The Semigroups package is a GAP package containing methods for semigroups, monoids, and inverse semigroups. There are particularly efficient methods for semigroups or ideals consisting of transformations, partial permutations, bipartitions, partitioned binary relations, subsemigroups of regular Rees 0-matrix semigroups, and matrices of various semirings including boolean matrices, matrices over finite fields, and certain tropical matrices. Semigroups contains efficient methods for creating semigroups, monoids, and inverse semigroup, calculating their Green’s structure, ideals, size, elements, group of units, small generating sets, testing membership, finding the inverses of a regular element, factorizing elements over the generators, and so on. It is possible to test if a semigroup satisfies a particular property, such as if it is regular, simple, inverse, completely regular, and a variety of further properties. There are methods for finding presentations for a semigroup, the congruences of a semigroup, the normalizer of a semigroup in a permutation group, the maximal subsemigroups of a finite semigroup, smaller degree partial permutation representations, and the character tables of inverse semigroups. There are functions for producing pictures of the Green’s structure of a semigroup, and for drawing graphical representations of certain types of elements.
半群软件包是一个 GAP 软件包，包含半群、单体和逆半群方法。对于由变换、部分排列、二元分区、二元分区关系、正则里斯 0 矩阵半群的子半群，以及包括布尔矩阵、有限域上的矩阵和某些热带矩阵在内的各种语义的矩阵，都有特别有效的方法。半群包含创建半群、单体和逆半群、计算其格林结构、理想、大小、元素、单位群、小生成集、测试成员资格、查找正则元素的逆、在生成器上因式分解元素等的有效方法。可以测试一个半群是否满足特定的性质，如是否正则、简单、逆、完全正则，以及其他各种性质。有一些方法可以找到半群的呈现、半群的同序、半群在置换群中的归一化、有限半群的最大子半群、较小程度的部分置换表示，以及逆半群的特征表。还有一些函数用于生成半群格林结构的图片，以及绘制某些类型元素的图形表示。
{"title":"Semigroups","authors":"T. Whitelaw","doi":"10.1201/9780203750230-5","DOIUrl":"https://doi.org/10.1201/9780203750230-5","url":null,"abstract":"The Semigroups package is a GAP package containing methods for semigroups, monoids, and inverse semigroups. There are particularly efficient methods for semigroups or ideals consisting of transformations, partial permutations, bipartitions, partitioned binary relations, subsemigroups of regular Rees 0-matrix semigroups, and matrices of various semirings including boolean matrices, matrices over finite fields, and certain tropical matrices. Semigroups contains efficient methods for creating semigroups, monoids, and inverse semigroup, calculating their Green’s structure, ideals, size, elements, group of units, small generating sets, testing membership, finding the inverses of a regular element, factorizing elements over the generators, and so on. It is possible to test if a semigroup satisfies a particular property, such as if it is regular, simple, inverse, completely regular, and a variety of further properties. There are methods for finding presentations for a semigroup, the congruences of a semigroup, the normalizer of a semigroup in a permutation group, the maximal subsemigroups of a finite semigroup, smaller degree partial permutation representations, and the character tables of inverse semigroups. There are functions for producing pictures of the Green’s structure of a semigroup, and for drawing graphical representations of certain types of elements.","PeriodicalId":130083,"journal":{"name":"Introduction To Abstract Algebra","volume":"79 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141214029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Semigroups","authors":"T. A. Whitelaw","doi":"10.1007/bfb0089803","DOIUrl":"https://doi.org/10.1007/bfb0089803","url":null,"abstract":"","PeriodicalId":130083,"journal":{"name":"Introduction To Abstract Algebra","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124152826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mappings","authors":"T. A. Whitelaw","doi":"10.1201/b19047-13","DOIUrl":"https://doi.org/10.1201/b19047-13","url":null,"abstract":"","PeriodicalId":130083,"journal":{"name":"Introduction To Abstract Algebra","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120939754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An Introduction to Groups","authors":"T. A. Whitelaw","doi":"10.1201/9780203750230-6","DOIUrl":"https://doi.org/10.1201/9780203750230-6","url":null,"abstract":"","PeriodicalId":130083,"journal":{"name":"Introduction To Abstract Algebra","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130735801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sets and Logic","authors":"T. A. Whitelaw","doi":"10.1201/9780203750230-1","DOIUrl":"https://doi.org/10.1201/9780203750230-1","url":null,"abstract":"","PeriodicalId":130083,"journal":{"name":"Introduction To Abstract Algebra","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128117105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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