Structure of the (Total) Transformation Monoids Under Rank N Generators

Structure of the (Total) Transformation Monoids... | 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/15-170" }, "headline": "Structure of the (Total) Transformation Monoids Under Rank N Generators", "datePublished": "2026-02-02T18:01:12", "dateModified": "2026-04-30T07:37:45", "author": [ { "@type": "Person", "name": "Hala M. Sulaiman" }, { "@type": "Person", "name": "Asawer Al-Aadhami" } ], "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": "Throughout this paper our study focuses on transformation semigroups. These kinds of semigroups are the corner stone of semigroup theory. This is because every semigroup is isomorphic to transformation semigroup. The (total) transformation monoid T X n on a finite set X n = { 1 , 2 , … , n } where n ≥ 0 , n ∈ Z , is a semigroup of mapping that takes a set X n into itself, under the operation of composition of mapping with identity I X n . In this paper, we use an algebraic method for considering the monoid T ( Fl ) n ( G ) , where an independence algebra ( Fl ) n ( G ) is a disjointed union of sets of the form G x i for all 1 ≤ i ≤ n . Firstly, particular attention is paid to find the isomorphism between T ( Fl ) n ( G ) and the endomorphism monoid End ( F ℓ ) n ( G ) . Secondly, the embeddedness of T ( Fl ) n ( G ) in (full) wreath product of T n by G n has been found. Finally, the description of Green’s relation of T ( Fl ) n ( G ) has been provided." } { "@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/15-170/v1", "name": "Structure of the (Total) Transformation Monoids Under Rank N Generators" } } ] } Home Browse Structure of the (Total) Transformation Monoids Under Rank N Generators ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article M. Sulaiman H and Al-Aadhami A. Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.12688/f1000research.173831.1 ) 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 Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] Hala M. Sulaiman 1,2 , Asawer Al-Aadhami https://orcid.org/0000-0002-4414-9144 1 Hala M. Sulaiman 1,2 , Asawer Al-Aadhami https://orcid.org/0000-0002-4414-9144 1 PUBLISHED 02 Feb 2026 Author details Author details 1 Department of Mathematics, College of Science, University of Baghdad, Baghdad, Baghdad Governorate, Iraq 2 Department of Administrative and Financial Affairs, University of Technology, Baghdad, Iraq Hala M. Sulaiman Roles: Conceptualization, Formal Analysis, Methodology, Resources, Validation, Writing – Original Draft Preparation Asawer Al-Aadhami Roles: Conceptualization, Formal Analysis, Investigation, Resources, Supervision, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract Throughout this paper our study focuses on transformation semigroups. These kinds of semigroups are the corner stone of semigroup theory. This is because every semigroup is isomorphic to transformation semigroup. The (total) transformation monoid T X n on a finite set X n = { 1 , 2 , … , n } where n ≥ 0 , n ∈ Z , is a semigroup of mapping that takes a set X n into itself, under the operation of composition of mapping with identity I X n . In this paper, we use an algebraic method for considering the monoid T ( Fl ) n ( G ) , where an independence algebra ( Fl ) n ( G ) is a disjointed union of sets of the form G x i for all 1 ≤ i ≤ n . Firstly, particular attention is paid to find the isomorphism between T ( Fl ) n ( G ) and the endomorphism monoid End ( F ℓ ) n ( G ) . Secondly, the embeddedness of T ( Fl ) n ( G ) in (full) wreath product of T n by G n has been found. Finally, the description of Green’s relation of T ( Fl ) n ( G ) has been provided. READ ALL READ LESS Keywords A finite group, Semigroup, Wreath product, A free (left) G-act, Total transformation semigroup, Endomorphism monoid, Semigroup morphism. Corresponding Author(s) Asawer Al-Aadhami ( [email protected] ) Close Corresponding author: Asawer Al-Aadhami Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2026 M. Sulaiman H and Al-Aadhami A. 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: M. Sulaiman H and Al-Aadhami A. Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.12688/f1000research.173831.1 ) First published: 02 Feb 2026, 15 :170 ( https://doi.org/10.12688/f1000research.173831.1 ) Latest published: 30 Apr 2026, 15 :170 ( https://doi.org/10.12688/f1000research.173831.2 )  There is a newer version of this article available. Suppress this message for one day. 1. Introduction In this paper G will be a finite monoid (group) and ( F ℓ ) n ( G ) be a free (left)- G -act of rank n . It is clear that ( F ℓ ) n ( G ) is an independence algebra, 1 which is a class of universal algebras of rank n generators that includes free (left)- G -acts for a finite group G . For a positive integer n , we write the (total) transformation T X n to be T n which is the monoid of n degree, under the operation of a compositions of functions, such that for all σ , τ ∈ T n and x ∈ X n we have x ( σ ∘ τ ) = x ( στ ) = ( x σ ) τ . As well as, it is a set that maps every transformations into itself (i.e., X n ⟶ X n , ∀ n ≥ 0 , n ∈ Z ) . Clearly, if n = 0 , then X n = ∅ , and T 0 becomes a monoid that consists of only one map, which is the (empty) map. All transformations of this paper are composed from left to right, so for any σ ∈ T n and for any i ∈ X n , then iσ will be written as the image of i , and for any σ ∈ T n , we define Im { σ } = { iσ : i ∈ X n } and Ker { σ } = { ( i , j ) ∈ X n × X n : iσ = jσ } and ℘ ( σ ) = rank { σ } = | Im { σ } | . A number of authors appeared in 