All \(2\)-solid varieties of semigroups
type: Item
instance of: scholarly article
title: All \(2\)-solid varieties of semigroups
zbMATH Open document ID: 0960.20036
author: Jörg Koppitz
published in: Semigroup Forum
publication date: 2001-05-10T00:00:00Z
review text: For a given type \(\tau\), a mapping from a set of operation symbols into the set of all terms of type \(\tau\) is called a hypersubstitution. The set \(\text{Hyp}(\tau)\) of all hypersubstitutions is a monoid with respect to composition and the identity map. For a variety \(V\) of type \(\tau\) an equivalence relation \(\sim_V\) on \(\text{Hyp}(V)\) of \(V\)-equivalent substitutions is a congruence if and only if \(V\) has a certain solidity property which for semigroup varieties is known to be equivalent to a condition known as 2-solidity. This paper characterizes all 2-solid varieties of semigroups as of two types: (i) band varieties containing all rectangular bands and (ii) varieties \(V\) containing the variety defined by the identity \(xyz=xz\), contained in another known finitely based variety \(W_2\), and whose set of identities is closed under duals of its 2-variable identities.
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 1445559
zbMATH Keywords: Plonka equivalences
zbMATH Keywords: hypersubstitutions
zbMATH Keywords: \(2\)-solid varieties of semigroups
zbMATH Keywords: band varieties
zbMATH Keywords: rectangular bands
zbMATH Keywords: finitely based varieties
zbMATH Keywords: identities
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
full work available at URL: s002339910033
OpenAlex ID: W1981563003
DOI: 10.1007/S002339910033
description: scientific article; zbMATH DE number 1445559