Associativity of products of existence varieties of regular semigroups
type: Item
instance of: scholarly article
title: Associativity of products of existence varieties of regular semigroups
zbMATH Open document ID: 0934.20044
DOI: 10.1016/S0022-4049(97)00135-7
author: Norman R. Reilly
author: Shuhua Zhang
published in: Journal of Pure and Applied Algebra
publication date: 2000-04-12T00:00:00Z
review text: The concept of an e-variety -- a class of regular semigroups that is closed under products, quotients and regular subsemigroups -- has allowed some of the techniques of universal algebra to be applied to regular semigroups. Motivated by the theory of semigroup varieties, the reviewer and \textit{P. G. Trotter} [Trans. Am. Math. Soc. 349, No. 11, 4265-4310 (1997; Zbl 0892.20037)] introduced a partial operation on the set of all e-varieties, as follows: if \(\mathcal{U,V}\) are e-varieties, at least one of which is contained in the e-variety \(\mathcal{CS}\) of completely simple semigroups, then \({\mathcal U}*{\mathcal V}\) consists of the regular parts of the wreath products of members of \(\mathcal U\) with members of \(\mathcal V\). One direction taken by the paper under review is to extend the work of the cited paper and the authors' paper [Semigroup Forum 53, No. 1, 1-24 (1996; Zbl 0854.20069)] on the properties of this operation. Associativity of the product is proven under some broad conditions; for instance if \(\mathcal{U,V}\) are group varieties and \(\mathcal W\) is ``monoidal'', \({\mathcal U}*({\mathcal V}*{\mathcal W})=({\mathcal U}*{\mathcal V})*{\mathcal W}\). This enables properties of such products as \(\mathcal{CS}*{\mathcal V}\) to be found for various \(\mathcal V\). A second direction is to study a different ``wreath product'', introduced by the authors in the cited paper, related to certain Malcev products of e-varieties. Reviewer's remark: In a subsequent paper, \textit{B. Billhardt} and \textit{M. B. Szendrei} [Associativity of regular semidirect product of existence varieties (to appear)] have extended the main associativity result to a broader class of cases.
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 1308297
zbMATH Keywords: Malcev products of e-varieties
zbMATH Keywords: regular semigroups
zbMATH Keywords: completely simple semigroups
zbMATH Keywords: wreath products
zbMATH Keywords: group varieties
zbMATH Keywords: associativity
reviewed by: Peter R. Jones
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00135-7
OpenAlex ID: W2031953278
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description: scientific article; zbMATH DE number 1308297