{"title":"Yang-Baxter方程集论解的结构单群","authors":"F. Cedó, E. Jespers, C. Verwimp","doi":"10.5565/PUBLMAT6522104","DOIUrl":null,"url":null,"abstract":"Given a set-theoretic solution $(X,r)$ of the Yang--Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A'=A'(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A'$ and 1-cocycles $\\pi$ and $\\pi'$ of $M$ with coefficients in $A$ and in $A'$ with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case $X$ is finite, it turns out that $\\pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $\\pi'$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $\\pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(\\bar M, \\bar r)$ on the least cancellative image $\\bar M= M(X,r)/\\eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/\\eta$, for example when $(X,r)$ is irretractable, then $\\bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":"{\"title\":\"Structure monoids of set-theoretic solutions of the Yang-Baxter equation\",\"authors\":\"F. Cedó, E. Jespers, C. Verwimp\",\"doi\":\"10.5565/PUBLMAT6522104\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given a set-theoretic solution $(X,r)$ of the Yang--Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A'=A'(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A'$ and 1-cocycles $\\\\pi$ and $\\\\pi'$ of $M$ with coefficients in $A$ and in $A'$ with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case $X$ is finite, it turns out that $\\\\pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $\\\\pi'$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $\\\\pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(\\\\bar M, \\\\bar r)$ on the least cancellative image $\\\\bar M= M(X,r)/\\\\eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/\\\\eta$, for example when $(X,r)$ is irretractable, then $\\\\bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2019-12-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"11\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.5565/PUBLMAT6522104\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.5565/PUBLMAT6522104","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Structure monoids of set-theoretic solutions of the Yang-Baxter equation
Given a set-theoretic solution $(X,r)$ of the Yang--Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A'=A'(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A'$ and 1-cocycles $\pi$ and $\pi'$ of $M$ with coefficients in $A$ and in $A'$ with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case $X$ is finite, it turns out that $\pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $\pi'$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $\pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(\bar M, \bar r)$ on the least cancellative image $\bar M= M(X,r)/\eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/\eta$, for example when $(X,r)$ is irretractable, then $\bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.