{"title":"Regular semigroups weakly generated by idempotents","authors":"Lu'is Oliveira","doi":"10.1142/s0218196723500388","DOIUrl":null,"url":null,"abstract":"A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation $\\langle G(X),\\rho_e\\cup\\rho_s\\rangle$ and its structure is studied. Although each of the sets $G(X)$, $\\rho_e$, and $\\rho_s$ is infinite for $|X|\\geq 2$, we show that the word problem is decidable as each congruence class has a canonical form. If $FI_n$ denotes FI(X) for $|X|=n$, we prove also that $FI_2$ contains copies of all $FI_n$ as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide $FI_2$; and (ii) all finite semigroups divide $FI_2$.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"12 1","pages":"851-891"},"PeriodicalIF":0.0000,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Int. J. Algebra Comput.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0218196723500388","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation $\langle G(X),\rho_e\cup\rho_s\rangle$ and its structure is studied. Although each of the sets $G(X)$, $\rho_e$, and $\rho_s$ is infinite for $|X|\geq 2$, we show that the word problem is decidable as each congruence class has a canonical form. If $FI_n$ denotes FI(X) for $|X|=n$, we prove also that $FI_2$ contains copies of all $FI_n$ as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide $FI_2$; and (ii) all finite semigroups divide $FI_2$.