Type-II conjecture is true for finite \(\mathcal J\)-trivial monoids
type: Item
instance of: scholarly article
title: Type-II conjecture is true for finite \(\mathcal J\)-trivial monoids
zbMATH Open document ID: 0773.20029
DOI: 10.1016/0021-8693(92)90141-8
author: Karsten Henckell
author: John L. Rhodes
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: A relation \(\varphi: M\to G\) from a finite monoid \(M\) to a finite group \(G\) is called a relational morphism if its graph is a submonoid of \(M\times G\) and it projects surjectively on the first component. The submonoid \(M_{II}\) of the ``type-II elements'' is defined as the set of all elements of \(M\) which are related with \(1\in G\) for every relational morphism \(\varphi: M\to G\). The ``weak type-II conjecture'' states that the function \(M\to M_{II}\) is computable. An approach to this conjecture proposed by \textit{J. Karnofsky} and \textit{J. Rhodes} [Semigroup Forum 24, 55-66 (1982; Zbl 0503.20028)] consists in taking the submonoid \(M_{II'}\) of constructible type-II elements which is defined as the least submonoid of \(M\) containing the idempotents and closed under the operation \(m\mapsto amb\) for any \(a,b\in M\) such that \(aba=a\) or \(bab=b\). While it is clear that \(M_{II'}\subseteq M_{II}\), the ``strong type- II conjecture'' asserts that \(M_{II'}=M_{II}\) for any finite monoid \(M\). The main theorem in the paper states that if every regular element of \(M\) lies in \(M_{II'}\), then \(M_{II'}=M_{II}\). As an application, it is shown that the strong type-II conjecture holds for the power monoid \({\mathcal P}(G)\) of any finite group \(G\). (The strong type-II conjecture has since been established by \textit{C. J. Ash} [Int. J. Algebra Comput. 1, No. 1, 127-146 (1991; Zbl 0722.20039)]).
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 130658
zbMATH Keywords: weak type-II conjecture
zbMATH Keywords: computable function
zbMATH Keywords: finite monoid
zbMATH Keywords: finite group
zbMATH Keywords: relational morphism
zbMATH Keywords: constructible type-II elements
zbMATH Keywords: idempotents
zbMATH Keywords: strong type-II conjecture
zbMATH Keywords: power monoid
Wikidata QID: Q123017095
reviewed by: Jorge Almeida
MaRDI profile type: Publication
full work available at URL: 0021-8693(92)90141-8
OpenAlex ID: W1986686753
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description: scientific article; zbMATH DE number 130658