A unified approach to biidentities for e-varieties
type: Item
instance of: scholarly article
title: A unified approach to biidentities for e-varieties
zbMATH Open document ID: 0953.20048
author: Yanyan Li
published in: Semigroup Forum
publication date: 2000-06-07T00:00:00Z
review text: An existence variety (or e-variety) of regular semigroups is a class of such semigroups closed under direct products, regular subsemigroups and morphic images. Two e-varieties are critical in this context: \(L{\mathcal I}\), the e-variety of all locally inverse semigroups and \(\mathcal{ES}\), the e-variety of all \(E\)-solid semigroups. Precisely these e-varieties (and their sub-e-varieties) contain bifree objects on any set \(X\). These are objects behaving like free objects do in usual varieties of universal algebras: for example, each member of the e-variety is a morphic image of some bifree object. The existence of these has been discovered by \textit{Y.~T.~Yeh} [Int. J. Algebra Comput. 2, No. 4, 471-484 (1992; Zbl 0765.20030)]. Yeh's results have been generalized by \textit{J.~Kadourek} and \textit{M.~B.~Szendrei} [Semigroup Forum 59, No. 3, 470-521 (1999; Zbl 0931.20049)] (for the \(E\)-solid case) and the reviewer [J. Algebra 166, No. 3, 630-650 (1994; Zbl 0806.20052)] (for the locally inverse case) in that suitable concepts of equational logic (including a Birkhoff type theorem) have been developed for either of these cases (and this also included a new proof of the existence of bifree objects). The language for these respective equational logics is different for the respective cases: in the \(E\)-solid case we have a doubled alphabet \(X\cup X'\) (representing elements and their inverses), a binary symbol \(\cdot\) (representing multiplication) and a unary symbol \(^{-1}\) (being interpreted as the partial unary operation of forming the group inverse inside the selfconjugate core of an \(E\)-solid semigroup). The set of terms has to be restricted appropriately according to the partiality of the unary operation \(^{-1}\). In the locally inverse case, the language consists of \(X\cup X'\) and \(\cdot\) as above and another binary symbol \(\wedge\) (being interpreted as the sandwich operation on a locally inverse semigroup). The paper under review consists of…
reviewed by: K. Auinger
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 1423169
zbMATH Keywords: e-varieties
zbMATH Keywords: bifree objects
zbMATH Keywords: \(E\)-solid semigroups
zbMATH Keywords: locally inverse semigroups
zbMATH Keywords: varieties of regular semigroups
zbMATH Keywords: Birkhoff type theorems
zbMATH Keywords: equational logics
zbMATH Keywords: partial binary operations
zbMATH Keywords: sandwich elements
zbMATH Keywords: idempotents
zbMATH Keywords: trifree objects
MaRDI profile type: Publication
full work available at URL: s002339910015
OpenAlex ID: W2071949427
DOI: 10.1007/S002339910015
description: scientific article; zbMATH DE number 1423169