is a fuzzy pseudo-UP ideal of a pseudo-UP algebra X . Theorem 3.4. If κ c is a fuzzy PUP ideal of a PUP algebra X and a fuzzy subset κ ν (x) = min{1 − κ(x), 1 − ν} , for all x ∈ X and ν ∈ [0, 1] is a fuzzy PUP ideal of X . Proof. Let κ c be a fuzzy pseudo-UP ideal of X. Define κ ν (x) = min{1 − κ(x), 1 − ν} . Since κ c is a fuzzy PUP ideal of X, we have κ c (0) ≥ κ c (x), for all x ∈ X. Then κ ν (0) = min{1 − κ(0), 1 − ν} ≥ min{1 − κ(x), 1 − ν} = κ ν (x), for all x ∈ X and ν ∈ [0, 1]. Next, suppose that κ c is a fuzzy PUP ideal of X that is κ c (x · z) ≥ min{κ c (x · (y ٨ z)), κ c (y)} and κ c (x ٨ z) ≥ min{κ c (x ٨ (y · z)), κ c (y)} , for all x, y, z ∈ X. Now, κ ν ( x ⋅ z ) = min { 1 − κ ( x ⋅ z ) , 1 − ν } = min { κ c ( x ⋅ z ) , 1 − ν } ≥ min { min { κ c ( x ⋅ ( y ٨ z ) ) , κ c ( y ) } , 1 − ν } = min { min { 1 − κ ( x ⋅ ( y ٨ z ) ) , 1 − ν } , min { 1 − κ ( y ) , 1 − ν } } = min { κ ν ( x ⋅ ( y ٨ z ) ) , κ ν ( y ) } . ⇒ κ ν ( x ⋅ z ) ≥ min { κ ν ( x ⋅ ( y ٨ z ) ) , κ ν ( y ) } . Moreover, κ ν (x ٨ z) ≥ min{κ ν (x ٨ (y · z)), κ ν (y)} , for all x, y, z ∈ X and ν ∈ [0, 1]. κ ν ( x ٨ z ) = min { 1 − κ ( x ٨ z ) , 1 − ν } = min { κ c ( x ٨ z ) , 1 − ν } ≥ min { min { κ c ( x ٨ ( y ⋅ z ) ) , κ c ( y ) } , 1 − ν } = min { min { 1 − κ ( x ٨ ( y ⋅ z ) ) , 1 − ν } , min { 1 − κ ( y ) , 1 − ν } } = min { κ ν ( x ٨ ( y ⋅ z ) ) , κ ν ( y ) } . ⇒ κ ν ( x ٨ z ) ≥ min { κ ν ( x ٨ ( y ⋅ z ) ) , κ ν ( y ) } . This implies that κ ν is a fuzzy PUP ideal of a PUP algebra X, for ν ∈ [0, 1]. □ Remark 3.5. This fact is evidenced by the following example, showing that the converse of the above theorem need not be true. Example 3.6. From Example 3.3 , we define the fuzzy subset κ ( x ) = { 0.7 , if x = 0 0.6 , if x = a 0.5 , if x = b 0.5 , otherwise . Since κ c (0) = 0.2 < 0.3 = κ c (a) . It follows that κ c is not a fuzzy pseudo-UP ideal of X due to its failure to comply with the Definition 2.4 (i). If ν = 0.8 , then κ ν (x) = 0.2 , for all x ∈ X . Thus, through standard calculations, we obtain κ ν is a fuzzy pseudo-UP ideal of a pseudo-UP algebra X . Theorem 3.7. Let κ α and ξ α be a fuzzy PUP ideals of X . Then κ α ∩ ξ α is a fuzzy PUP ideal of X . Proof. Let κ α and ξ α be a fuzzy PUP ideals of the universe X. First we need to show that (κ ∩ ξ) α = κ α ∩ ξ α . Now, ( κ ∩ ξ ) α ( x ) = min { α , ( κ ∩ ξ ) ( x ) } = min { α , min { κ ( x ) , ξ ( x ) } } = min { min { α , κ ( x ) } , min { α , ξ ( x ) } } = min { κ α ( x ) , ξ α ( x ) } = ( κ α ∩ ξ α ) ( x ) . ⇒ ( κ ∩ ξ ) α = κ α ∩ ξ α . Then, ( κ ∩ ξ ) α ( 0 ) = min { α , ( κ ∩ ξ ) ( 0 ) } = min { min { α , κ ( x ) } , min { α , ξ ( x ) } } ≥ min { min { α , κ ( x ) } , min { α , ξ ) ( x ) } } = ( κ α ∪ ξ α ) ( x ) . Next, ( κ ∩ ξ ) α ( x ⋅ z ) = min { α , ( κ ∩ ξ ) ( x ⋅ z ) } = min { κ α ( x ⋅ z ) , ξ α ( x ⋅ z ) } ≥ min { min { κ α ( x ⋅ ( y ٨ z ) ) , κ α ( y ) } , min { ξ α ( x ⋅ ( y ٨ z ) ) , ξ α ( y ) } } = min { min { κ α ( x ⋅ ( y ٨ z ) ) , ξ α ( x ⋅ ( y ٨ z ) ) } , min { κ α ( y ) , ξ α ( y ) } } = min { ( κ α ∩ ξ α ) ( x ⋅ ( y ٨ z ) ) , ( κ α ∩ ξ α ) ( y ) } . Moreover, ( κ ∩ ξ ) α ( x ٨ z ) = min { α , ( κ ∩ ξ ) ( x ٨ z ) } = min { κ α ( x ٨ z ) , ξ α ( x ٨ z ) } ≥ min { min { κ α ( x ٨ ( y ⋅ z ) ) , κ α ( y ) } , min { ξ α ( x ٨ ( y ⋅ z ) ) , ξ α ( y ) } } = min { min { κ α ( x ٨ ( y ⋅ z ) ) , ξ α ( x ٨ ( y ⋅ z ) ) } , min { κ α ( y ) , ξ α ( y ) } } = min { ( κ α ∩ ξ α ) ( x ٨ ( y ⋅ z ) ) , ( κ α ∩ ξ α ) ( y ) } . Consequently, κ α ∩ ξ α is a fuzzy PUP ideal of X. □ Corollary 3.8. Let {κ α /i ∈ Λ} be a family of fuzzy PUP ideals of X . Then ∩ i∈ Λ κ α is also fuzzy iiz PUP ideal of X . Remark 3.9. Let κ α and ξ α be fuzzy PUP ideals of X . Then κ α ∪ ξ α may not qualify as a fuzzy PUP ideal of X . Example 3.10. Let X = {0, 1, 2, 3} be a set with a binary operation “ · ” and “*” defined by the following cayley table: Table 3. · 0 1 2 3 0 0 1 2 3 1 0 0 3 3 2 0 1 0 0 3 0 1 3 0 Table 4. ٨ 0 1 2 3 0 0 1 2 3 1 0 0 3 3 2 0 1 0 0 3 0 2 3 0 See 4 (X , ⋅ , ∗ , 0 ) is PUP algebra . κ ( x ) = { 0.9 , if x = 0 0.6 , if x = 1 0.5 , if x = 2 0.4 , if x = 3 . ξ ( x ) = { 1 , if x = 0 0.4 , if x = 1 0.3 , if x = 2 0.1 , if x = 3 . Clearly, κ and ξ are fuzzy PUP ideals of X . Thus by Theorem 3.1 κ α and ξ α are fuzzy