The bifree regular \(E\)-solid semigroups
type: Item
instance of: scholarly article
title: The bifree regular \(E\)-solid semigroups
zbMATH Open document ID: 0841.20053
DOI: 10.1007/BF02574082
published in: Semigroup Forum
publication date: 1996-07-14T00:00:00Z
full work available at URL: https://eudml.org/doc/135434
review text: Bifree objects have been introduced by \textit{J. Kaďourek} and \textit{M. Szendrei} for e-varieties (equivalently: bivarieties) of orthodox semigroups [Semigroup Forum 40, No. 3, 257-296 (1990; Zbl 0705.20052)], and this concept has turned out to be the adequate notion of free object in the context of e-varieties of regular semigroups (that is, classes of regular semigroups closed under the formation of direct products, regular subsemigroups and homomorphic images). Bifree objects do not exist in arbitrary e-varieties, however. \textit{Y. Yeh} [Int. J. Algebra Comput. 2, No. 4, 471-484 (1992; Zbl 0765.20030)] has shown that an e-variety \(\mathbf V\) contains bifree objects on any non-empty set \(X\) if and only if \(\mathbf V\) is contained in \(\mathbf L I\) (the e-variety of all locally inverse semigroups) or in \(\mathbf E S\) (the e-variety of all \(E\)-solid semigroups). To give concrete descriptions of the bifree objects in the critical e-varieties \(\mathbf L I\) and \(\mathbf E S\) is therefore of particular interest. The case \(\mathbf L I\) has been dealt with by the reviewer [J. Algebra 166, No. 3, 630-650 (1994; Zbl 0806.20052)], the present paper is devoted to the case \(\mathbf E S\). The construction of the bifree \(E\)-solid semigroup on a set \(X\) roughly is as follows: Let \(X\) be any non-empty set, let \(G = F \mathbf G(X)\) be the free group on \(X\) and let \(\mathcal G\) be the Cayley graph of \(G\). Let \(U^{\text{loc}}({\mathcal G})\) be the free locally unary semigroupoid on the graph \(\mathcal G\) (i.e. the free semigroupoid on \(\mathcal G\) such that a unary operation is defined on all loops). It is clear that the set of all edges of \(U^{\text{loc}}({\mathcal G})\) is contained in the free unary semigroup on the set \(A = \text{Arr}({\mathcal G}) = G \times X\) of all edges of \(\mathcal G\). Let \(\xi\) be the congruence on \(U^{\text{loc}}({\mathcal G})\) which arises if one restricts the fully invariant congruence on the free una…
reviewed by: K. Auinger
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 836809
zbMATH Keywords: e-varieties
zbMATH Keywords: orthodox semigroups
zbMATH Keywords: regular semigroups
zbMATH Keywords: bifree objects
zbMATH Keywords: locally inverse semigroups
zbMATH Keywords: \(E\)-solid semigroups
zbMATH Keywords: semidirect products
zbMATH Keywords: completely regular semigroups
author: Mária B. Szendrei
MaRDI profile type: Publication
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OpenAlex ID: W2335294072
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description: scientific article; zbMATH DE number 836809