2 – 9 obtained and described various transformation semigroups, such as the description of the singular transformation Sn ℊ n , 10 and they study the structure of the partial endomorphism monoids of the independence algebra like a free acts and other set such as vector spaces and modules. When G is a finite monoids the endomorphism monoids of free (left)- G -acts played a major part of semigroup theory, and it can be found in. 11 It has long been known that set of the endomorphism monoid of the free (left)- G -acts of rank n is isomorphic to the (full) wreath product G ≀ n T n , 12 , 13 or, more generally S ≀ n T n for arbitrary semigroup S . Clearly, if G = { e } , then G ≀ n T n ≅ T n . Where S is a finite semigroup, the (total) transformation monoid T ( F ℓ ) n ( S ) of a free (left) S − act of rank n has been described in, 12 for further information we recommend the reader see. 14 – 21 Through this work, the discerption of T ( F ℓ ) n ( M ) , has established in particular when M is a finite group, and many properties of this concepts have been studied such as the isomorphic of T ( F ℓ ) n ( M ) to End ( F ℓ ) n ( M ) and the embeddedness of it in the (total) wreath product M ≀ n T n . Furthermore, in the last section of this work the discerption of Green’s relation of T ( F ℓ ) n ( M ) has been found. 2. Preliminaries Definition 2.1: 12 A mapping σ from a monoid A to a monoid B is named (morphism) if 1- ( a a ¯ ) σ = ( a ) σ ( a ¯ ) σ , for all a , a ¯ ∈ A ; 2- ( 1 A ) σ = 1 B . Definition 2.2: 12 Let B be a non-empty set and A be a monoid, then B is refed to a (left)- A -act if there is a mapping α : A × B ⟶ B , defined by α ( a , b ) = ab , such the following conditions are satisfied: 1 - a ( rb ) = ( ar ) b . 2 - 1 A b = b for all b ∈ B and a , r ∈ A . Definition 2.3: 12 Suppose that Y and T are two (left)- A -acts, then a mapping α : Y ⟶ T is named an A - acts morphism if ( ay ) α = a ( yα ) , ∀ a ∈ A , y ∈ Y . Definition 2.4: 12 , 13 The (total) transformation semigroup on a non-empty set X is denoted by T X = { σ | σ : X → X } . If n ≥ 0 , such that n ∈ Z , where X = X n = { 1 , 2 , … , n } , we write T X to be T n . With respect to the semigroup operation of composition that x ( δ ∘ γ ) = x ( δγ ) = ( xδ ) γ , for all x ∈ X and δ , γ ∈ T x . Clearly, T n is monoid with identity transformation I X = ( 1 2 … n 1 2 … n ) . Remark 2.5: 12 , 13 1 - If σ ∈ T n , n ≥ 0 , n ∈ Z , then σ can be written as σ = ( 1 … n 1 σ … nσ ) . 2 - For any σ ∈ T n , the second row of σ is not a permutation for the first row. Since not all element σ ∈ T n is injective. 3 - The number of elements of T n is equal to n n . Recall that, if S is a semigroup and if t ∈ S , the (principal) left ideal generated by t is the smallest left ideal of S containing t , and it is denoted by S 1 t . Dually, for tS 1 . 22 Now, let’s have the following definition: Definition 2.6: 22 Let S be any semigroup, the Green’s preorder binary relations are defined as follows: 1. s ≤ L t if and only if s ∈ S 1 t . 2. s ≤ R t if and only if s ∈ tS 1 . 3. s ≤ J t if and only if s ∈ S 1 tS 1 . From the above definition we can define the Green’s relations by: Definition 2.7: 22 Let S be any semigroup, the Green’s relations are defined by: 1. R = ≤ R ∩ ≥ R . 2. L = ≤ L ∩ ≥ L . 3. J = ≤ J ∩ ≥ J . 4. H = R ∩ L . 5. D = R ∘ L . Clearly, if S is finite, then D = J . So, the above definition can be translated as follows: Definition 2.8: 22 Let S be a finite semigroup, the Green’s relations can be defined as: 1. s L t if and only if S 1 s = S 1 t . 2. s R t if and only if sS 1 = tS 1 . 3. s J t if and only if S 1 sS 1 = S 1 tS 1 . 3. Free (left)- G -act structure ( F ℓ ) n ( G ) This section is devoted to give an explicit description of the set of the free (left)- G -acts, when G is a finite group. Definition 3.1: 12 , 22 Let G be a finite group and X be a non-empty set, then ( F ℓ ) X ( G ) is known as a free (left)- G -acts on X if: 1 - For any mapping σ : X ⟶ ( F ℓ ) X ( G ) ; 2 - For all G -acts Y and every map δ : X ⟶ Y , there exists a unique morphism ϕ : ( F ℓ ) X ( G ) ⟶ Y such that the following diagram is commute: If X = X n = { 1 , 2 , … , n } where n ≥ 0 and n ∈ Z , the free (left)- G -act can be written as ( F ℓ ) n ( G ) . It is clear that ( F ℓ ) n ( G ) consists elements of the form g x , where g ∈ G and x ∈ X i.e., ( F ℓ ) n ( G ) = ∪ ̇ i = 1 i = n G x i , such that G x i = { g x i : g ∈ G , 1 ≤ i ≤ n , x i ∈ X } . 1 , 12 It is well-known that the set of all the morphisms σ : B ⟶ B is called the endomorphism monoid, and it is denoted by End ( B ) . Therefore, the set of all morphisms from the free (left)- G -acts into itself is an endomorphism monoid and it is denoted by End ( F ℓ ) n ( G ) such that End ( F ℓ ) n ( G ) = { σ | σ : ( F ℓ ) n ( G ) ⟶ ( F ℓ ) n ( G ) ; and σ is a G − morphism}. If σ ∈ End ( F ℓ ) n ( G ) , we define σ to be x j σ = g j σ x j σ ´ , where 1 ≤ j ≤ n , σ ´ ∈ T n , and g j σ ∈ G n . As σ is a G − morphism and for g ∈ G and 1 ≤ j ≤ n we have ( g x j ) σ = g ( x j σ ) = g ( g j σ x j σ ´ ) . 4. Properties of the (full) wreath product multiplication The (full) wreath product multiplication in semigroup theory is a multiplication comes from a semidirect product between two semigroups. The next definition gives an abstract construction of the (full) wreath product multiplication. Definition 4.1: 12 , 13 Let G be a finite group and T n be the (total) transformation on X n , where n ≥ 0 , n ∈ Z . Then, the (full) wreath product of G n by T n is ( g 1 δ , … , g n δ , δ ) ( g 1 σ , … , g n σ , σ ) = ( g 1 δ g 1 δ σ , … , g n δ g nδ σ , δσ ) , where δ , σ ∈ T n and g j δ , g j σ ∈ G n . Under this multiplication, the set ( G n × T n ) is a monoid which is denoted by G ≀ n T n , with identity ( e , e , … , e n , I n ) , when I n is the identity of T n . 