PUP ideals of X , for α ∈ [0, 1] . If α = 0.7 , then κ α and ξ α are given by κ α ( x ) = { 0.7 , if x = 0 0.5 , if x = 1 0.5 , if x = 2 0.4 , if x = 3 . ξ α ( x ) = { 0.6 , if x = 0 0.4 , if x = 1 0.3 , if x = 2 0.1 , if x = 3 . Now, (κ α ∪ ξ α )(1 · 3) = max{κ α (1 · 3), ξ α (1 · 3)} = max{κ α (3), ξ α (3)} = max{0.4, 0.1} = 0.1 . Hence , ( κ α ∪ ξ α ) ( 1 ⋅ 3 ) = 0.1 ............. ( ٨ ) . ( κ α ∪ ξ α ) ( 1 ⋅ 3 ) = max { κ α ( 1 ⋅ 3 ) , ξ α ( 1 ⋅ 3 ) } ≥ max { min { κ α ( 1 ⋅ ( 2 ٨ 3 ) ) , κ α ( 2 ) } , min { ξ α ( 1 ⋅ ( 2 ٨ 3 ) ) , ξ α ( 2 ) } } = max { min { κ α ( 1 ⋅ 0 ) , ξ α ( 1 ⋅ 0 ) } , min { κ α ( 2 ) , ξ α ( 2 ) } } = max { min { κ α ( 0 ) , ξ α ( 0 ) } , min { κ α ( 2 ) , ξ α ( 2 ) } } = max { min { 0.7 , 0.7 } , min { 0.6 , 0.4 } } = max { 0.7 , 0.4 } = 0.7 . From (٨) we get 0.1 ≥ 0.7 which contradicts the Definition 2.4 . Theorem 3.11. Any PUP subalgebra of a PUP algebra X can be realized as a level PUP subalgebra of some fuzzy PUP subalgebra of X . Theorem 3.12. Let κ be a fuzzy PUP subalgebra of a PUP algebra X . Two upper level PUP subalgebra U(κ, t) and U(κ, s) (with t < s ) of a fuzzy PUP subalgebra are equal if and only if there is no x ∈ X such that t ≤ κ(x) < s . Proof. Let κ be a fuzzy PUP subalgebra of X. Suppose that U(κ, t) = U(κ, s). Then we claim that there is no x ∈ X such that t ≤ κ(x) < s. Assume that there exist x ∈ X such that t ≤ κ(x) < s. ⇒ x ∈ U ( κ , t ) , but x / ∈ U ( κ , s ) ⇒ U ( κ , s ) is a proper subset of U ( κ , s ) . This contradicts to the assumption that U(κ, t) = U(κ, s). Conversely, suppose that there is no x ∈ X such that t ≤ κ(x) < s. Then we have to prove that (1) U ( κ , t ) = U ( κ , s ) . Since t s (since κ(x) does not lie between t and s) ⇒ x ∈ U(κ, s) (2) Hence , U ( κ , t ) ⊆ U ( κ , s ) From (1) and (2) we get U(κ, t) = U(κ, s). □ Corollary 3.13. Let X be a finite PUP algebra and κ be a fuzzy PUP subalgebra of X . If Im (κ) = {t 1 , …, t n } , then the family of PUP subalgebra { U (κ, t i ) | 1 ≤ i ≤ n} , constitutes all the upper level PUP subalgebras of κ in X . Lemma 3.14. Let X be a PUP algebra and κ be a fuzzy PUP subalgebra of X . If Im(κ) is finite, say {t 1 , t 2 …, t n }, then for any t i , t j ∈ Im(κ), U (κ, t i ) = U (κ, t j ) ⇒ t i = t j . Theorem 3.15. Let κ and ξ be two fuzzy PUP subalgebra of a PUP algebra X with identical family of level PUP Subalgebras. If Im(κ) = {t 1 , t 2 …, t r } and Im(ξ) = {q 1 , q 2 …, q k } where t 1 ≥ t 2 ≥ ··· ≥ t r and q 1 ≥ q 2 ≥ ··· ≥ q k . Then 1) k = r 2) U (κ, t i ) = U (ξ, q i ), i = 1, 2, …, r . 3) If x ∈ X such that κ(x) = t i then ξ(x) = q i , i = 1, 2, …, r . Corollary 3.16. Let κ and ξ be two fuzzy PUP subalgebras of a PUP algebra X with identical family of level PUP subalgebras. Then Im(κ) = Im(ξ) ⇒ κ = ξ . Proof. Let Im(κ) = Im(ξ) = {q 1 , q 2 , …, q r } where q 1 ≥ q 2 ≥ · · · ≥ q r . By Theorem 3.15 , for any x ∈ X there exists q j such that κ(x) = q i = ξ(x). Thus κ(x) = ξ(x) for all x ∈ X. This implies κ = ξ. □ Theorem 3.17. Every PUP ideal of X can be realized as a level PUP ideal of some fuzzy PUP ideal of X . Proof. Let J be a PUP ideal of a PUP algebra X and κ be a fuzzy subset in X defined by κ ( x ) = { t , if x ∈ J 0 , otherwise . Clearly, U(κ, t) = J. Since 0 ∈ J, we have κ(0) = t. Thus κ(0) ≥ κ(x), for all x ∈ X. Now, consider the following cases: Case i) If y ∈ J, x · (y ٨ z) ∈ J, then x · z ∈ J and x ٨ (y · z) ∈ J then x ٨ z ∈ J. Since J is a PUP ideal of X such that κ(x · z) = κ(y) = κ(x · (y ٨ z)) = t and κ(x ٨ z) = κ(y) = κ(x ٨ (y · z)) = t. ⇒ κ(x · z) = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) = min{κ(x ٨ (y · z)), κ(y)} . Case ii) If x · (y ٨ z), x ٨ (y · z) ∈ J, y ∈ J, then κ(x · (y ٨ z)) = t, κ(y) = 0 and κ(x ٨ (y · z)) = t. Then we have κ(x · z) ≥ 0 = min{0, t} = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) ≥ 0 = min{0, t} = min{κ(x ٨ (y · z)), κ(y)} . Case iii) If x · (y ٨ z), x ٨ (y · z) ∈ J, y ∈ J, then κ(x · (y ٨ z)) = 0, κ(y) = t and κ(x ٨ (y · z)) = 0. Then we get similar resualt as in case(ii). Case iv) If y ∈ J, x · (y ٨ z), x ٨ (y · z) ∈ J, then κ(y) = 0 = κ(x · (y ٨ z)) = 0 and κ(x ٨ (y · z)) = 0. ⇒ κ(x · z) ≥ 0 = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) ≥ 0 = min{κ(x ٨ (y · z)), κ(y)} , for all x, y, z ∈ X. Thus κ is a fuzzy PUP ideal of a PUP algebra X. Hence, J is a level PUP ideal of X corresponding to a fuzzy PUP ideal of X. □ Theorem 3.18. Let {J i : i = 0, 1, 2, …, n} be any family of a PUP ideal of a PUP algebra X such that J 0 ⊂ J 1 ⊂ J 2 ⊂ … ⊂ J n = X , then there exist a fuzzy PUP ideal of X . Theorem 3.19. Let κ be a fuzzy PUP ideal of X , then the upper level PUP ideals U(κ, s) and U(κ, t) , with (s < t) of a fuzzy PUP ideals κ of X are equal if and only if there is no x ∈ X such that s ≤ κ(x) < t . Proof. Consider κ is a fuzzy PUP ideal of X and U(κ, s) = U(κ, t), for some s < t. We need to show that there is no x ∈ X such that s ≤ κ(x) < t. Assume that there exist x ∈ x ∈ X such that s ≤ κ(x) < t. This implies that x ∈ U(κ, s) and x ∈ U(κ, t), which contradicts the assumption U(κ, s) = U(κ, t). Hence, there is no x ∈ X such that s ≤ κ(x) < t. Conversely, assume that there is no x ∈ X such that s ≤ κ(x) < t. We need to prove that U(κ, s) = U(κ, t). Since s < t, we get U(κ, s) ⊆ U(κ, t) (٨). Let x ∈ U(κ, s) then κ(x) ≥ s. By our assumption there is no x ∈ X such that κ(x) < t. Then κ(x) ≥ t ⇒ x ∈ U(κ, t) because κ(x) does not lie between s and t. Hence U(κ, s) ⊆ U(κ, t)................................ (٨ ٨). From (٨) and (٨٨), it results in U(κ, s) = U(κ, t). □ Corollary 3.20. Let κ be a fuzzy PUP ideal of X with finite image. If U(κ, s) = U(κ, t) , for some s, t ∈ Im(κ) , then s = t . Theorem 3.21. Let κ be a fuzzy PUP ideal of a PUP algebra X and let x ∈ X . Then κ(x) = s if and only if x ∈ U(κ, s) but x /∈ U(κ, t) , for all t > s . Definition 3.22. Let (X, · X , ٨ X , 0 X ) and (Y, · Y , ٨ Y , 0 Y ) be PUP algebras. Let f: X → Y be homomorphism on PUP algebra. Then any fuzzy set κ in Y , we define a new fuzzy set κ f in X by κ f (x) = κ(f(x)). Theorem 3.23. Let f: X → Y be a homomorphism of PUP algebra. 1) If a fuzzy subset κ is a fuzzy PUP subalgebra of Y , then a fuzzy set κ f in X is a fuzzy PUP subalgebra of X . 2) If a fuzzy subset κ is a fuzzy PUP ideal of Y , then a fuzzy set κ f in X is a fuzzy PUP ideal of X. Proof. 1) Let f be a homomorphism of a PUP algebra and let κ be a fuzzy PUP subalgebra of Y. Let x, y ∈ X. Then κ f ( x ⋅ y ) = κ ( f ( x ⋅ y ) ) = κ ( f ( x ) ⋅ f ( y ) ) ≥ min { κ ( f ( x ) ) , κ ( f ( y ) ) } = min { κ f ( x ) , κ f ( y ) } . κ f ( x ٨ y ) = κ ( f ( x ٨ y ) ) = κ ( f ( x ) ٨ f ( y ) ) ≥ min { κ ( f ( x ) ) , κ ( f ( y ) ) } = min { κ f ( x ) , κ f ( y ) } . Hence, κ f is a fuzzy PUP subalgebra of X. 2) Assume f is a homomorphism of a PUP algebras and κ is a fuzzy PUP ideal of X, then κ f (0) = κ(f(0)) = κ(0) ≥ κ(x) = κ(f(x)) = κ f (x), for all x ∈ X. Additionally, let x, y, z ∈ X, then κ f ( x ⋅ z ) = κ ( f ( x ⋅ z ) ) = κ ( f ( x ) ⋅ f ( z ) ) ≥ min { κ ( f ( x ) ⋅ ( f ( y ) ٨ f ( z ) ) ) , κ ( f ( y ) ) } = min { κ ( f ( x ⋅ ( y ٨ z ) ) ) , κ ( f ( y ) ) } = min { κ f ( x ⋅ ( y ٨ z ) ) , κ ( f ( x ) ) } = min { κ f ( x ⋅ ( y ٨ z ) ) , κ f ( y ) } . Moreover, κ f ( x ٨ z ) = κ ( f ( x ٨ z ) ) = κ ( f ( x ) ٨ f ( z ) ) ≥ min { κ ( f ( x ) ٨ ( f ( y ) ⋅ f ( z ) ) ) , κ ( f ( y ) ) } = min { κ ( f ( x ٨ ( y ⋅ z ) ) ) , κ ( f ( y ) ) } = min { κ f ( x ٨ ( y ⋅ z ) ) , κ ( f ( x ) ) } = min { κ f ( x ٨ ( y ⋅ z ) ) , κ f ( y ) } . Therefore, κ f is a fuzzy PUP ideal of X. □ Theorem 3.24. Let f: X → Y be an epimorphism of PUP algebra. Then κ f is a fuzzy PUP subalgebra of X if and only if κ is a fuzzy PUP subalgebra of Y . Definition 3.25. Let f be a mapping defined on a PUP Algebra X . If κ is a fuzzy set in X , then the fuzzy set ξ in f(X) defined by ξ ( y ) = sup x ∈ f − 1 ( y ) κ ( x ) for all y ∈ f(X) is called image of κ under f . If ξ is a fuzzy set in f(X) , then the fuzzy set κ = ξ ◦f in X i.e. the fuzzy set defined by κ(x) = ξ(f(x)) for all x ∈ X is called the pre-image of ξ under f . Theorem 3.26. Let f: X → Y be an epimorphism of PUP algebras. If ξ is a fuzzy PUP subalgebra of Y and let κ is the pre-image of ξ under f , then κ is a fuzzy PUP subalgebra of X . Proof. Suppose ξ is a fuzzy PUP subalgebra of Y and κ is pre-image of ξ under f. For x, y ∈ X, we need to show that κ is a fuzzy PUP subalgebra of X. By Definition 3.25 κ = ξ ◦ f defined as