Pythagorean  fuzzy deductive system of BCL-algebra

Pythagorean fuzzy deductive system of... | 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/14-64" }, "headline": "Pythagorean fuzzy deductive system of BCL-algebra", "datePublished": "2025-01-10T15:00:02", "dateModified": "2025-01-10T15:00:02", "author": [ { "@type": "Person", "name": "Asmamaw Abebe Biabeyin" }, { "@type": "Person", "name": "Berhanu Assaye Alaba" }, { "@type": "Person", "name": "Yohannes Gedamu Wondifraw" } ], "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 The deductive system of BCL-algebra has not been thoroughly explored, and its fuzzification remains undefined. This study aims to the association of this gap by introducing a fuzzy extension of the deductive system of BCL-algebra using Pythagorean fuzzy sets. Methods We define a Pythagorean fuzzy set as a pair of membership and non-membership functions, where the sum of their squares lies between 0 and 1. These functions are used to extend the classical deductive system of BCL-algebra, allowing for reasoning based on fuzzy degrees. The system incorporates fuzzy operations such as union, intersection, and complement, while maintaining the structure of BCL-algebra. We also introduce key components such as the accuracy function, score function, degree of indeterminacy, and square deviation to measure the certainty, truth, uncertainty, and deviation of fuzzy sets. Results We prove that the intersection of two Pythagorean fuzzy deductive systems remains a valid Pythagorean fuzzy deductive system within BCL-algebra. However, we show that the union of such systems does not necessarily form a valid fuzzy deductive system. The study also provides detailed proofs using induction, logical derivations, and algebraic techniques. Conclusion The results disclose that while the intersection of Pythagorean fuzzy deductive systems preserves the system's structure, the union does not, offering new insights into the behavior and limitations of fuzzy systems in BCL-algebra. " } { "@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/14-64/v1", "name": "Pythagorean fuzzy deductive system of BCL-algebra" } } ] } Home Browse Pythagorean fuzzy deductive system of BCL-algebra ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Biabeyin AA, Alaba BA and Wondifraw YG. Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.12688/f1000research.159263.1 ) 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 Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] Asmamaw Abebe Biabeyin https://orcid.org/0009-0001-4337-6804 1 , Berhanu Assaye Alaba 2 , Yohannes Gedamu Wondifraw 3 Asmamaw Abebe Biabeyin https://orcid.org/0009-0001-4337-6804 1 , Berhanu Assaye Alaba 2 , Yohannes Gedamu Wondifraw 3 PUBLISHED 10 Jan 2025 Author details Author details 1 Mathematics, Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia 2 Mathematics, Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia 3 Mathematics, Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Asmamaw Abebe Biabeyin Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Berhanu Assaye Alaba Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Yohannes Gedamu Wondifraw Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS Abstract Background The deductive system of BCL-algebra has not been thoroughly explored, and its fuzzification remains undefined. This study aims to the association of this gap by introducing a fuzzy extension of the deductive system of BCL-algebra using Pythagorean fuzzy sets. Methods We define a Pythagorean fuzzy set as a pair of membership and non-membership functions, where the sum of their squares lies between 0 and 1. These functions are used to extend the classical deductive system of BCL-algebra, allowing for reasoning based on fuzzy degrees. The system incorporates fuzzy operations such as union, intersection, and complement, while maintaining the structure of BCL-algebra. We also introduce key components such as the accuracy function, score function, degree of indeterminacy, and square deviation to measure the certainty, truth, uncertainty, and deviation of fuzzy sets. Results We prove that the intersection of two Pythagorean fuzzy deductive systems remains a valid Pythagorean fuzzy deductive system within BCL-algebra. However, we show that the union of such systems does not necessarily form a valid fuzzy deductive system. The study also provides detailed proofs using induction, logical derivations, and algebraic techniques. Conclusion The results disclose that while the intersection of Pythagorean fuzzy deductive systems preserves the system's structure, the union does not, offering new insights into the behavior and limitations of fuzzy systems in BCL-algebra. READ ALL READ LESS Keywords BCL-Algebra, Deductive system, Fuzzy set, Fuzzy Deductive system, Pythagorean Fuzzy Deductive system, Pythagorean Fuzzy set. Corresponding Author(s) Asmamaw Abebe Biabeyin ( [email protected] ) Close Corresponding author: Asmamaw Abebe Biabeyin Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2025 Biabeyin 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: Biabeyin AA, Alaba BA and Wondifraw YG. Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.12688/f1000research.159263.1 ) First published: 10 Jan 2025, 14 :64 ( https://doi.org/10.12688/f1000research.159263.1 ) Latest published: 10 Jan 2025, 14 :64 ( https://doi.org/10.12688/f1000research.159263.1 ) 1. Introduction Georg Cantor (1874) 3 had introduced the concept of set theory (crisp set or a classical set) in mathematics as a fundamental theory and had defined a set as a collection of distinguishable and definite objects that contain elements which satisfy precise properties of membership. In such set theory, a subset U of a non-void set A could be defined by a characteristic function. L. A. Zadeh (1965) 20 had introduced a generalization of classical sets called the concept of fuzzy set to handle mathematically vague or uncertain but Cantorian set could not address. A fuzzy set U in a non-empty set X is defined as: A = { ⟨ x , μ U ( x ) ⟩ : x ∈ A}, where the mapping μ U : A → [0, 1] defines the degree of membership of the element x in A to set U. The values ‘0’ and ‘1’are used to represent complete non-membership and complete membership, respectively, and values in between ‘0’ and ‘1’ are used to represent intermediate degrees of membership. As an extension of a fuzzy set, K. T. Atanassov (1986) 1 introduced the concept of an intuitionistic fuzzy set to better deal with uncertainties, later following K. T. Atanassov, R. R. Yager (2013) 18 launched Pythagorean fuzzy sets to alleviate the constraint on the total of membership degrees and non-membership degrees in intuitionistic fuzzy sets and introduced it as a new class of non-standard fuzzy subsets and the related idea of Pythagorean membership grades and non-membership grades. An innovative algorithm stated by X. Fuyuan, D. Weiping 10 (2019) established by the Pythagorean fuzzy set distance degree is proposed to explain the problems of medical diagnosis. By involving the different methods in the medical diagnosis application, it is found that the new algorithm is capable of the other methods. These results ascertain that this method is practical in dealing with the medical diagnosis complications. Y. H. Liu in 2011 14 introduced notions of algebras, one of which is BCL-algebra, with the partial orders and different algebraic structures. However, in this algebra, deductive system of BCL-algebra has not been introduced nor has been fuzzifeid and hence we are initiated to fill these gaps, and further we are motivated to introduce the Pythagorean fuzzy deductive system of BCL-algebra in depth under this manuscript. Thus in this paper, we define deductive system of BCL-algebra and then fuzzify it. We use the idea of Pythagorean fuzzy set to deductive system of in BCL-algebra. The concept of a Pythagorean fuzzy deductive system in BCL-algebra is given with some important characteristics. We have also examined the accuracy function, score function, degree of indeterminacy and square deviation in Pythagorean fuzzy deductive system of BCL-algebra and some interesting results have been delivered. 2. Preliminaries Under this section, we recall some basic concepts of algebras, deductive systems and fuzzy deductive systems of a few algebras, discuss basic concepts of BCL-algebra and Pythagorean fuzzy sets that are related to our study. Definition 2.1. (Ref. 11 ) A BCI-algebra is an algebra ( X ; ∗ , 0) of type (2, 0) satisfying the following conditions; ∀ x , y , z ∈ X.: (i) x ⊛ x = 0, (ii) ( ( x ⊛ y ) ⊛ ( x ⊛ z ) ) ⊛ ( z ⊛ y ) = 0, (iii) ( x ⊛ ( x ⊛ y ) ) ⊛ y = 0 (iv) x ⊛ y = 0 and y ⊛ x = 0 ⇒ x = y . Definition 2.2. (Ref. 5 ) An algebra ( A, ∗ , 0) is called a BCC-algebra if it satisfies the following axioms: (1) ( ∀ x , y , z ∈ A )((( x ∗ y ) ∗ ( z ∗ y )) ∗ ( x ∗ z ) = 0), (2) ( ∀ x ∈ A )(0 ∗ x = 0), (3) ( ∀ x ∈ A ) ( x ∗ 0 = x ), (4) ( ∀ x , y ∈ A)(( x ∗ y = 0 and y ∗ x = 0) ⇒ x = y ). Definition 2.3. (Ref. 13 ) By a GE-algebra we mean a non-empty set X with a constant 1 and a binary operation “ ⊛ ” satisfying the following axioms: (1) u ⊛ u = 1, (2) 1 ⊛ u =u, (3) u ⊛ ( v ⊛ w ) = u ⊛ ( v ⊛ ( u ⊛ w )), for all u , v , w ∈ X. Definition 2.4. (Refs. 4 , 10 ) A Hilbert algebra H is an algebra ( H , ⊛ , 1) satisfying the following conditions; for all x , y , z in H : (i) x ⊛ ( y ⊛ x ) = 1, (ii) ( x ⊛ ( y ⊛ z )) ⊛ (( x ⊛ y ) ⊛ ( x ⊛ z )) = 1, (iii) if x ⊛ y = y ⊛ x = 1, then x = y . Definition 2.5. (Ref. 2 ) A nonempty subset D of GE-algebra X is called a deductive system of X if it satisfies, ∀ x , y , z ∈ X: (i) X ⊛ D := { x ⊛ a :, a ∈ D } ⊆ D. (ii) ( x , y ∈ D ⇒ ( x ⊛ ( y ⊛ z )) ⊛ z ∈ D ). Definition 2.6. (Ref. 9 ) A subset A of a Hilbert algebra H is called a deductive system of H if it satisfies; for all x , y in H : (i) 1 ∈ A , (ii) x , x ⊛ y ∈ A ⇒ y ∈ A . Definition 2.7. (Refs. 15 , 16 ) An algebra ( B ; ⊛ , 0) of type (2, 0) is said to be a BCL-algebra if and only if for any x , y , z ∈ B , the following conditions are satisfied: (1) x ⊛ x = 0, (2) x ⊛ y = 0 and y ⊛ x = 0 imply x = y , (3) [ ( ( x ⊛ y ) ⊛ z ) ⊛ ( ( x ⊛ z ) ⊛ y ) ] ⊛ ( ( z ⊛ y ) ⊛ x ) = 0. Definition 2.8. (Ref. 17 ) Let ( B ; ⊛ , 0) be a BCL-algebra. A binary relation ≤ on B by which x ≤ y if and only if x ⊛ y = 0 for any x , y ∈ B , we call the BCL‐ordering ≤ is partial ordering on B . Definition 2.9. (Ref. 6 ) Let x ≤ y if and only if x ⊛ y = 0. Then the definition for BCL-algebra above can be rewritten as: (1) x ≤ x , (2) x ≤ y and y ≤ x ⇒ x = y , (3) [( x ⊛ y ) ⊛ z ] ⊛ [( x ⊛ z ) ⊛ y ] ≤ [( z ⊛ y ) ⊛ x ] . Definition 2.10. (Ref. 21 ) A fuzzy set μ in a Hilbert algebra H is called a fuzzy deductive system of H if (i) μ (1) ≥ μ ( x ) for x ∈ H, (ii) μ ( y ) ≥ min { μ (x), μ ( x ⊛ y )} for all x , y ∈ H . Definition 2.11. (Ref. 7 , 12 ) Let X ≠ ∅ . A fuzzy subset A of the set X is defined as: A = { ⟨ x , μ A ( x ) ⟩ | x ∈ X } , where the mapping μ A ( x ) : X → [0, 1] defines the degree of membership and the complement of μ A denoted by μ ¯ A , is the fuzzy set in X given by μ ¯ A ( x ) = 1 − μ A ( x ) , for all x ∈ X . Definition 2.12. (Ref. 8 , 11 ) An intuitionistic fuzzy set I in non-empty set X is an object having the form { x , μ I ( x ) , ν I ( x ) : x ∈ X }, where the function μ I ( x ) : X → [0, 1] and the function ν I ( x ) : X → [0, 1] define the degree of membership and the degree of non − membership respectively satisfying the condition: 0 ≤ μ I ( x ) + ν I ( x ) ≤ 1 Definition 2.13. (Ref. 19 , 22 ) A Pythagorean fuzzy set P in a non-empty set X is an object having the form: P = { ⟨ x , μ P ( x ) , ν P ( x ) ⟩ : x ∈ X } or simply P = ( μ P , ν P ) , where the function μ P ( x ) : X → [0, 1] and ν P ( x ) : X → [0, 1] define the degree of membership and the degree of non-membership, respectively satisfying the condition: 0 ≤ ( μ P ( x ) ) 2 + ( ν P ( x ) ) 2 ≤ 1 . Definition 2.14. (Refs. 18 , 22 ) Let P be Pythagorean fuzzy set. Then, the score function s of P is defined by: s ( P ) = ( μ P ( x ) ) 2 − ( ν P ( x ) ) 2 , where s ( P ) ∈ [ - 1 , 1 ] . Definition 2.15. (Ref. 18 ) Let P be a Pythagorean fuzzy set. Then, the accuracy function a of P is defined by: a ( P ) = ( μ P ( x ) ) 2 + ( ν P ( x ) ) 2 and hence a ( P ) ∈ [ 0 , 1 ] 3. Main Results 3.1 Pythagorean fuzzy BCL-structures on BCL-algebra Under this section, we define deductive system of BCL-algebra which has not been defined so far under BCL-algebra and similarly we introduce fuzzy deductive system of the BCL-algebra and then discuss some properties and theorems on fuzzy deductive system of BCL-algebra accompanied by corresponding proofs. Furthermore, we introduce Pythagorean fuzzy deductive system in BCL-algebra, state and prove different properties and theorems which no one has tried, yet. For this section; unless otherwise specified, B , DS . and B P denote “BCL-algebra ( B ; ⊛ , 0)”, the word “deductive system”, and “Pythagorean fuzzy set ( η P , τ P ) or { : x ∈ X } ” respectively, where the functions η P ( x ) : X → [0, 1] and τ P ( x ) : X → [0, 1] define the degree of membership and the degree of non-membership respectively , satisfying the condition: 0 ≤ ( η P ( x ) ) 2 + ( τ P ( x ) ) 2 ≤ 1. 3.2 Deductive system − Some properties and fuzzy structures in BCL-algebra Under this subsection, we define deductive systems of the BCL-algebra and fuzzy deductive systems of the BCL-algebra setting the stage for the subsequent development of Pythagorean fuzzy DS. of BCL-algebra where each concept is explained with examples for clarity. Remark 3.1. For BCL-algebra, ( B ; ⊛ , 0), one can easily prove that the following equations hold, ∀ m , n ∈ B : (1) 0 ⊛ m = 0, (2) m ⊛ n = m ⇒ m = 0, (3) m ⊛ n = n ⇒ m = 0 Definition 3.2. A non-empty subset D of BCL-algebra ( B ; ⊛ , 0) is called deductive system ( DS .) of B if it satisfies, ∀ x , y , z ∈ B: (i) x ∈ D ⇒ ( x ⊛ z ) ⊛ z ∈ D , (ii) x , y ∈ D ⇒ x ⊛ ( y ⊛ z ) ∈ D . Example 3.3. Let B = {0, p , q , r } and define a binary operation ⊛ on B by the Cayley Table 1 as follows: Table 1. A Cayley table of BCL-algebra, ( B ; ⊛ , 0). ⊛ 0 p q r 0 0 0 0 0 p p 0 r p q q r 0 q r r p q 0 In the Table 1 , it can be easily seen that B is a BCL-algebra and, (1) { 0 } , { 0 , r } , { 0 , p , q } and B are deductive systems of B (2) { p } , { q } , { r } , { 0 , p } , { 0 , q } , { p , q } , { p , r } , { q , r } , { 0 , q , r } , { 0 , p , r } , { p , q , r } are not deductive systems of B . Proposition 3.4. If D is deductive system of B , where ( B ; ⊛ , 0) is BCL-algebra, then the following hold true: (i) 0 ∈ D , (ii) D has never exactly two elements. Proof. Suppose D is deductive system of B . (i) D ≠ ∅ ⇒ ∃ x ∈ D such that ( x ⊛ x ) ⊛ x ∈ D ⇒ 0 ⊛ x = 0 ∈ D. (ii) Let D = {0, x } ⇒ x ⊛ (0 ⊛ z ) ∈ D ⇒ x ⊛ 0 ∈ D ⇒ x ⊛ 0 = 0 or x ⊛ 0 = x But by a remark above, x ⊛ 0 = 0 ⇒ x = 0 and also, x ⊛ 0 = x ⇒ x = 0 ⇒ in any case, D = {0} ◻ Definition 3.5. A fuzzy set η B in a BCL-algebra B is called a fuzzy deductive system (fuzzy DS .) of B , if the following axioms are satisfied, for all x , y , z ∈ B : (i) η B ( ( x ⊛ y ) ⊛ y ) ≥ η B ( x ) (ii) η B ( x ⊛ ( y ⊛ z ) ) ≥ min { η B ( x ) , η B ( y ) } Example 3.6. Suppose B = {0, q , r , p } and the binary operation ⊛ on B is as given by the Table 1 above: Define a fuzzy set η B : B → [0. 