5. The structure of semigroup T ( F ℓ ) n ( G ) This section is devoted to describe the set T ( F ℓ ) n ( G ) , when ( F ℓ ) n ( G ) is an independence algebra. 5.1 The monoids T Y and T Y Suppose B be an algebra and B be a universe of B , 1 , 12 , 13 The sets T B = { σ | σ : B ⟶ B , ∀ σ ∈ T B } , and T B = { τ | τ : B ⟶ B , such that τ is a morphism , ∀ τ ∈ T B } are monoids. Furthermore, T B is submonoid of T B , this result has been proved by Al-Aadhami in, 12 and this can be shown by the following lemma: Lemma 5.1.1: The semigroup T B is a submonoid of T B . Proof: See. 12 5.2 The monoid T ( F ℓ ) n ( G ) In this subsection we give an explicit description to the (total) transformation monoid T ( F ℓ ) n ( G ) . Definition 5.2.1: The set of all morphisms T ( F ℓ ) n ( G ) = { σ | σ : ( F ℓ ) n ( G ) ⟶ ( F ℓ ) n ( G ) , such that σ is morphism} is a monoid. If γ ∈ T ( F ℓ ) n ( G ) , then γ can be expressed as: γ = ( x i 1 … x i m ℊ i 1 γ x i 1 γ ´ … ℊ i m γ x i m γ ´ ) . Where γ ´ ∈ T n , ℊ i 1 γ , … , ℊ i m γ ∈ G such that 1 ≤ i 1 < … < i m ≤ n , m ≥ 0 and x i k γ = ℊ i k γ x i k γ ´ . Notice, for every selection of σ ´ ∈ T n with Dom ( σ ´ ) = { ℓ 1 , … . , ℓ r } for 1 ≤ ℓ 1 < … < ℓ r ≤ n , r ≥ 0 and ℊ ℓ 1 σ , … , ℊ ℓ r σ ∈ G this gives σ = ( x ℓ 1 … x ℓ r ℊ ℓ 1 σ x ℓ 1 σ ´ … ℊ ℓ r σ x ℓ r σ ´ ) ∈ T ( F ℓ ) n ( G ) . Clearly, where G = { e } , i.e., ( G is a trivial set), then T ( F ℓ ) n ( G ) will be isomorphic to T n . The next lemma is analogous to Lemma 4.1. 12 However, the following lemma a semigroup S is assumed to be a finite group G . Theorem 5.2.2: If n ≥ 0 , n ∈ Z , then T ( F ℓ ) n ( G ) ≅ End ( F ℓ ) n ( G ) . Proof: Suppose σ be a mapping such that σ : T ( F ℓ ) n ( G ) ⟶ End ( F ℓ ) n ( G ) defined by λσ = λ ¯ when ( g x i ) λ ¯ = ( g x i ) λ , for all g ∈ G . We must show that λ ¯ is a G -acts morphism, i.e., we want to show that g ( b λ ¯ ) = ( gb ) λ ¯ , for all g ∈ G and b ∈ ( F ℓ ) n ( G ) . Let b = k x i . Clearly, an element x i ∈ Dom ( λ ) if and only if k x i ∈ Dom ( λ ) for all k ∈ G . Where k x i ∈ Dom ( λ ) , that means for all g ∈ G , gk x i ∈ Dom ( λ ) then ( gk x i ) λ ¯ = ( gk x i ) λ = g ( ( k x i ) λ ) = g ( ( k x i ) λ ¯ ) , since λ is a G -acts morphism. To show σ is injective. Assume that λσ = γσ , then for any g x i ∈ ( F ℓ ) n ( G ) obtaining ( g x i ) λσ = ( g x i ) γσ , ∀ i . Notice that, ( g x i ) λ = ( g x i ) λ ¯ = ( g x i ) λσ = ( g x i ) γσ = ( g x i ) γ ¯ = ( g x i ) γ , so λ = γ , as required. To prove σ is onto. Let β ¯ ∈ End ( F ℓ ) n ( G ) and let β ∈ T ( F ℓ ) n ( G ) defined from ( F ℓ ) n ( G ) to ( F ℓ ) n ( G ) by ( g x i ) β = ( g x i ) β ¯ , for each g x i ∈ ( F ℓ ) n ( G ) . As β ∈ T ( F ℓ ) n ( G ) that implies β is a G -acts morphism, therefore for some g x i ∈ Dom ( β ) = ( F ℓ ) n ( G ) and k ∈ G , we have k ( ( g x i ) β ) = k ( ( g x i ) β ¯ ) = ( kg x i ) β ¯ = ( kg x i ) β . Since, ( g x i ) β = ( g x i ) β ¯ = ( g x i ) βσ , for all g x i ∈ ( F ℓ ) n ( G ) , g ∈ G . Therefore, βσ = β ¯ , that gives σ is onto. To prove σ is a homomorphism. Let δ , γ ∈ T ( F ℓ ) n ( G ) .We need ( δγ ) σ = δσγσ . From the definition of σ and from our assumption we have ( g x i ) ( δγ ) σ = ( g x i ) δγ ¯ = ( g x i ) δγ ∀ i . Conversely, ( g x i ) ( δσγσ ) = ( g x i ) δ ¯ γ ¯ . Obviously, δ , ¯ γ ¯ ∈ End ( F ℓ ) n ( G ) , so δ , ¯ and γ ¯ are a G -acts morphisms. Then, ( g x i ) δ ¯ γ ¯ = ( ( g x i ) δ ¯ ) γ ¯ = ( ( g x i ) δ ) γ ¯ = ( ( g x i ) δ ) γ = ( g x i ) δγ = ( g x i ) δγ ¯ . So, ( δγ ) σ = δγ ¯ = δ ¯ γ ¯ = δγ , and this complete the proof. Theorem 5.2.3: For all n ≥ 0 , n ∈ Z , T ( F ℓ ) n ( G ) ↪ G ≀ n T n . Proof: Assume that σ ∈ T ( F ℓ ) n ( G ) , then σ can be written as = ( x i 1 … x i m g i 1 σ x i 1 σ ´ … g i m σ x i m σ ´ ) , such that σ ´ ∈ T n , g i 1 σ , … , g i m σ ∈ G for 1 ≤ i 1 < … < i m ≤ n , m ≥ 0 . Let α : T ( F ℓ ) n ( G ) ⟶ G ≀ n T n defined by μα = ( g 1 μ , … , g n μ , μ ´ ) . In order to prove α is embedding, we have to prove α is one to one and a homomorphism map. Let δ , γ ∈ T ( F ℓ ) n ( G ) , we need show ( δγ ) α = δαγα . Now, δαγα = ( g 1 δ , … , g n δ , δ ´ ) ( g 1 γ , … , g n γ , γ ´ ) = ( g 1 δ g 1 δ ´ γ , … , g n δ g n δ ´ γ , δ ´ γ ´ ) . The other side gives ( δγ ) α = ( g 1 δγ , … , g n δγ , ( δγ ) ′ ) . As δ , γ ∈ T ( F ℓ ) n ( G ) then δ , γ are G -acts morphisms such that they take T ( F ℓ ) n ( G ) to itself. Hence, ( x i ) δγ = ( x i δ ) γ = ( g i δ x i δ ´ ) γ = g i δ ( x i δ ´ γ ) = g i δ g i δ ´ γ x i δ ´ γ ´ ∀ i , and ( x i ) δγ = g i δγ x i ( δγ ) ′ . From that, ∀ i we get g i δγ x i ( δγ ) ′ = g i δ g i δ ´ γ x i δ ´ γ ´ , so we obtain g i δγ = g i δ g i δ ´ γ and ( δγ ) ′ = δ ´ γ ´ . This implies, α is homomorphism. To prove α is one-to-one. Let δ , γ ∈ T ( F ℓ ) n ( G ) such that δα = γα that leads to ( g 1 δ , … , g n δ , δ ´ ) = ( g 1 γ , … , g n γ , γ ´ ) . Then for any g x i ∈ ( F ℓ ) n ( G ) , we have ( g x i ) δ = g ( x i δ ) = g ( g i δ x i δ ´ ) = g ( g i γ x i γ ´ ) = g ( x i γ ) = ( g x i ) γ , and then δ = γ and α is one-to-one. 6. Green’s relations properties of T ( F ℓ ) n ( G ) Where B is an independence algebra V. Gould, 1 gave the characterization of Green’s relations on End ( B ) , after that A. Al-Aadhami, 12 described the Green’s relations on End ( F ℓ ) n ( S ) , where S is a finite semigroup. In this section the characterization of Green’s relations on T ( F ℓ ) n ( G ) have