κ(x) = ξ(f(x)). κ ( x ⋅ y ) = ξ ( f ( x ⋅ y ) ) = ξ ( f ( x ) ⋅ f ( y ) ) ≥ min { ξ ( f ( x ) ) , ξ ( f ( y ) ) } = min { κ ( x ) , κ ( y ) } . κ ( x ٨ y ) = ξ ( f ( x ٨ y ) ) = ξ ( f ( x ) ٨ f ( y ) ) ≥ min { ξ ( f ( x ) ) , ξ ( f ( y ) ) } = min { κ ( x ) , κ ( y ) } . Hence, κ is a fuzzy PUP subalgebra of X. □ Theorem 3.27. Let f: X → Y be an onto homomorphism of PUP algebra. If κ is a fuzzy PUP subalgebra of Y , then f − 1 (κ c ) = (f − 1 (κ)) c . Proof. Let κ be a fuzzy PUP subalgebra of Y. Then, for x ∈ X. f − 1 ( κ c ) ( x ) = κ c ( f ( x ) ) = 1 − κ ( f ( x ) ) = 1 − f − 1 ( κ ) ( x ) = ( f − 1 ( κ ) ) c ( x ) . Thus f − 1 ( κ c ) = ( f − 1 ( κ ) ) c . □ Definition 3.28. An ideal J of a PUP algebra X is said to be characteristic if f(J) = J , for all f ∈ Aut (X) , where Aut (X) is the set of all automorphisms of X . Similarly, a fuzzy PUP ideal κ of a PUP algebra X is said to be fuzzy characteristic if κ f (x) = κ(x) , for all x ∈ X and f ∈ Aut (X) . Theorem 3.29. A subset J of X is a PUP ideal if and only if the characteristic function of J is a fuzzy PUP ideal of X . Proof. Suppose that κ is a fuzzy characteristic, and let s ∈ Im(κ), f ∈ Aut(X) and x ∈ U(κ, s). Then κ f ( x ) = κ ( x ) implies κ ( f ( x ) ) ≥ s , whence f ( x ) ∈ U ( κ , s ) . Thus f ( U ( κ , s ) ) ⊆ U ( κ , s ) ( ٨ ) . Let x ∈ U(κ, s) and y ∈ X such that f(y) = x. Then κ(y) = κ f (y) = mu(x) ≥ s. Consequnetly, y ∈ U(κ, s). So, x = f(y) ∈ U(κ, s). Thus, U ( κ , s ) ⊆ f ( U ( κ , s ) ) ( ٨ ٨ ) . From (٨) and (٨٨) we get f(U(κ, s)) = U(κ, s). Therefore, U(κ, s) is characteristic. Conversely, suppose thta each level PUP ideal of κ is characteristic, and let x ∈ X, f ∈ Aut(X), κ(x) = s. Then by virtue of Theorem 3.21 , x ∈ U(κ, s) and x ∈ U(κ, t), for all t > s. It follows from the assumption that f(x) ∈ f(U(κ, s)) = U(κ, s), so that κ f (x) = κ(f(x)) ≥ s. Let t = κ f (x) and assume that t > s. Then, f(x) ∈ U(κ, t) = f(U(κ, t)), which implies from the injectivety of f that x ∈ U(κ, t), a contradiction. Hence, κ f (x) = κ(f(x)) = s = κ(x) show that κ is a fuzzy characteristic. □ Theorem 3.30. Let f: X → Y be an epimorphism of a PUP algebra. κ is a fuzzy PUP ideal of X with ker f ⊆ J , then f − 1 (f(κ)) = κ. □ Proof. Let y ∈ Y, then there exist x ∈ X such that f(x) = y. Then f − 1 (f(κ))(x) = f(κ)(f(x)) = κ ( y ) = sup x ∈ f − 1 ( y ) { κ ( x ) } . For any z ∈ X, z ∈ f − 1 (y) ⇒ f(z) = y ⇒ f(z) = f(x) implies f(z)·f(x) = 0 and f(z) ٨ f(x) = 0. Therefore, f(z · x) = 0 and f(z ٨ x) = 0, since f is homomorphism. ⇒ z ⋅ x , z ٨ x ∈ ker f ⇒ z ⋅ x , z ٨ x ∈ J , since ker f ⊆ J . ⇒ κ ( z ⋅ x ) = κ ( 0 ) and κ ( z ٨ x ) = κ ( 0 ) . ⇒ κ ( z ) ≥ min { κ ( z ⋅ x ) , κ ( x ) } = min { κ ( 0 ) , κ ( x ) } = mu ( x ) . ⇒ κ ( z ) ≥ min { κ ( z ٨ x ) , κ ( x ) } = min { κ ( 0 ) , κ ( x ) } = mu ( x ) . ⇒ κ ( z ) ≥ κ ( x ) . Again we can prove that κ(x) ≥ κ(z) ⇒ κ(z) = κ(x). Thus, f − 1 ( f ( κ ) ) ( x ) = sup z ∈ f − 1 ( y ) {κ(z)} = κ(x), for some x ∈ X. Hence, f − 1 (f(κ)) = κ. □ Theorem 3.31. Let f: X → Y be an epimorphism of PUP algebras and α ∈ [0, 1] . If κ α is a fuzzy PUP ideal of X , then f(κ α ) is a fuzzy PUP ideal of Y . Theorem 3.32. Let f: X → Y be a homomorphism of fuzzy PUP algebra X into Y and α ∈ [0, 1]. If κ α is a fuzzy PUP ideal of Y , then the pre-image f − 1 (κ α ) is a fuzzy PUP ideal of X . Theorem 3.33. Let f: X → Y be an onto homomorphism of PUP algebras and κ α is a fuzzy subset in Y . If f − 1 (κ α ) is a fuzzy PUP ideal of X , then κ α is a fuzzy PUP ideal of Y , for α ∈ [0, 1]. Theorem 3.34. Let f: X → Y be an epimorphism of PUP algebras, for all ν ∈ [0, 1] , and we define κ ν (x) = min{1 − κ(x), 1 − ν} is a fuzzy PUP ideal of X , then f(κ ν ) is a fuzzy PUP ideal of Y . Proof. Since f: X → Y is an epimorphism of PUP algebras for any y ∈ Y, there exist x ∈ X such that f(x) = y. First we need to show that (f(κ)) ν = f(κ ν ). Now f ( κ ν ) ( y ) = sup { κ ν ( x ) : f ( x ) = y } = sup { min { 1 − κ ( x ) , 1 − ν } : f ( x ) = y } = min { sup { 1 − κ ( x ) , sup 1 − ν } } = min { 1 − f ( κ ) ( y ) , 1 − ν } = ( f ( κ ) ) ν , for all y ∈ Y . Hence, (f(κ)) ν = f(κ ν ). Then, (f(κ)) ν (0) = f(κ ν )(x) = f(κ ν )(f(0)) = f − 1 (f(κ ν ))(0) ≥ κ ν (0) ≥ κ ν (x), by Theorem 