1] by: η B ( x ) = { 0.8 , if x = 0 , 0.6 , if x = p , q , 0.2 , if x = r . Then it is easy to check that η B is a fuzzy DS . of the BCL-algebra, B . 3.3 Pythagorean fuzzy deductive system of BCL-algebra Definition 3.7. A Pythagorean fuzzy set B P = ( η P , τ P ) , where the functions: η P ( x ) : B → [0, 1] and τ P ( x ) : B → [0, 1] define the degree of membership and the degree of non‐membership , respectively in B is called a Pythagorean fuzzy DS . of B if the following axioms are satisfied, for all x , y , z ∈ B : (i) ( η B ( ( x ⊛ y ) ⊛ y ) ) 2 ≥ ( η B ( x ) ) 2 and ( τ B ( ( x ⊛ y ) ⊛ y ) ) 2 ≤ ( τ B ( x ) ) 2 (ii) ( η B ( x ⊛ ( y ⊛ z ) ) ) 2 ≥ min { ( η B ( x ) ) 2 , ( η B ( y ) ) 2 } and ( τ B ( x ⊛ ( y ⊛ z ) ) ) 2 ≤ max { ( τ B ( x ) ) 2 , ( τ B ( y ) ) 2 } Example 3.8. Let B and the binary operation ⊛ be as defined as in Table 1 and let the fuzzy sets η B : B → [0. 1] and τ B : B → [0. 1] be defined as follows: η B ( x ) = { 0.83 , if x = 0 , 0.54 , if x = p , q , 0.16 , if x = r and τ B ( x ) = { 0.24 , if x = 0 , 0.67 , if x = p , q 0.76 , if x = r Then by routine calculations, it can be easily seen that B P = ( η B , τ B ) is a Pythagorean fuzzy DS . of B ( but not intuitionistic fuzzy DS . of B ) . Lemma 3.9. Let B P = ( η B , τ B ) be Pythagorean fuzzy set in B . If B P is Pythagorean fuzzy DS . of B , then the following hold, ∀ m , n , u ∈ B : (i) ( η B ( 0 ) ) 2 ≥ ( η B ( m ) ) 2 and ( τ B ( 0 ) ) 2 ≤ ( τ B ( m ) ) 2 (ii) m ⊛ n = n ⇒ ( η B ( u ) ) 2 ≥ ( η B ( m ) ) 2 and ( τ B ( u ) ) 2 ≤ ( τ B ( m ) ) 2 (Independent of n ) And, ( η B ( u ) ) 2 ≥ min{ ( η B ( m ) ) 2 , ( η B ( n ) ) 2 } and ( τ B ( u ) ) 2 ≤ max{ ( τ B ( m ) ) 2 , ( τ B ( n ) ) 2 } (iii) m ⊛ n = n ⊛ m ⇒ ( η B ( ( m ⊛ u ) ⊛ u ) ) 2 ≥ ( η B ( u ) ) 2 and ( τ B ( ( m ⊛ u ) ⊛ u ) ) 2 ≤ ( τ B ( u ) ) 2 Proof. Let B P = ( η B , τ B ) be Pythagorean fuzzy DS. in B . (i) Straight forward. (ii) Let m ⊛ n = n ⇒ ( η B ( m ⊛ u ) ⊛ u ) 2 ≥ ( η B ( m ) ) 2 and ( τ B ( m ⊛ u ) ⊛ u ) ) 2 ≤ ( τ B ( m ) ) 2 ⇒ ( η B ( u ) ) 2 ≥ ( η B ( m ) ) 2 and ( τ B ( u ) ) 2 ≤ ( τ B ( m ) ) 2 (iii) Let m ⊛ n = n ⊛ m (Independent of n since: ( η B ( m ⊛ u ) ⊛ u ) ) 2 = ( η B ( u ⊛ ( m ⊛ u ) ) ) 2 = ( η B ( u ⊛ ( u ⊛ m ) ) ) 2 ≥ min { ( η B ( u ) ) 2 , ( η B ( u ) ) 2 } = ( η B ( u ) ) 2 , and ⇒ ( τ B ( m ⊛ u ) ⊛ u ) ) 2 ≤ ( η B ( u ) ) 2 (Independent of n ) ◻ Definition 3.10. For membership fuzzy set; μ B : B → [0, 1], and square subtrahend μ ¯ ¯ B : B → [0, 1] from 1 such that ( μ ¯ ¯ B ( m ) ) 2 = 1 − ( μ ( m ) ) 2 , we call such fuzzy set, μ ¯ ¯ B , the square deviation of μ B . Theorem 3.11. Let η B and its square deviation η ¯ ¯ B be fuzzy sets in B such that ( η B ( m ⊛ n ) ) 2 = ( η B ( n ) ) 2 . Then ( η ¯ ¯ B ( m ⊛ n ) ) 2 = ( η ¯ ¯ B ( n ) ) 2 , ∀ m , n ∈ B . Then, B P = ( η B , η ¯ ¯ B ) is Pythagorean fuzzy DS . of B . iff η B and then η ¯ ¯ B are constants. Furthermore, he accuracy function a B ( m ) , the score function s B ( m ) and the degree of indeterminacy π B ( m ) are respectively given as: ∀ m ∈ B : (a) a B ( m ) = 1, (b) s B ( m ) = 2 ( η B ( m ) ) 2 − 1, (c) π B ( m ) = 0. Proof. Suppose ( η B ( m ⊛ n ) ) 2 = ( η B ( n ) ) 2 Then ( η ¯ ¯ B ( m ⊛ n ) ) 2 = 1 − ( η B ( m ⊛ n ) ) 2 = 1 − ( η B ( n ) ) 2 = ( η ¯ ¯ B ( n ) ) 2 and Let B P = ( η B , η ¯ ¯ B ) be Pythagorean fuzzy DS .. of B , a nd then ( η ¯ ¯ B ( m ⊛ n ) ) 2 = ( η ¯ ¯ B ( n ) ) 2 , Now we claim to verify that η B and η ¯ ¯ B are constants, ( or ( η B ( m ) ) 2 = ( η B ( n ) ) 2 and then ( η ¯ ¯ B ( m ) ) 2 = ( η ¯ ¯ B ( n ) ) 2 , ∀ m , n ∈ B . ) As B P = ( η B , η ¯ ¯ B ) is a Pythagorean fuzzy DS . of B , η B is a fuzzy DS . of B , and hence by one of the axioms, we have η B ( 0 ) = η B ( 0 ⊛ m ) , ∀ m ∈ B , ⇒ ( η B ( 0 ) ) 2 = ( η B ( ( 0 ⊛ m ) ⊛ m ) ) 2 = ( η B ( m ) ) 2 , ∀ m ∈ B , and again ( η B ( 0 ) ) 2 = ( η B ( ( 0 ⊛ n ) ⊛ n ) ) 2 = ( η B ( n ) ) 2 , ∀ n ∈ B ⇒ ( η B ( 0 ) ) 2 = ( η B ( m ) ) 2 = ( η B ( n ) ) 2 , ∀ m , n ∈ B , Or ( η B ( m ) ) 2 = ( η B ( n ) ) 2 , ∀ m , n ∈ B and hence, η B is constant, and analogously, η ¯ ¯ B is, too. Conversely, suppose η B and η ¯ ¯ B are constants, or: η B ( m ) = η B ( n ) and η ¯ ¯ B ( m ) = η ¯ ¯ B ( n ) , ∀ m , n ∈ B ( η B ( m ⊛ n ) ) 2 = ( η B ( n ) ) 2 and ( η ¯ ¯ B ( m ⊛ n ) ) 2 = ( η ¯ ¯ B ( n ) ) 2 , and To prove: B P = ( η B , η ¯ ¯ B ) is a Pythagorean fuzzy DS . of B , (i) ( η B ( m ) ) 2 = ( η B ( n ) ) 2 = ( η B ( ( m ⊛ n ) ⊛ n ) ) 2 ⇒ ( η B ( ( m ⊛ n ) ⊛ n ) ) 2 ≥ ( η B ( m ) ) 2 , ( and then { ( η ¯ ¯ B ( ( m ⊛ n ) ⊛ n ) ) 2 ≥ ( η ¯ ¯ B ( m ) ) 2 ) (ii) ( η B ( m ⊛ ( n ⊛ u ) ) ) 2 = ( η B ( m ) ) 2 = ( η B ( n ) ) 2 = ( η B ( u ) ) 2 ≥ min { ( η B ( m ) ) 2 , ( η B ( n ) ) 2 } and then ( η ¯ ¯ B ( m ⊛ ( n ⊛ u ) ) ) 2 = ( η ¯ ¯ B ( m ) ) 2 = ( η ¯ ¯ B ( n ) ) 2 = ( η ¯ ¯ B ( u ) ) 2 ≤ max { ( η ¯ ¯ B ( m ) ) 2 , ( η ¯ ¯ B ( n ) ) 2 } Therefore, by (i) and (ii) above, B P = ( η B , η ¯ ¯ B ) is Pythagorean fuzzy DS . of B . Furthermore: (a) a B ( m ) = ( η B ( m ) ) 2 + ( η ¯ ¯ B ( m ) ) 2 = ( η B ( m ) ) 2 + ( 1 − ( η B ( m ) ) 2 ) = 1 (b) s(m) = ( η B ( m ) ) 2 − ( η ¯ ¯ B ( m ) ) 2 = ( η B ( m ) ) 2 − ( 1 − ( η B ( m ) ) 2 ) = 2 ( η B ( m ) ) 