been illustrated. Theorem 6.1: For all σ , τ ∈ T ( F ℓ ) n ( G ) , we get the following identities: I. σ ≤ L τ if and only if Im { σ } ⊆ Im { τ } . II. σ ≤ R τ if and only if Ker { τ } ⊆ Ker { σ } . III. ℘ ( στ ) ≤ ℘ ( σ ) , ℘ ( τ ) . Proof: I. ⟹ ) Let σ ≤ L τ in T ( F ℓ ) n ( G ) that implies σ = δτ for some δ ∈ T ( F ℓ ) n ( G ) and so by, 12 [Proposition 5.5] we have Im { σ } = Im { δτ } ⊆ Im { τ } , and hence Im { σ } ⊆ Im { τ } . ⇐ ) Assume that Im { σ } ⊆ Im { τ } , so for all j ∈ { 1 , 2 , … , n } , we get x j σ ∈ Im { σ } ⊆ Im { τ } . If we select b j ∈ ( F ℓ ) n ( G ) such that x j σ = b j τ , then by defining υ ∈ T ( F ℓ ) n ( G ) such that x j υ = b j for j ∈ { 1 , 2 , … , n } . It is clear that x j υτ = b j τ = x j σ and therefore we can obtain υτ = σ that means σ ≤ L τ . II. ⟹ (Where σ ≤ R τ in T ( F ℓ ) n ( G ) that implies σ = τλ for any λ ∈ T ( F ℓ ) n ( G ) . Assume ( x , y ) ∈ Ker { τ } , hence x τ = y τ . Therefore, x σ = x ( τλ ) = ( x τ ) λ ( as τ , λ are G − acts morphism ) = ( y τ ) λ = y ( τλ ) ( as τ , λ are G − acts morphism ) = y σ . That means, ( x , y ) ∈ Ker { σ } , and so Ker { τ } ⊆ Ker { σ } . ⇐ ) If Ker { τ } ⊆ Ker { σ } , let ϖ ∈ T ( F ℓ ) n ( G ) such that ϖ : ( F ℓ ) n ( G ) ⟶ ( F ℓ ) n ( G ) . Since Im { τ } = G x j 1 ∪ ̇ ⋯ ∪ ̇ G x j n by, 12 [Lemma 5.1], define x j ı ϖ = y ı σ , such that y ı τ = x j ı , and x j ϖ = x j , for all j ∉ { j 1 , j 2 , …, j m } . Now, let y ı τ = y i ´ τ = x j ı then ( y ı , y i ´ ) ∈ Ker { τ } ⊆ Ker { σ } means y ı σ = y i ´ σ that implies y ı ϖ = y i ´ ϖ , therefore, ϖ is well –defined. Since ( F ℓ ) n ( G ) is free on X n = { x 1 , … . , x n }, and as ϖ ∈ T ( F ℓ ) n ( G ) then ϖ must be a G -acts morphism. Let z ∈ ( F ℓ ) n ( G ) be such that z = g x k , also let x k τ = h x j ı , that implies zτ = ( g x k ) τ = g ( x k τ ) = g ( h x j ı ) = gh ( x j ı ) = gh ( y ı τ ) = ( gh y ı ) τ . Now, zτϖ = ( gh x j i ) ϖ = ( gh ) ( x j ı ϖ ) = ( gh ) ( y ı σ ) = ( gh y ı ) σ . Because zτ = ( gh y ı ) τ , and Ker { τ } ⊆ Ker { σ } we must have z σ = ( gh y ı ) σ = zτϖ . Hence, σ = τϖ . III. If we suppose that for any σ , τ ∈ T ( F ℓ ) n ( G ) then ℘ ( στ ) ≤ ℘ ( σ ) , and ℘ ( στ ) ≤ ℘ ( τ ) , by, 12 [Remark 5.2]. Recall, ℘ ( σ ) = ℘ ( Im { σ } ) and let Im { σ } = ∪ ̇ z ∈ Z Gz , where Z ⊆ X n , so that, ℘ ( σ ) = | Z |. Because Im { στ } = Im { σ } τ = ( ∪ ̇ z ∈ Z Gz ) τ = ∪ ̇ z ∈ Z G ( zτ ) this implies that ℘ ( στ ) ≤ | Z | = ℘ ( σ ) . Notice, ℘ ( τσ ) = ℘ ( Im { τσ } ) = ℘ ( ( Im { τ } ) σ ) . As Im { τσ } ⊆ Im { σ } we obtain ℘ ( Im { τσ } ) ≤ ℘ ( Im { σ } ) and then ℘ ( τσ ) ≤ ℘ ( τ ) . Therefore, ℘ ( στ ) ≤ ℘ ( σ ) , ℘ ( τ ) . Theorem 6.2: For all σ , τ ∈ T ( F ℓ ) n ( G ) , we have the following: 1) σ L τ if and only if Im { σ } = Im { τ } . 2) σ R τ if and only if Ker { τ } = Ker { σ } . 3) σ H τ if and only if Im { σ } = Im { τ } and Ker { τ } = Ker { σ } . 4) σ D τ if and only if ℘ ( σ ) = ℘ ( τ ) . 5) σ ≤ J τ if and only if ℘ ( σ ) ≤ ℘ ( τ ) . 6) σ J τ if and only if ℘ ( σ ) = ℘ ( τ ) . 7) D = J . Proof: The point (1) and (2) can be easily be verified using Theorem 6.1 . 3) It is direct result from (1) and (2). 4 ) ⟹ ) Let σ D τ that means σ R δ L τ for any δ ∈ T ( F ℓ ) n ( G ) . From (1) and (2) we get Im { δ } = Im { τ } and Ker { σ } = Ker { δ } . Now, Im { σ } ≅ ( F ℓ ) n ( G ) / Ker { σ } (by the Fundamental Theorem of Semigroup), therefore, m { σ } ≅ ( F ℓ ) n ( G ) / Ker { δ } ≅ Im { δ } , and so ℘ ( σ ) = ℘ ( Im { σ } ) = ℘ ( Im { δ } ) = ℘ ( δ ) , and then we get ℘ ( σ ) = ℘ ( δ ) , furthermore, ℘ ( δ ) = ℘ ( Im { δ } ) = ℘ ( Im { τ } ) = ℘ ( τ ) . Hence, ℘ ( σ ) = ℘ ( τ ) . ⇐ ) Let ℘ ( σ ) = ℘ ( τ ) . As Im { σ } = ∪ ̇ y ∈ Y Gy , Im { τ } = ∪ ̇ z ∈ Z G z for some Y , Z ⊆ X n . With | Y |=| Z | = ℘ ( σ ) = ℘ ( τ ) . If we let φ : Y ⟶ Ζ be a bijection and by defining ϕ : Im { σ } ⟶ Im { τ } by ( g y ) ϕ = g ( y φ ) , for all g ∈ G and y ∈ Y . Clearly, ϕ is one to one as where ( g y ) ϕ = ( h y ′ ) ϕ that implies g ( y φ ) = h ( y ′ φ ) , for all g , h ∈ G and y , y ´ ∈ Y . Then we must have g = h and ( y φ ) = ( y ´ φ ) , this is because they are in Im { τ } and as Im { τ } is free (left)- G -act and φ is one-one, then y = y ´ . Furthermore, ϕ is onto since from definition of ϕ we get, ( g y ) ϕ = g ( y φ ) for all g ∈ G , y ∈ Y and because φ is bijection hence for all gz ∈ Im { τ } , choose y ∈ Y with y φ = z , then g y ∈ Im { σ } and gz = g ( y ) φ = ( g y ) ϕ . Suppose ξ = σϕ , where ξ ∈ T ( F ℓ ) n ( G ) . We have Im { ξ } = Im { σϕ } = Im { σ } ϕ = Im { τ } , therefore, τ L ξ . Let c , d ∈ ( F ℓ ) n ( G ) . Clearly, c σ = d σ if and only if ( c σ ) ϕ = ( d σ ) ϕ . Since ϕ is one to one, then Ker { σ } = Ker { σϕ } = Ker { ξ } , therefore, σ R ξ . Hence, σ D τ . 5) ⟹ ) Let σ ≤ I τ that means σ = ζτη , for any ζ , η ∈ T ( F ℓ ) n ( G ) . From Theorem 6.1 , (III), we have ℘ ( σ ) = ℘ ( ζτη ) ≤ ℘ ( ζτ ) ≤ ℘ ( τ ) . ⇐ ) If ℘ ( σ ) ≤ ℘ ( τ ) , Im { σ } = ∪ ̇ y ∈ Y Gy and Im { τ } = ∪ ̇ z ∈ Z G z for some Y , Ζ ⊆ X n , with ℘ ( σ ) = | Y | and ℘ ( τ ) = | Z | . Because ℘ ( σ ) ≤ ℘ ( τ ) then by, 12 [Remark 5.7] there is a one to one map ϕ : Y ⟶ Ζ . Let P = Im { ϕ } , that means P ⊆ Z and | Y | = | P | . Fix p 0 ∈ P and define π : Z ⟶ P by z π = z , for all z ∈ P , and zπ = p 0 for all z ∈ Z \ P , then Im { π } = P . Define β : ∪ ̇ z ∈ Z G z ⟶ ∪ ̇ p ∈ P G p by zβ = zπ . Obviously, β extends to a G -acts morphism therefore, βπ ∈ T ( F ℓ ) n ( G ) . Since Im { βπ } = ( Im { β } ) π = ( ∪ ̇ z ∈ Z Gz ) π = ∪ ̇ z ∈ Z Gzπ = ∪ ̇ z ∈ Z Gzβ = ∪ ̇ p ∈ P Gp , we obtain ℘ ( βπ ) = | P | = | Y | = ℘ ( σ ) so, by (4) we get βπ D σ and then βπ J σ because D ⊆ J . Therefore, σ ≤ I τ . 