3.31 (٨). Implies κ ν (x) = f − 1 (f(κ ν ))(x) = f(κ ν )(f(x)) = f(κ ν )(y). Thus (f(κ)) ν (0) ≥ (f(κ)) ν (y), y ∈Y. Also, let x, y, z ∈ Y, since f: X → Y is an epimorphisms of PUP algebras, there exist a, b, c ∈ X, such that f(a) = x, f(b) = y, f(c) = z. Then ( f ( κ ) ) ν ( x ⋅ z ) = f ( κ ν ) ( x ⋅ z ) = f ( κ ν ) ( f ( a ) ⋅ f ( c ) ) = f ( κ ν ) ( f ( a ⋅ c ) ) ≥ κ ν ( a ⋅ c ) , by Theorem 3.31 ( ٨ ) ≥ min { κ ν ( a ⋅ ( b ٨ c ) ) , κ ν ( b ) } = min { f − 1 ( f ( κ ν ) ) , f − 1 ( f ( κ ν ) ) ( b ) } = min { f ( κ ν ) ( f ( a ⋅ ( b ٨ c ) ) ) , f ( κ ν ) ( f ( b ) ) } = min { f ( κ ν ) ( f ( a ) ⋅ ( f ( b ) ٨ f ( c ) ) ) , f ( κ ν ) f ( b ) } = min { f ( κ ν ) ( x ⋅ ( y ٨ z ) ) , f ( κ ν ) ( y ) } = min { ( f ( κ ) ) ν ( x ⋅ ( y ٨ z ) ) , ( f ( κ ) ) ν ( y ) } . And also, ( f ( κ ) ) ν ( x ٨ z ) = f ( κ ν ) ( x ٨ z ) = f ( κ ν ) ( f ( a ) ٨ f ( c ) ) = f ( κ ν ) ( f ( a ٨ c ) ) ≥ κ ν ( a ٨ c ) , by Theorem 3.31 ( ٨ ) ≥ min { κ ν ( a ٨ ( b ⋅ c ) ) , κ ν ( b ) } = min { f − 1 ( f ( κ ν ) ) , f − 1 ( f ( κ ν ) ) ( b ) } = min { f ( κ ν ) ( f ( a ٨ ( b ⋅ c ) ) ) , f ( κ ν ) ( f ( b ) ) } = min { f ( κ ν ) ( f ( a ) ٨ ( f ( b ) ⋅ f ( c ) ) ) , f ( κ ν ) f ( b ) } = min { f ( κ ν ) ( x ٨ ( y ⋅ z ) ) , f ( κ ν ) ( y ) } = min { ( f ( κ ) ) ν ( x ٨ ( y ⋅ z ) ) , ( f ( κ ) ) ν ( y ) } . Hence, f(κ ν ) is a fuzzy PUP ideal of Y. □ Theorem 3.35. Let f: X → Y be a homomorphism of fuzzy PUP algebra X into Y and ν ∈ [0, 1]. If κ ν is a fuzzy PUP ideal of Y , then the pre-image f − 1 (κ ν ) is a fuzzy PUP ideal of X . Proposition 3.36. Let κ and ξ be fuzzy sets on a PUP algebra X . Then 1. κ × ξ is a fuzzy relation on X . 2. U(κ × ξ, t) = U(κ, t) × U(ξ, t). 3. U + (κ × ξ, t) = U + (κ, t) × U + (ξ, t). 4. L(κ × ξ, s) = L(κ, s) × L(ξ, s) . 5. L + (κ × ξ, s) = L + (κ, s) × L(ξ, s). Lemma 3.37. Let κ and ξ be any two fuzzy PUP subalgebras of X and Y , respectively. Then (κ × ξ)(0, 0) ≥ (κ × ξ)(x, y) , for all (x, y) ∈ X × Y. Proof. Let (x, y) ∈ X × Y. Then by definition, it follows that (κ × ξ)(0, 0) = min{κ(0), ξ(0)} ≥ min{κ(x), ξ(y)} = (κ × ξ)(x, y). □ Theorem 3.38. Let κ 1 and κ 1 be any two fuzzy PUP subalgebras of X 1 and X 2 , respectively. Then the cartesian product κ 1 × κ 2 of κ 1 and κ 2 defined as (κ 1 × κ 2 )(x 1 , x 2 ) = min{κ 1 (x 1 ), κ 2 (x 2 )} is also a fuzzy PUP subalgebra of X 1 × X 2 . Proof. Let X = X 1 × X 2 and κ = κ 1 × κ 2 . For x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ X, we have κ ( x ⋅ y ) = κ ( ( x 1 , x 2 ) ⋅ ( y 1 , y 2 ) ) = κ ( x 1 ⋅ y 1 , x 2 ⋅ y 2 ) = ( κ 1 × κ 2 ) ( x 1 ⋅ y 1 , x 2 ⋅ y 2 ) = min { κ 1 ( x 1 ⋅ y 1 ) , κ 2 ( x 2 ⋅ y 2 ) } ≥ min { min { κ 1 ( x 1 ) , κ 1 ( y 1 ) } , min { κ 2 ( x 2 ) , κ 2 ( y 2 ) } } = min { min { κ 1 ( x 1 ) , κ 2 ( x 2 ) } , min { κ 1 ( y 1 ) , κ 2 ( y 2 ) } } = min { ( κ 1 × κ 2 ) ( x 1 , x 2 ) , ( κ 1 × κ 2 ) ( y 1 , y 2 ) } = min { κ ( x , y ) } . Also, κ ( x ٨ y ) = κ ( ( x 1 , x 2 ) ٨ ( y 1 , y 2 ) ) = κ ( x 1 ٨ y 1 , x 2 ٨ y 2 ) = ( κ 1 × κ 2 ) ( x 1 ٨ y 1 , x 2 ٨ y 2 ) = min { κ 1 ( x 1 ٨ y 1 ) , κ 2 ( x 2 ٨ y 2 ) } ≥ min { min { κ 1 ( x 1 ) , κ 1 ( y 1 ) } , min { κ 2 ( x 2 ) , κ 2 ( y 2 ) } } = min { min { κ 1 ( x 1 ) , κ 2 ( x 2 ) } , min { κ 1 ( y 1 ) , κ 2 ( y 2 ) } } = min { ( κ 1 × κ 2 ) ( x 1 , x 2 ) , ( κ 1 × κ 2 ) ( y 1 , y 2 ) } = min { κ ( x , y ) } . This shows that, κ 1 × κ 2 is a fuzzy PUP subalgebra of X 1 × X 2 . □ Theorem 3.39. Let κ and ξ be fuzzy subsets of the PUP algebras X and Y , respectively. Suppose 0 X and 0 Y are the constant elements of X and Y , respectively. If κ × ξ is a fuzzy PUP subalgebra of X × Y then at least one of the following two statements holds. i. κ(x) ≤ ξ(0 Y ) , for all x ∈ X ii. ξ(y) ≤ κ(0 X ) , for all y ∈ Y. Theorem 3.40. Let κ be a fuzzy subset of a PUP algebra X . Let ξ be a strong fuzzy relation on X . If ξ is a fuzzy PUP subalgebra of X × X , then κ(0) ≥ κ(x) , for all x ∈ X. Proof. Since ξ is a fuzzy PUP subalgebra of X × X and it follows that ξ(x, y) = min{κ(x), κ(y)} . By Lemma 3.37 ξ(0, 0) ≥ ξ(x, x). Then we have κ(0) = min{κ(0), κ(0)} = ξ(0, 0) ≥ ξ(x, x) = min{κ(x), κ(x)} = κ(x). Hence, κ(0) ≥ κ(x), for all x ∈ X. □ Theorem 3.41. Let κ be a fuzzy subset of a PUP algebra X and let ξ be the strongest fuzzy relation