2 − 1 (c) π B ( m ) = 1 − a P ( m ) = 0 ◻ Proposition 3.12. Let U be a non-void subset of B such that χ U is characteristic function and χ ¯ ¯ U is its square deviation. Then B P = ( χ U , χ ¯ ¯ U ) is Pythagorean fuzzy DS . of B if and only if U is DS . of B . Proof. Let χ U : U → [0, 1] be characteristic function and the square deviation χ ¯ ¯ U : U → [0, 1] be defined as: χ U ( x ) = { 1 , if x ∈ U , 0 , if x ∉ U and then ( χ ¯ ¯ U ) 2 ( x ) = { 1 , if x ∉ U , 0 , if x ∈ U Suppose B P = ( χ U , χ ¯ ¯ U ) ⇒ ( χ U ( ( m ⊛ n ) ⊛ n ) ) 2 ≥ ( χ U ( m ) ) 2 and ( χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) ) 2 ≤ ( χ ¯ ¯ U ( m ) ) 2 Let m ∈ U ⇒ χ U ( m ) = 1 and χ ¯ ¯ U ( m ) = 0 Then χ U ( ( m ⊛ n ) ⊛ n ) ≥ χ U ( m ) = 1 and χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) ≤ χ ¯ ¯ U ( m ) = 0 But χ U ( ( m ⊛ n ) ⊛ n ) ≤ 1 and χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) ≥ 0 ⇒ χ U ( ( m ⊛ n ) ⊛ n ) = 1 and χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) = 0 ⇒ ( m ⊛ n ) ⊛ n ) ∈ U and the other axiom can be checked similarly. ⇒ U is DS . of B . Conversely, suppose U is DS . of B . We claim that B P = ( χ U , χ ¯ ¯ U ) is Pythagorean fuzzy DS . of B , which means we need to show: (i) ( χ U ( ( m ⊛ n ) ⊛ n ) ) 2 ≥ χ U ( m ) 2 and ( χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) ) 2 ≤ ( χ ¯ ¯ U ( m ) ) 2 For the following we follow steps without squaring as it holds since each are members of [0, 1] and the result is either 1 or 0 (ii) χ U ( m ⊛ ( n ⊛ u ) ≥ min { χ U ( m ) , χ U ( n ) } and χ ¯ ¯ U ( m ⊛ ( n ⊛ u ) ≤ max { χ ¯ ¯ U ( m ) , χ U ( n ) } (i) Now we prove this case by taking the following three cases: Case (1) Let m ∈ U ( ⇒ ( m ⊛ n ) ⊛ n ∈ U by hypothesis ) ⇒ χ U ( ( m ⊛ n ) ⊛ n ) = χ U ( m ) = 1 ⇒ χ U ( ( m ⊛ n ) ⊛ n ) ≥ χ U ( m ) and ⇒ χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) = χ ¯ ¯ U ( m ) = 0 ⇒ χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) ≤ χ ¯ ¯ U ( m ) Case (2) Let m ∉ U and ( m ⊛ n ) ⊛ n ∈ U ⇒ χ U ( ( m ⊛ n ) ⊛ n ) = 1 and χ U ( m ) = 0 ⇒ χ U ( ( m ⊛ n ) ⊛ n ) ≥ χ U ( m ) and χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) = 0 and χ ¯ ¯ U ( m ) = 1 ⇒ χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) ≤ χ ¯ ¯ U ( m ) The above steps hold for squaring (for Pythagorean) each term (and the steps hereunder also hold) Case (3) Let m ∉ U ( m ⊛ n ) ⊛ n ∉ U ⇒ χ U ( ( m ⊛ n ) ⊛ n ) = χ U ( m ) = 0 ⇒ χ U ( ( m ⊛ n ) ⊛ n ) ≥ χ U ( m ) and ⇒ χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) = χ ¯ ¯ U ( m ) = 1 ⇒ χ ¯ ¯ U ( ( m ⊛ n ) ⊛ n ) ≤ χ ¯ ¯ U ( m ) (ii) By following similar steps for this as ( i ) above, where here the cases are: Case (1) For m , n ∈ U ( ⇒ m ⊛ ( n ⊛ u ) ∈ U by hypothesis ) Case (2) For ( m ∉ U or n ∉ U ) and ( m ⊛ n ) ⊛ n ∈ U Case (3) For ( m ∉ U or n ∉ U ) and ( m ⊛ n ) ⊛ n ∉ U we arrive at the result: χ U ( m ⊛ ( n ⊛ u ) ≥ min { χ U ( m ) , χ U ( n ) } and χ ¯ ¯ U ( m ⊛ ( n ⊛ u ) ≤ max { χ ¯ ¯ U ( m ) , χ U ( n ) } Therefore, B P is Pythagorean fuzzy DS . of B . ◻ Theorem 3.13. The intersection, B 1 ∩ B 2 , of any two Pythagorean fuzzy DS.s , B 1 and B 2 , of B is also a Pythagorean fuzzy DS . of B . Proof. Let B 1 = ( η B 1 , τ B 1 ) and B 2 =( η B 2 , τ B 2 ) be any two Pythagorean fuzzy DSs . of B . Then we need to prove that B 1 ∩ B 2 is a Pythagorean fuzzy DS . of B . Let m , n , u ∈ B . (i) ( η B 1 ∩ B 2 ( ( m ⊛ n ) ⊛ n ) ) 2 = min { ( η B 1 ( ( m ⊛ n ) ⊛ n ) ) 2 , ( η B 2 ( ( m ⊛ n ) ⊛ n ) ) 2 } ≥ min { ( η B 1 ( m ) ) 2 , ( η B 2 ( m ) ) 2 } = ( η B 1 ∩ B 2 ( m ) ) 2 and ( τ B 1 ∩ B 2 ( ( m ⊛ n ) ⊛ n ) ) 2 = max { ( τ B 1 ( ( m ⊛ n ) ⊛ n ) ) 2 , ( τ B 2 ( ( m ⊛ n ) ⊛ n ) ) 2 } ≤ max { ( τ B 1 ( m ) ) 2 , ( τ B 2 ( m ) ) 2 } = ( τ B 1 ∩ B 2 ( m ) ) 2 (ii) ( η B 1 ∩ B 2 ( m ⊛ ( n ⊛ u ) ) ) 2 = min { ( η B 1 ( m ⊛ ( n ⊛ u ) ) ) 2 , ( η B 2 ( m ⊛ ( n ⊛ u ) ) ) 2 } ≥ min { min { ( η B 1 ( m ) ) 2 , ( η B 1 ( n ) ) 2 } , min { ( η B 2 ( ( m ) ) ) 2 , ( η B 2 ( n ) ) 2 } } = min { min { ( η B 1 ( m ) ) 2 , ( η B 2 ( ( m ) ) ) 2 , min { ( η B 1 ( n ) ) 2 } , ( η B 2 ( n ) ) 2 } } = min { ( η B 1 ∩ B 2 ( m ) ) 2 , ( η B 1 ∩ B 2 ( n ) ) 2 } and Similarly, we have: ( τ B 1 ∩ B 2 ( m ⊛ ( n ⊛ u ) ) ) 2 ≤ max { ( η B 1 ∩ B 2 ( m ) ) 2 , ( η B 1 ∩ B 2 ( n ) ) 2 } and Hence by (i) and (ii) above, B 1 ∩ B 2 is a Pythagorean fuzzy DS . of B . The above theorem can also be generalized to any set of Pythagorean fuzzy DS.s as in the following corollary. ◻ Corollary 3.14. The intersection, ⋂ i ∈ I B i , of any family of Pythagorean fuzzy DSs. , { B i : i ∈ I } , in B is also a Pythagorean fuzzy DS . of B , where, ⋂ i ∈ I ( η B i ( m ) ) 2 = inf i ∈ I ( η B i ( m ) ) 2 and ⋂ i ∈ I ( τ B i ( m ) ) 2 = sup i ∈ I ( τ B i ( m ) ) 2 . Remark 3.15. The union of any two Pythagorean fuzzy DS . s of B is not necessarily Pythagorean fuzzy DS . of B . Example 3.16. Let ( B ; ⊛ , 0) be a BCL-algebra, where B = { p , q , r , 0} and let “ ⊛ ” be as defined in the Table 1 at the bottom, and define two Pythagorean fuzzy DSs . B 1 = ( ( η B 1 ) 2 , ( τ B 1 ) 2 ) and B 2 = ( ( η B 2 ) 2 , ( τ B 2 ) 2 ) as follows: ( η B 1 ( m ) ) 2 = { 0.8 , if m = 0 