6) ⟹ ) If σ J τ , then σ = δτλ , τ = ασυ , for some δ , λ , α , υ ∈ T ( F ℓ ) n ( G ) . By Theorem 6.1 , (III), ℘ ( σ ) = ℘ ( δτλ ) ≤ ℘ ( δτ ) ≤ ℘ ( τ ) , as well as ℘ ( τ ) = ℘ ( ασυ ) ≤ ℘ ( ασ ) ≤ ℘ ( σ ) , that gives ℘ ( σ ) = ℘ ( τ ) . ⇐ ) Let ℘ ( σ ) = ℘ ( τ ) , then σ D τ by using (4), so that σ J τ as D ⊆ J . 7) This is an immediate consequence of (4) and (6). 7. Conclusions Throughout this paper, the independence algebra ( F ℓ ) n ( G ) has been studied, and we show that it is free (left)- G -acts under specific conditions. Moreover, the structure of T ( F ℓ ) n ( G ) have been considered, and the we show that T ( F ℓ ) n ( G ) isomorphic to the endomorphism monoid End ( F ℓ ) n ( G ) , with embeddedness in the (full) wreath product of T n by G n . For future work we may consider the structure of PT ( F ℓ ) n ( S ) , where S is a finite semigroup. Data availability Data sharing is not applicable to this article as no new data were created or analyzed in this study. References 1. Gould V: Independence algebras. Algebra Univers. 1995; 33 (3): 294–318. Publisher Full Text 2. Li DB, Fernandes VH: Endomorphisms of semigroups of oriented transformations. Semigroup Forum. 2023; 106 (1): 184–210. Publisher Full Text 3. Oliveira L: A combinatorial approach to the structure of locally inverse semigroups. Semigroup Forum. 2021; 103 (2): 575–621. Publisher Full Text 4. Fernandes VH: On the monoid of order-preserving transformations of a finite chain whose ranges are intervals. Semigroup Forum. 2024; 109 : 336–346. Publisher Full Text 5. Kudryavtsev D: Semigroups locally embeddable into the class of finite semigroups. International Journal of Algebra and Computation. 2023; 33 (05): 969–988. Publisher Full Text 6. Sarkar M, Singh SN: On certain semigroups of transformations whose restrictions belong to a given semigroup. Semigroup Forum. 2024; 108 : 707–723. Publisher Full Text 7. Grau A: The semigroup of endomorphisms with restricted range of an independence algebra. Semigroup Forum. 2023; 106 : 128–159. Publisher Full Text 8. Konieczny J: Semigroups of transformations whose restrictions belong to a given semigroup. Semigroup Forum. 2022; 104 (1): 109–124. Publisher Full Text 9. Sarkar M, Singh SN: Comments on Semigroups of transformations whose restrictions belong to a given semigroup. Semigroup Forum. 2022; 104 (3): 758–759. Publisher Full Text 10. Howie J: The subsemigroup generated by the idempotents of a full transformation semigroup. J. Lond. Math. Soc. 1966; s1-41 (1): 707–716. Publisher Full Text 11. Kilp M, Knauer U, Mikhalev A: Monoids, Acts and Categories: With Applications to Wreath Products and Graphs. A Handbook for Students and Researchers. Walter Gruyter; 2000; volume 29 . . Publisher Full Text 12. Al-Aadhami A: Full Transformation Semigroup of A Free Left S-Act on N-Generators. Iraqi Journal of Science. 2022; 63 (5): 2194–2202. Publisher Full Text 13. Al-Aadhami A: Ideals of the full transformation semigroup of a free left G-act on n-generators. AIP Conference Proceedings, AIP Publishing LLC. 2024; vol. 2922 (1): p. 090002. Publisher Full Text 14. Dağdeviren A, Ayık G, Ayık H: Combinatorial results for semigroups of orientation-preserving and order-decreasing transformations. Semigroup Forum. 2024; 108 : 43–55. Publisher Full Text 15. Mesyan Z: Conjugacy and Least Commutative Congruences in Semigroups. Semigroup Forum. 2025; 110 : 655–690. Publisher Full Text 16. Sarkar M, Singh SN: On certain semigroups of transformations with an invariant set. Asian-European Journal of Mathematics. 2022; 15 (11): 2250198. Publisher Full Text 17. Kudryavtseva G, Laan V: Globalization of partial actions of semigroups. Semigroup Forum. 2023; 107 : 200–217. Publisher Full Text 18. Guba VS: On diagram groups over Fibonacci-like semigroup presentations and their generalizations. Semigroup Forum. 2022; 104 : 320–329. Publisher Full Text 19. Szendrei MB: Normal extensions and full restricted semidirect products of inverse semigroups. Semigroup Forum. 2025; 111 : 820–837. Publisher Full Text 20. Yang X: Transformation semigroups generated by semi-complete digraphs. Semigroup Forum. 2022; 104 : 195–206. Publisher Full Text 21. Kudryavtseva G: Two-sided expansions of monoids. International Journal of Algebra and Computation. 2019; 29 (08): 1467–1498. Publisher Full Text 22. Howie J: Fundamentals of semigroup theory. Oxford: Clarendon; 1995. Publisher Full Text Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 02 Feb 2026 ADD YOUR COMMENT Comment Author details Author details 1 Department of Mathematics, College of Science, University of Baghdad, Baghdad, Baghdad Governorate, Iraq 2 Department of Administrative and Financial Affairs, University of Technology, Baghdad, Iraq Hala M. Sulaiman Roles: Conceptualization, Formal Analysis, Methodology, Resources, Validation, Writing – Original Draft Preparation Asawer Al-Aadhami Roles: Conceptualization, Formal Analysis, Investigation, Resources, Supervision, Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (2) version 2 Revised Published: 30 Apr 2026, 15:170 https://doi.org/10.12688/f1000research.173831.2 version 1 Published: 02 Feb 2026, 15:170 https://doi.org/10.12688/f1000research.173831.1 Copyright © 2026 M. Sulaiman H and Al-Aadhami A. 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 M. Sulaiman H and Al-Aadhami A. Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.12688/f1000research.173831.1 ) 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 02 Feb 2026 Views 0 Cite How to cite this report: Fuentes KB. Reviewer Report For: Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.5256/f1000research.191682.r470859 ) The direct URL for this report is: https://f1000research.com/articles/15-170/v1#referee-response-470859 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 15 Apr 2026 Katrina Belleza Fuentes , University of San Carlos, Cebu City, Central Visayas, Philippines Approved VIEWS 0 https://doi.org/10.5256/f1000research.191682.r470859 The paper discusses the structure of transformation monoids and how they relate to endomorphisms, wreath products, and Green’s relations. Comments: 1) It is weird to have sections 3,4, and 5 containing only 1 entry ... Continue reading READ ALL The paper discusses the structure of transformation monoids and how they relate to endomorphisms, wreath products, and Green’s relations. Comments: 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections. 2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term. 3) Perhaps include an end of proof symbol to formally mark the end of a proof. 4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format 5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated 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? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Algebra, Algebra and Music Theory, Topological Algebra, Algebraic Topology 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 Fuentes KB. Reviewer Report For: Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.5256/f1000research.191682.r470859 ) The direct URL for this report is: https://f1000research.com/articles/15-170/v1#referee-response-470859 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 Author Response 30 Apr 2026 Asawer Duraid Hamdi , Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq 30 Apr 2026 Author Response The comments and suggestion made by the referee are gratefully acknowledged 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend ... Continue reading The comments and suggestion made by the referee are gratefully acknowledged 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections. Answer: we prefere to unchange the order of the definitons becouse we aim the reader to be connected with the context and the proof. 2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term. Answer: Done 3) Perhaps include an end of proof symbol to formally mark the end of a proof. Answer: Done 4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format Answer: We check the format and it is correct. 5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated Answer: Thank you for your opinion. The comments and suggestion made by the referee are gratefully acknowledged 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections. Answer: we prefere to unchange the order of the definitons becouse we aim the reader to be connected with the context and the proof. 2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term. Answer: Done 3) Perhaps include an end of proof symbol to formally mark the end of a proof. Answer: Done 4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format Answer: We check the format and it is correct. 5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated Answer: Thank you for your opinion. Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 30 Apr 2026 Asawer Duraid Hamdi , Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq 30 Apr 2026 Author Response The comments and suggestion made by the referee are gratefully acknowledged 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend ... Continue reading The comments and suggestion made by the referee are gratefully acknowledged 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections. Answer: we prefere to unchange the order of the definitons becouse we aim the reader to be connected with the context and the proof. 2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term. Answer: Done 3) Perhaps include an end of proof symbol to formally mark the end of a proof. Answer: Done 4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format Answer: We check the format and it is correct. 5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated Answer: Thank you for your opinion. The comments and suggestion made by the referee are gratefully acknowledged 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections. Answer: we prefere to unchange the order of the definitons becouse we aim the reader to be connected with the context and the proof. 2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term. Answer: Done 3) Perhaps include an end of proof symbol to formally mark the end of a proof. Answer: Done 4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format Answer: We check the format and it is correct. 5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated Answer: Thank you for your opinion. Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Ahmed Bajri DS. Reviewer Report For: Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.5256/f1000research.191682.r469306 ) The direct URL for this report is: https://f1000research.com/articles/15-170/v1#referee-response-469306 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 03 Apr 2026 Dr. Sanaa Ahmed Bajri , Department of Mathematical Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia Approved VIEWS 0 https://doi.org/10.5256/f1000research.191682.r469306 This paper addresses a study on an interesting topic in semigroup theory which is the transformation semigroups. It provides all the required facts and properties concerning transformation semigroups. The presentation and the structure of the paper is clear and accurate. ... Continue reading READ ALL This paper addresses a study on an interesting topic in semigroup theory which is the transformation semigroups. It provides all the required facts and properties concerning transformation semigroups. The presentation and the structure of the paper is clear and accurate. Thus, I have found it excellent paper, however, I have some minor comments throughout the paper to improve it. The following are the minor comments that I have suggested: First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab". Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it. Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following". Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ... Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line. 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? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: My area is algebra and its applications. 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 Ahmed Bajri DS. Reviewer Report For: Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.5256/f1000research.191682.r469306 ) The direct URL for this report is: https://f1000research.com/articles/15-170/v1#referee-response-469306 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 Author Response 30 Apr 2026 Asawer Duraid Hamdi , Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq 30 Apr 2026 Author Response The comments and suggestions made by the referee are gratefully acknowledged First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper ... Continue reading The comments and suggestions made by the referee are gratefully acknowledged First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab". Answer: Done Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it. Answer: Done Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following". Answer: Done Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ... Answer: Done Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line. Answer: Done The comments and suggestions made by the referee are gratefully acknowledged First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab". Answer: Done Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it. Answer: Done Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following". Answer: Done Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ... Answer: Done Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line. Answer: Done Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 30 Apr 2026 Asawer Duraid Hamdi , Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq 30 Apr 2026 Author Response The comments and suggestions made by the referee are gratefully acknowledged First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper ... Continue reading The comments and suggestions made by the referee are gratefully acknowledged First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab". Answer: Done Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it. Answer: Done Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following". Answer: Done Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ... Answer: Done Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line. Answer: Done The comments and suggestions made by the referee are gratefully acknowledged First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab". Answer: Done Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it. Answer: Done Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following". Answer: Done Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ... Answer: Done Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line. Answer: Done Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 02 Feb 2026 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 2 (revision) 30 Apr 26 read Version 1 02 Feb 26 read read Dr. Sanaa Ahmed Bajri , Princess Nourah bint Abdulrahman University, P.O. Box 84428, Saudi Arabia Katrina Belleza Fuentes , University of San Carlos, Cebu City, Philippines 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 © 2026 Ahmed Bajri D. 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. 13 May 2026 | for Version 2 Dr. Sanaa Ahmed Bajri , Department of Mathematical Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia 0 Views copyright © 2026 Ahmed Bajri D. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The manuscript has been reviewed, and it appears that the authors have made the suggested amendments. I approve of this manuscript. Competing Interests No competing interests were disclosed. Reviewer Expertise My area is algebra and its applications. 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) Ahmed Bajri DS. Peer Review Report For: Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.5256/f1000research.199004.r480569) 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/15-170/v2#referee-response-480569 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Fuentes K. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 15 Apr 2026 | for Version 1 Katrina Belleza Fuentes , University of San Carlos, Cebu City, Central Visayas, Philippines 0 Views copyright © 2026 Fuentes K. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The paper discusses the structure of transformation monoids and how they relate to endomorphisms, wreath products, and Green’s relations. Comments: 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections. 2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term. 3) Perhaps include an end of proof symbol to formally mark the end of a proof. 4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format 5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated 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? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Algebra, Algebra and Music Theory, Topological Algebra, Algebraic Topology 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 (1) Author Response 30 Apr 2026 Asawer Duraid Hamdi, Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq The comments and suggestion made by the referee are gratefully acknowledged 1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections. Answer: we prefere to unchange the order of the definitons becouse we aim the reader to be connected with the context and the proof. 2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term. Answer: Done 3) Perhaps include an end of proof symbol to formally mark the end of a proof. Answer: Done 4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format Answer: We check the format and it is correct. 5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated Answer: Thank you for your opinion. View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Fuentes KB. Peer Review Report For: Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.5256/f1000research.191682.r470859) 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/15-170/v1#referee-response-470859 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Ahmed Bajri D. 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. 03 Apr 2026 | for Version 1 Dr. Sanaa Ahmed Bajri , Department of Mathematical Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia 0 Views copyright © 2026 Ahmed Bajri D. 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 (1) 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 paper addresses a study on an interesting topic in semigroup theory which is the transformation semigroups. It provides all the required facts and properties concerning transformation semigroups. The presentation and the structure of the paper is clear and accurate. Thus, I have found it excellent paper, however, I have some minor comments throughout the paper to improve it. The following are the minor comments that I have suggested: First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab". Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it. Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following". Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ... Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line. 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? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise My area is algebra and its applications. 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 (1) Author Response 30 Apr 2026 Asawer Duraid Hamdi, Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq The comments and suggestions made by the referee are gratefully acknowledged First Section (Preliminaries): Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab". Answer: Done Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it. Answer: Done Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following". Answer: Done Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ... Answer: Done Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line. Answer: Done View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Ahmed Bajri DS. Peer Review Report For: Structure of the (Total) Transformation Monoids Under Rank N Generators [version 1; peer review: 2 approved] . F1000Research 2026, 15 :170 ( https://doi.org/10.5256/f1000research.191682.r469306) 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/15-170/v1#referee-response-469306 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 = "Structure of the (Total) Transformation Monoids...".replace("'", ''); var linkedInUrl = "http://www.linkedin.com/shareArticle?url=https://f1000research.com/articles/15-170/v1" + "&title=" + encodeURIComponent(lTitle) + "&summary=" + encodeURIComponent('Read the article by '); var deliciousUrl = "https://del.icio.us/post?url=https://f1000research.com/articles/15-170/v1&title=" + encodeURIComponent(lTitle); var redditUrl = "http://reddit.com/submit?url=https://f1000research.com/articles/15-170/v1" + "&title=" + encodeURIComponent(lTitle); linkedInUrl += encodeURIComponent('M. Sulaiman H and Al-Aadhami A'); 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/15-170/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/15-170", templates : { twitter : "Structure of the (Total) Transformation Monoids Under Rank N.... M. Sulaiman H and Al-Aadhami A, published by " + "@F1000Research" + ", https://f1000research.com/articles/15-170/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/173831/191682") new F1000.Clipboard(); new F1000.ThesaurusTermsDisplay("articles", "article", "191682"); $(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 = { "460055": 0, "460054": 0, "460053": 0, "460052": 0, "460051": 0, "460050": 0, "460049": 0, "460048": 0, "460057": 0, "460056": 0, "470831": 0, "457526": 0, "469303": 0, "457527": 0, "469302": 0, "457524": 0, "457525": 0, "457522": 0, "457523": 0, "470834": 0, "470833": 0, "457521": 0, "470832": 0, "469311": 0, "469310": 0, "469309": 0, "469308": 0, "457530": 0, "469307": 0, "469306": 15, "480569": 0, "457528": 0, "469305": 0, "480568": 0, "457529": 0, "469304": 0, "455366": 0, "455367": 0, "455364": 0, "455365": 0, "455362": 0, "455363": 0, "455361": 0, "470863": 0, "470862": 0, "470861": 0, "470860": 0, "455370": 0, "470859": 11, "470858": 0, "455368": 0, "455369": 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 = "b872b300-e56b-4137-ba63-5b69c0c29e8d"; 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(",","/")); } }); });