on a PUP algebra X . Then κ is a fuzzy PUP subalgebra of X if and only if ξ is a fuzzy PUP subalgebra of X × X. Theorem 3.42. If κ and ξ are fuzzy PUP ideals of X and Y , respectively, then κ × ξ is a fuzzy PUP ideal of X × Y . Theorem 3.43. Let κ and ξ be two fuzzy subsets of X and Y , resepecively such that κ × ξ is a fuzzy PUP ideal of κ × ξ of X × Y . Then either κ is a fuzzy PUP ideal of X or ξ is a fuzzy PUP ideal of Y . Theorem 3.44. Let κ and ξ be fuzzy subsets of X and Y such that κ × ξ is a fuzzy PUP ideal of X × Y . Then i) either κ(0) ≥ κ(x) or ξ(0) ≥ ξ(y) , for all (x, y) ∈ X × Y. ii) If κ(0) ≥ κ(x) , for all x ∈ X , then either ξ(0) ≥ ξ(y) or ξ(0) ≥ κ(x) , for all (x, y) ∈ X × Y. iii) If ξ(0) ≥ ξ(y) , for all y ∈ Y , then either κ(0) ≥ κ(x) or κ(0) ≥ ξ(y) , for all (x, y) ∈ X × Y. Proof. Let κ × ξ be a fuzzy PUP ideal of X × Y. Then i) Suppose that κ(0) < κ(x), for some x ∈ X and ξ(0) min{κ(0), ξ(0)} = (κ × ξ)(0, 0). This is a contradiction to the fact that κ × ξ is a fuzzy PUP ideal of X × Y. Therefore, either κ(0) ≥ κ(x) or ξ(0) ≥ ξ(y), for all (x, y) ∈ X × Y. ii) Suppose that κ(0) ≥ κ(x), for all x ∈ X. Assume that there exist x ∈ X and y ∈ Y such that ξ(0) < κ(x) and ξ(0) < ξ(y). Then ξ(0) min{ξ(0), ξ(0)} = ξ(0) = min{κ(0), ξ(0)} = (κ × ξ)(0, 0). This is a contradiction since κ × ξ is a fuzzy PUP ideal of X × Y. Hence, either ξ(0) ≥ ξ(y), for all y ∈ Y or ξ(0) ≥ κ(x), for all x ∈ X. iii) Assume that ξ(0) ≥ ξ(y), for all y ∈ Y. Assume that there exist x ∈ X and y ∈ Y such that κ(0) < κ(x) and κ(0) < ξ(y). Then κ(0) < ξ(y) ≤ ξ(y). Thus (κ × ξ)(x, y) = min{κ(x), ξ(y)} ≥ minκ(0), κ(0) = κ(0) = min{κ(0), ξ(0)} = (κ × ξ)(0, 0) which is a contradiction. Hence, either κ(0) ≥ κ(x), for all x ∈ X or κ(0) ≥ ξ(y), for all y ∈ Y. □ Theorem 3.45. Let κ and Q be fuzzy PUP ideals of X and Y , respectively. Then κ × Q is a fuzzy PUP ideal of X × Y if and only if each non-empty upper t -level set U(κ × Q, t) is a PUP ideal of X × Y , for all t ∈ [0, 1] . Theorem 3.46. Let κ be a fuzzy set of X and let ξ be the strongest fuzzy relation on X . If ξ , we define ξ(x, y) = min{κ(x), κ(y)} is a fuzzy PUP ideal X × X , then κ(0) ≥ κ(x) , for all x, y ∈ X . Proof. Suppose ξ is a fuzzy PUP ideal of X × X, it follows that ξ(0, 0) ≥ ξ(x, x), for all x ∈ X. Then we have κ(0) = min{κ(0), κ(0)} = ξ(0, 0) ≥ ξ(x, x) = min{κ(x), κ(x)} = κ(x). Hence, κ(0) ≥ κ(x), for all x ∈ X. □ Theorem 3.47. Let κ be a fuzzy subset of X and let ξ be the strongest fuzzy relation on X , then κ is a fuzzy PUP ideal of X if and only if ξ is a fuzzy PUP ideal of X × X . Conclusion In this paper, we introduced the concept of the Cartesian product of fuzzy PUP ideals and sub-algebras of a PUP algebra X. We also investigated the effects on the image and inverse image of κ α and κ ν of fuzzy PUP ideal under PUP homomorphism. The fuzzy set κ α and κ ν generalize the notion of classical fuzzy sets, aiming to evaluate the level of ambiguity in a fuzzy situation. Finally, we discussed the concept of the strongest fuzzy PUP relation on a fuzzy PUP subalgebra and ideal of a PUP algebra and examined some of its properties. Author contributions All the authors are contributed equally in this manuscript and also both authors read and approved the final manuscript. Ethics and consent Ethical approval and consent were not required. Data availability No data are associated with this article. Acknowledgements The authors thank the editors and the anonymous reviewers for their valuable suggestions in improving this paper. References 1. Zadeh LA: Fuzzy sets. Inf. Control. 1965; 8 (3): 338–353. Publisher Full Text 2. Rosenfeld A: Fuzzy groups. J. Math. Anal. Appl. 1971; 35 (3): 512–517. Publisher Full Text 3. Akram M: Fuzzy lie algebras. Singapore: Springer; 2018. 4. Romano DA: PUP algebras, An introduction. Bull. Int. Math. Virtual Inst. 2020; 10 (2): 349–355. 5. Romano DA: PUP ideals and PUP filters in PUP algebras. Math. Sci. Appl. E-Notes. 2020; 8 (1): 155–158. Publisher Full Text 6. Romano DA: Homomorphism of PUP algebras. Bull. Inter. Math. Virtual Inst. 2021; 11 (1): 47–53. 7. Choudhury FP, Chakraborty AB, Khare SS: A note on fuzzy subgroups and fuzzy homomorphism. J. Math. Anal. Appl. 1988; 131 (2): 537–553. 