The proof and the generalization of Higgins' theorem of divisors of semigroups of order-preserving mappings
type: Item
instance of: scholarly article
title: The proof and the generalization of Higgins' theorem of divisors of semigroups of order-preserving mappings
zbMATH Open document ID: 0844.20038
author: A. S. Vernitskiĭ
author: M. V. Volkov
published in: Russian Mathematics
publication date: 1996-05-21T00:00:00Z
review text: The main result of this paper is that every finite semigroup \(S\) the idempotents of which form an ideal is isotone, in the sense that \(S\) is a homomorphic image of a semigroup of order-preserving transformations. This generalizes a result of the reviewer [Int. J. Algebra Comput. 5, No. 6, 725-742 (1995; Zbl 0842.20054)] that any finite band is isotone, a fact there proved by implementation of the solution of the word problem for free bands by isotone automata. The proof of the authors is shorter and is based on construction of \(S_\ell\), a semigroup which extends \(S\) by adjoining a copy of the set \(S^1\) to act as a left zero ideal in a certain fashion. It transpires that if \(S\) is isotone then so is \(S_\ell\). This, together with the self-duality of the pseudovariety \(\mathcal O\) of all semigroups generated by semigroups of order-preserving mappings, allows the authors to devise a contradiction argument based on Gerard's solution to the word problem for semigroups satisfying the identity \(x_1x_2\dots x_n=(x_1x_2\dots x_n)^2\).
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 862276
zbMATH Keywords: finite semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: homomorphic images
zbMATH Keywords: semigroup of order-preserving transformations
zbMATH Keywords: finite bands
zbMATH Keywords: word problem
zbMATH Keywords: free bands
zbMATH Keywords: pseudovariety
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 862276