0.5 , if m = p , q , 0.1 , if m = r and ( τ B 1 ( m ) ) 2 = { 0.2 , if m = 0 0.4 , if m = p , q , 0.7 , if m = r ( η B 2 ( m ) ) 2 = { 0.5 , if m = 0 0.4 , if m = p , q , 0.2 , if m = r and ( τ B 2 ( m ) ) 2 = { 0.1 , if m = 0 0.2 , if m = p , q 0.6 , if m = r It is easy to check that B 1 and B 2 are Pythagorean fuzzy DS . s of B but to show that their union is not necessarily a Pythagorean fuzzy DS . of B , we justify it as follows using the above pairs of Pythagorean fuzzy DS.s of B which are B 1 and B 2 , where ( B ; ⊛ , 0 ) is as defined in Table at the bottom: ( η B 1 ∪ B 2 ( m ) ) 2 = { 0.8 , if m = 0 0.5 , if m = p , q , 0.2 , if m = r and ( τ B 1 ∪ B 2 ( m ) ) 2 = { 0.1 , if m = 0 0.2 , if m = p , q 0.6 , if m = r Take m = r , n = p and u = r ⇒ ( q ⊛ ( p ⊛ r ) ) = ( q ⊛ p ) ) = r Then (i) ( η B 1 ∪ B 2 ( q ⊛ ( p ⊛ r ) ) ) 2 = ( η B 1 ∪ B 2 ( r ) ) 2 = 0.2 ≥ min { ( η B 1 ∪ B 2 ( p ) ) 2 , ( η B 1 ∪ B 2 ( q ) ) 2 } = min{0.5, 0.5 } = 0.5 is not true. Thus, by the above justifications, the union of any two Pythagorean fuzzy DS . s of B is not necessarily a Pythagorean fuzzy DS . of B . Theorem 3.17. Let η , be a fuzzy set such that η B is a membership function and η ¯ ¯ B is its square deviation in B . Suppose ( η B b ) 2 = { x ∈ B : ( η B ( x ) ) 2 ≥ ( η B ( b ) ) 2 ∀ b ∈ B } and then ( η ¯ ¯ B b ) 2 = { x ∈ B : ( η ¯ ¯ B ( x ) ) 2 ≤ ( η ¯ ¯ B ( b ) ) 2 , ∀ b ∈ B } . Then B P = ( η B , η ¯ ¯ B ) is Pythagorean fuzzy DS . of B iff B b P = ( η B b , η ¯ ¯ B b ) is Pythagorean fuzzy DS . of B . Proof. Suppose B P = ( η B , η ¯ ¯ B ) is Pythagorean fuzzy DS . of B , or ∀ m , n , u , ∈ B : We need to show that B b P = ( η B b , η ¯ ¯ B b ) is Pythagorean fuzzy DS . of B . (1) For m ∈ ( η B b ) 2 ; ( η B ( ( m ⊛ n ) ⊛ n ) ) 2 ≥ ( η B ( m ) ) 2 ≥ ( η B ( b ) ) 2 , and ( η ¯ ¯ B ( ( m ⊛ n ) ⊛ n ) ) 2 ≤ ( η ¯ ¯ B ( m ) ) 2 ≤ ( η ¯ ¯ B ( b ) ) 2 ; ∀ b ∈ B (2) For m , n ∈ ( η B b ) 2 ; ( η B ( m ⊛ ( n ⊛ u ) ) ) 2 ≥ min { ( η B ( m ) ) 2 , ( η B ( n ) ) 2 } ≥ ( η B ( b ) ) 2 and ( η ¯ ¯ B ( m ⊛ ( n ⊛ u ) ) ) 2 ≤ max { ( η ¯ ¯ B ( m ) ) 2 , ( η ¯ ¯ B ( n ) ) 2 } ≤ ( η B ( b ) ) 2 ; ∀ b ∈ B Therefore, P b = ( η B b , η ¯ ¯ B b ) is a Pythagorean fuzzy DS . of B . Conversely, suppose ( η B b ) 2 = { x ∈ B: ( η B ( x ) ) 2 ≥ ( η B ( b ) ) 2 , ∀ b ∈ B } and ( η ¯ B b ) 2 = { x ∈ B: ( η ¯ B ( x ) ) 2 ≤ ( η ¯ B ( b ) ) 2 , ∀ b ∈ B } such that B B P = ( η B b , η ¯ B b ) is a Pythagorean fuzzy DS . of B . We need to prove that B P = ( η B , η ¯ ¯ B ) is Pythagorean fuzzy DS . of B . By the hypothesis we have the following: (1) m ∈ ( η B b ) 2 ⇒ ( ( m ⊛ n ) ⊛ n ) ∈ ( η B b ) 2 and m ∈ ( η ¯ ¯ B b ) 2 ⇒ ( ( m ⊛ n ) ⊛ n ) ∈ ( η ¯ ¯ B b ) 2 : ⇒ ( η B ( ( m ⊛ n ) ⊛ n ) ) 2 ≥ ( η B ( b ) ) 2 , ( η ¯ ¯ B ( ( m ⊛ n ) ⊛ n ) ) 2 ≤ ( η ¯ ¯ B ( b ) ) 2 ; ∀ b ∈ B (2) m n ∈ ( η B b ) 2 ⇒ m ⊛ ( n ⊛ u ) ∈ ( η B b ) 2 and m , n ∈ ( η ¯ B b ) 2 ⇒ m ⊛ ( n ⊛ u ) ∈ ( η ¯ B b ) 2 ⇒ ( η B b ( m ⊛ ( n ⊛ u ) ) ) 2 ≥ ( η B ( b ) ) 2 , and ( η ¯ ¯ B b ( m ⊛ ( n ⊛ u ) ) ) 2 ≤ ( η ¯ ¯ B ( b ) ) 2 , ∀ b ∈ B . Thus, by (1) & (2) above, B P = ( η B , η ¯ ¯ B ) is Pythagorean fuzzy DS . of B . ◻ Conclusion This Paper is the result of the initiations to define DS., to fuzzify it and to introduce the concept of Pythagorean fuzzy DS. of BCL-algebra following this definition and introductions, we state and prove new theorems of Pythagorean fuzzy DS. of BCL-algebra in depth which yield new fuzzified results which have not been addressed so far. The unique classifications: accuracy function, score function, degree of indeterminacy and square deviation are also characterised under the properties of the Pythagorean fuzzy DS. of BCL-algebra along with the corresponding proofs. Author contribution All authors have contributed equally to the completion and success of this manuscript at each step. Ethics and consent Ethical approval and consent were not required. Data availability statement Underlying data No data are associated with this article. No extended data. References 1. Atanassov KT: Intuitionistic Fuzzy sets. Fuzzy Sets Syst. 1986; 20 (1): 87–96. Publisher Full Text 2. Busneag D: A note on deductive systems of Hilbert algebra. Kobe J. Math. 1985; 2 : 29–35. 3. Cantor G: On a property of the class of all real algebraic numbers.Crelles J. Math.1874; 77 : 258262. Translated by C.P.Grant. 4. Chajda I, Halas R, Jun YB: Annihilators and deductive systems in commutative Hilbert algebras, Comment. Math. Univ. Carolin. 2002; 43 : 407–417. 5. Daniel RA: Notes on Ideals and Filters in BCC-ALGEBRAS. J. Univ. Math. 2018; 1 : 190–194. 6. Deena A, Rodyna H, On BCL-Algebra.World Appl. Sci. J. 27 ( 2013): 1752–1755, 1818-4952. 7. Derseh BL, Alaba BA, Wondifraw YG: Intuitionistic fuzzy PMS-ideals of a PMS-algebra. ai J. Math. 2022; 2022 : 1–13. Publisher Full Text 8. Derseh BL: Intuitionistic Fuzzy Structures on PMS-Algebras.2023. 9. Diego A: Sur les algebras de Hilbert, in Hermann, Ed. Collecfion de Logique Math. Serie A. 1966; 21 . 10. Fuyuan X, Weiping D: Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis. Appl. Soft Comput. 2019; 79 : 254–267. Publisher Full Text 11. Iseki K: On BCI-algebras. Math. Seminar Notes, bf. 1980; 8 : 125–130. 