8. Mostafa SM, Abd Elnaby MA, Yousef MM: Anti-fuzzy KU-ideals of KU-algebras. Int. J. Algebra Stat. 2012; 1 (1): 92–99. Publisher Full Text 9. Mechdesro AA, Alaba BA, Munie TM, et al. : Fuzzy PUP ideal of PUP algebra. Res. Math. 2023; 10 (1): 2234217. Publisher Full Text Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 20 Jan 2025 ADD YOUR COMMENT Comment Author details Author details 1 Department of Maths,, Kabridahar University, Kebri Dehar, Somali, Ethiopia 2 Mathematics, Bahir Dar University, Bahir Dar, Amhara, 6000, Ethiopia Alachew Amaneh Mechderso Roles: Conceptualization, Writing – Original Draft Preparation Berhanu Assaye Alaba Roles: Conceptualization, Supervision, Writing – Review & Editing Tilahun Mekonnen Munie Roles: Conceptualization, Supervision, Writing – Review & Editing 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: 20 Jan 2025, 14:115 https://doi.org/10.12688/f1000research.160312.1 Copyright © 2025 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 and Munie TM. On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra [version 1; peer review: 1 approved with reservations] . F1000Research 2025, 14 :115 ( https://doi.org/10.12688/f1000research.160312.1 ) 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 20 Jan 2025 Views 0 Cite How to cite this report: K K. Reviewer Report For: On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra [version 1; peer review: 1 approved with reservations] . F1000Research 2025, 14 :115 ( https://doi.org/10.5256/f1000research.176190.r371592 ) The direct URL for this report is: https://f1000research.com/articles/14-115/v1#referee-response-371592 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 16 Apr 2025 Kalaiarasi K , Cauvery College for Women, Tiruchirappalli, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.176190.r371592 The work of the concept of fuzzy PUP ideals in the context of PUP algebras will be extended The fuzzy homomorphism influence their structure should be expressed. The fuzzy relations in algebraic settings research into the properties and ... Continue reading READ ALL The work of the concept of fuzzy PUP ideals in the context of PUP algebras will be extended The fuzzy homomorphism influence their structure should be expressed. The fuzzy relations in algebraic settings research into the properties and applications of fuzzy ideals in PUP algebras will be discussed deeply. The Cartesian product of fuzzy PUP ideals forms a fuzzy pseudo-UP ideal will be added with more examples . The characterized through its level sets, providing a deeper understanding of the interaction between fuzzy PUP ideals and their algebraic structure will be included more . The strongest fuzzy PUP relation on a fuzzy PUP subalgebra , and several important properties should be derived. Is the work clearly and accurately presented and does it cite the current literature? Partly 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? Yes Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Partly References 1. Cauvery College for Women Autonomous, Tiruchirappalli, Tamil Nadu, India, Kalaiarasi K, Swathi S, Cauvery College for Women Autonomous, Tiruchirappalli, Tamil Nadu, India: Optimization of fuzzy inventory management in industrial processes using deep learning algorithms: a hybrid approach for enhancing demand forecasting and supply chain efficiency. Advanced Engineering Letters . 2024; 3 (4): 141-153 Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: fuzzy set theory, optimization of fuzzy inventory models, fuzzy algebra, fuzzy graph theory, fuzzy nuetrosophic sets , machine learning 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 K K. Reviewer Report For: On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra [version 1; peer review: 1 approved with reservations] . F1000Research 2025, 14 :115 ( https://doi.org/10.5256/f1000research.176190.r371592 ) The direct URL for this report is: https://f1000research.com/articles/14-115/v1#referee-response-371592 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 20 Jan 2025 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 Version 1 20 Jan 25 read Kalaiarasi K , Cauvery College for Women, Tiruchirappalli, 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 © 2025 K K. 