Bivarieties of orthodox semigroups generated by generalized Mal'cev products
type: Item
instance of: scholarly article
title: Bivarieties of orthodox semigroups generated by generalized Mal'cev products
zbMATH Open document ID: 0888.20040
DOI: 10.1007/PL00005928
author: Zhonghao Jiang
published in: Semigroup Forum
publication date: 1998-06-08T00:00:00Z
review text: The notion of Malcev product, originally defined in the context of varieties, has been extended to e-varieties of regular semigroups (classes closed under direct products, quotients and regular subsemigroups; sometimes called ``bivarieties'', as by the author): if \({\mathcal U},{\mathcal V}\) are e-varieties, then \(\langle{\mathcal U}{\mathbf m}{\mathcal V}\rangle\) is the e-variety generated by the regular semigroups \(S\) possessing a congruence whose quotient belongs to \(\mathcal V\) and such that the congruence classes of individual idempotents are semigroups belonging to \(\mathcal U\). The author introduces a variant, termed the generalized Malcev product: \(\langle{\mathcal U}{\mathbf g}{\mathbf m}{\mathcal V}\rangle\) is defined similarly to \(\langle{\mathcal U}{\mathbf m}{\mathcal V}\rangle\), except that only the regular elements of the congruence classes of idempotents (which in this context always form semigroups) need belong to \(\mathcal U\). An example is given to show that the products are different. It is also interpreted in terms of regular relational morphisms. Some interesting e-varieties are shown to arise in this way. For instance, if \(\mathcal U\) is any orthodox e-variety and \({\mathcal L}{\mathcal Z}\) denotes the (e-)variety of left zero semigroups, then \(\langle{\mathcal U}{\mathbf g}{\mathbf m}{\mathcal L}{\mathcal Z}\rangle\) comprises the orthodox semigroups with the property that the regular elements of each principal right ideal belong to \(\mathcal U\).
Mathematics Subject Classification ID: 20M19
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 1057432
zbMATH Keywords: orthodox e-varieties
zbMATH Keywords: e-varieties of regular semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: generalized Malcev products
zbMATH Keywords: regular elements
zbMATH Keywords: congruence classes
zbMATH Keywords: regular relational morphisms
zbMATH Keywords: left zero semigroups
zbMATH Keywords: orthodox semigroups
reviewed by: Peter R. Jones
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1057432