Abelian kernels of some monoids of injective partial transformations and an application
type: Item
instance of: scholarly article
title: Abelian kernels of some monoids of injective partial transformations and an application
zbMATH Open document ID: 0966.20030
author: Manuel Delgado
author: Vítor H. Fernandes
published in: Semigroup Forum
publication date: 2001-07-16T00:00:00Z
review text: The purpose of the paper is to calculate the Abelian kernels of the monoids \(\text{POI}_n\) and \(\text{POPI}_n\). The calculation relies on presentations for the monoids \(\text{POI}_n\) and \(\text{POPI}_n\) devised by the second author which appear elsewhere. In particular from the results we can infer that the Abelian kernel of \(\text{POPI}_4\) is not aperiodic so that the pseudovariety \(\mathbf{POPI}\) generated by the monoids \(\text{POPI}_n\) of all partial injective and orientation-preserving mappings on finite chains is not contained in the Malcev product of \(\mathbf{POI}\), the pseudovariety generated by the monoids of all partial injective and order-preserving mappings on finite chains, with \(\mathbf{Ab}\), the pseudovariety of all Abelian groups. More specifically we then have \(\mathbf{POPI}\nsubseteq\mathbf{POI}*\mathbf{Ab}\). Whether or not the negation of the reverse inclusion \(\mathbf{POI}*\mathbf{Ab}\nsubseteq\mathbf{POPI}\) is true remains unresolved but the left-hand side does lie inside the pseudovariety of all monoids with Abelian subgroups as the same is true of the Malcev product of all aperiodic monoids with Abelian groups.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 1545255
zbMATH Keywords: Abelian kernels
zbMATH Keywords: presentations
zbMATH Keywords: orientation-preserving mappings
zbMATH Keywords: finite chains
zbMATH Keywords: Malcev products
zbMATH Keywords: pseudovarieties of semigroups
zbMATH Keywords: order-preserving mappings
zbMATH Keywords: aperiodic monoids
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1545255