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. 16 Apr 2025 | for Version 1 Kalaiarasi K , Cauvery College for Women, Tiruchirappalli, India 0 Views copyright © 2025 K K. 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 The work of the concept of fuzzy PUP ideals in the context of PUP algebras will be extended The fuzzy homomorphism influence their structure should be expressed. The fuzzy relations in algebraic settings research into the properties and applications of fuzzy ideals in PUP algebras will be discussed deeply. The Cartesian product of fuzzy PUP ideals forms a fuzzy pseudo-UP ideal will be added with more examples . The characterized through its level sets, providing a deeper understanding of the interaction between fuzzy PUP ideals and their algebraic structure will be included more . The strongest fuzzy PUP relation on a fuzzy PUP subalgebra , and several important properties should be derived. Is the work clearly and accurately presented and does it cite the current literature? Partly 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? Yes Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Partly References 1. Cauvery College for Women Autonomous, Tiruchirappalli, Tamil Nadu, India, Kalaiarasi K, Swathi S, Cauvery College for Women Autonomous, Tiruchirappalli, Tamil Nadu, India: Optimization of fuzzy inventory management in industrial processes using deep learning algorithms: a hybrid approach for enhancing demand forecasting and supply chain efficiency. Advanced Engineering Letters . 2024; 3 (4): 141-153 Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise fuzzy set theory, optimization of fuzzy inventory models, fuzzy algebra, fuzzy graph theory, fuzzy nuetrosophic sets , machine learning 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) K K. Peer Review Report For: On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra [version 1; peer review: 1 approved with reservations] . F1000Research 2025, 14 :115 ( https://doi.org/10.5256/f1000research.176190.r371592) 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/14-115/v1#referee-response-371592 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 = "On homomorphism and Cartesian products of...".replace("'", ''); var linkedInUrl = "http://www.linkedin.com/shareArticle?url=https://f1000research.com/articles/14-115/v1" + "&title=" + encodeURIComponent(lTitle) + "&summary=" + encodeURIComponent('Read the article by '); var deliciousUrl = "https://del.icio.us/post?url=https://f1000research.com/articles/14-115/v1&title=" + encodeURIComponent(lTitle); var redditUrl = "http://reddit.com/submit?url=https://f1000research.com/articles/14-115/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/14-115/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/14-115", templates : { twitter : "On homomorphism and Cartesian products of fuzzy PUP ideal on.... Mechderso AA et al., published by " + "@F1000Research" + ", https://f1000research.com/articles/14-115/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/160312/176190") new F1000.Clipboard(); new F1000.ThesaurusTermsDisplay("articles", "article", "176190"); $(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 = { "371589": 0, "371588": 0, "371591": 0, "371590": 0, "371587": 0, "371596": 0, "371593": 0, "371592": 6, "371595": 0, "371594": 0, "400790": 0, "400791": 0, "400789": 0, "400798": 0, "400796": 0, "400797": 0, "400794": 0, "400795": 0, "400792": 0, "400793": 0, "362925": 0, "362924": 0, "362927": 0, "362926": 0, "362921": 0, "362923": 0, "362922": 0, "362929": 0, "362928": 0, "362930": 0, "370645": 0, "370644": 0, "370647": 0, "370646": 0, "370643": 0, "361181": 0, "370652": 0, "361180": 0, "361183": 0, "361182": 0, "370649": 0, "361177": 0, "370648": 0, "370651": 0, "361179": 0, "370650": 0, "361178": 0, "361185": 0, "361184": 0, "361186": 0, "369397": 0, "369396": 0, "369399": 0, "369398": 0, "369405": 0, "369404": 0, "369401": 0, "369400": 0, "369403": 0, "369402": 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 = "eae6474e-84de-4711-9cc2-e0269b32d3e0"; 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(",","/")); } }); });