12. Jun YB, Hong SM, Kim SJ, Song SZ, Fuzzy ideals and fuzzy subalgebras of BCK-algebras.J. Fuzzy Math.1999; 7 (2): 411–418. MR 2000c:06040. Zbl 943.06010. 13. Jun YB, Ravikumar B: Deductive System of GE-Algebras. Alg. Struc. Appl. 2022; 9 (1): 53–67. 14. Liu YH: A New Branch of the Pure Algebra: BCL- Algebras. Adv. Pure Math. 2011; 01 (5): 297–299. Publisher Full Text 15. Liu YH: On BCL + -Algebras. Adv. Pure Math. 2012; 2 : 59–61. 16. Liu YH: On Liu-algebras: A new Composite of the BCL + -algebra and the Semi-groups. J. Semigroup Theory Appl. 2017; 2 . 2051-2937. 17. Xi OG: Fuzzy BCK-algebras. Math. Japon. 1991; 36 : 935–942. 18. Yager RR: Pythagorean fuzzy subsets. Proceedings of the Joint IFSA World Congress NAFIPS Annual Meeting. 2013; pp. 57–61. 19. Yager RR: Pythagorean membership grades in multicriteria decision making. Technical report MII-3301. Iona College, New Rochelle: Machine Intelligence Institute; 2013. 20. Zadeh LA: Fuzzy sets. Inf. Control. 1965; 8 : 338–353. Publisher Full Text 21. Jianming Z, Xueling M: Fuzzy Deductive Systems of HS-Algebras. Sci. Math. Jpn. 2004; 155162 22. Zhang XL, Xu ZS: Extension of TOPSIS to multi-criteria decision making with Pythagorean fuzzy sets. Int. J. Intell Syst. 2014; 29 : 1061–1078. Publisher Full Text Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 10 Jan 2025 ADD YOUR COMMENT Comment Author details Author details 1 Mathematics, Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia 2 Mathematics, Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia 3 Mathematics, Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Asmamaw Abebe Biabeyin Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Berhanu Assaye Alaba Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Yohannes Gedamu Wondifraw Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, 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: 10 Jan 2025, 14:64 https://doi.org/10.12688/f1000research.159263.1 Copyright © 2025 Biabeyin 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 Biabeyin AA, Alaba BA and Wondifraw YG. Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.12688/f1000research.159263.1 ) 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 10 Jan 2025 Views 0 Cite How to cite this report: Derseh BL. Reviewer Report For: Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.5256/f1000research.174962.r370004 ) The direct URL for this report is: https://f1000research.com/articles/14-64/v1#referee-response-370004 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 10 Mar 2025 Beza Lamesgin Derseh , Department of Mathematics, Faculty of Natural and Computational Science, Debre Markos University, Debre Markos, Ethiopia,, Bahir Dar, Ethiopia Approved VIEWS 0 https://doi.org/10.5256/f1000research.174962.r370004 As I have thoroughly read this paper from the very beginning to the end, the manuscript introduces the novel concept of Pythagorean fuzzy deductive systems in BCL-algebras. It presents key theoretical findings that significantly enhance the understanding of such structures ... Continue reading READ ALL As I have thoroughly read this paper from the very beginning to the end, the manuscript introduces the novel concept of Pythagorean fuzzy deductive systems in BCL-algebras. It presents key theoretical findings that significantly enhance the understanding of such structures by extending existing algebraic frameworks with innovative approaches. The study contributes fresh insights and highlights potential applications in mathematical research, thereby advancing the field's development and expanding its practical relevance. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fuzzy algebra, Intuitionistic fuzzy structures, Pythagorean fuzzy structures (like subalgebras, ideals, filters and deductive systems), Boolean algebra, Bipolar fuzzy structures I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Derseh BL. Reviewer Report For: Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.5256/f1000research.174962.r370004 ) The direct URL for this report is: https://f1000research.com/articles/14-64/v1#referee-response-370004 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Mechderso AA. Reviewer Report For: Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.5256/f1000research.174962.r369997 ) The direct URL for this report is: https://f1000research.com/articles/14-64/v1#referee-response-369997 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 10 Mar 2025 Alachew Amaneh Mechderso , University of KabriDahar, Kabri Dahar, Ethiopia Approved VIEWS 0 https://doi.org/10.5256/f1000research.174962.r369997 This manuscript introduces the concept of Deductive system, Pythagorean fuzzy structure , combining Pythagoran fuzzy sets with Deductive systems in BCL- algebras. It presents several theoretical results that enhance understanding in this area. The study contributes to the field by ... Continue reading READ ALL This manuscript introduces the concept of Deductive system, Pythagorean fuzzy structure , combining Pythagoran fuzzy sets with Deductive systems in BCL- algebras. It presents several theoretical results that enhance understanding in this area. The study contributes to the field by extending existing algebraic structures with new insights and potential applications in mathematical research. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Reviewer Expertise: Fuzzy Sets, Bipolar Fuzzy Sets, Bipolar Soft Sets, Fuzzy algebra I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Mechderso AA. Reviewer Report For: Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.5256/f1000research.174962.r369997 ) The direct URL for this report is: https://f1000research.com/articles/14-64/v1#referee-response-369997 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 10 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 2 Version 1 10 Jan 25 read read Alachew Amaneh Mechderso , University of KabriDahar, Kabri Dahar, Ethiopia Beza Lamesgin Derseh , Department of Mathematics, Faculty of Natural and Computational Science, Debre Markos University, Debre Markos, Ethiopia,, Bahir Dar, Ethiopia 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 Derseh B. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 10 Mar 2025 | for Version 1 Beza Lamesgin Derseh , Department of Mathematics, Faculty of Natural and Computational Science, Debre Markos University, Debre Markos, Ethiopia,, Bahir Dar, Ethiopia 0 Views copyright © 2025 Derseh B. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions As I have thoroughly read this paper from the very beginning to the end, the manuscript introduces the novel concept of Pythagorean fuzzy deductive systems in BCL-algebras. It presents key theoretical findings that significantly enhance the understanding of such structures by extending existing algebraic frameworks with innovative approaches. The study contributes fresh insights and highlights potential applications in mathematical research, thereby advancing the field's development and expanding its practical relevance. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Fuzzy algebra, Intuitionistic fuzzy structures, Pythagorean fuzzy structures (like subalgebras, ideals, filters and deductive systems), Boolean algebra, Bipolar fuzzy structures I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Derseh BL. Peer Review Report For: Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.5256/f1000research.174962.r370004) 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/14-64/v1#referee-response-370004 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Mechderso A. 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. 10 Mar 2025 | for Version 1 Alachew Amaneh Mechderso , University of KabriDahar, Kabri Dahar, Ethiopia 0 Views copyright © 2025 Mechderso A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions This manuscript introduces the concept of Deductive system, Pythagorean fuzzy structure , combining Pythagoran fuzzy sets with Deductive systems in BCL- algebras. It presents several theoretical results that enhance understanding in this area. The study contributes to the field by extending existing algebraic structures with new insights and potential applications in mathematical research. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Reviewer Expertise: Fuzzy Sets, Bipolar Fuzzy Sets, Bipolar Soft Sets, Fuzzy algebra I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Mechderso AA. Peer Review Report For: Pythagorean fuzzy deductive system of BCL-algebra [version 1; peer review: 2 approved] . F1000Research 2025, 14 :64 ( https://doi.org/10.5256/f1000research.174962.r369997) 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/14-64/v1#referee-response-369997 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 = "Pythagorean\u00A0 fuzzy deductive system...".replace("'", ''); var linkedInUrl = "http://www.linkedin.com/shareArticle?url=https://f1000research.com/articles/14-64/v1" + "&title=" + encodeURIComponent(lTitle) + "&summary=" + encodeURIComponent('Read the article by '); var deliciousUrl = "https://del.icio.us/post?url=https://f1000research.com/articles/14-64/v1&title=" + encodeURIComponent(lTitle); var redditUrl = "http://reddit.com/submit?url=https://f1000research.com/articles/14-64/v1" + "&title=" + encodeURIComponent(lTitle); linkedInUrl += encodeURIComponent('Biabeyin 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/14-64/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/14-64", templates : { twitter : "Pythagorean\u00A0 fuzzy deductive system of BCL-algebra. Biabeyin AA et al., published by " + "@F1000Research" + ", https://f1000research.com/articles/14-64/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/159263/174962") new F1000.Clipboard(); new F1000.ThesaurusTermsDisplay("articles", "article", "174962"); $(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 = { "370004": 10, "370001": 0, "370000": 0, "370003": 0, "370002": 0, "369997": 11, "369996": 0, "369999": 0, "369998": 0, "369995": 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 = "1d6f4947-a4f5-4f84-91eb-d4518b7887e0"; 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(",","/")); } }); });