Matrix semigroup homomorphisms from dimension two to three
type: Item
instance of: scholarly article
title: Matrix semigroup homomorphisms from dimension two to three
zbMATH Open document ID: 0934.20042
DOI: 10.1016/S0024-3795(99)00105-6
published in: Linear Algebra and its Applications
publication date: 1999-11-29T00:00:00Z
review text: \({\mathcal M}_n(\mathbb{F})\) denotes the semigroup of all \(n\times n\) matrices with entries from a field \(\mathbb{F}\). One way to obtain a homomorphism from \({\mathcal M}_n(\mathbb{F})\) to \({\mathcal M}_m(\mathbb{F})\) is to take a group homomorphism \(\varphi\) from \(\text{GL}_n(\mathbb{F})\) to \(\text{GL}_m(\mathbb{F})\) and extend it over \({\mathcal M}_n(\mathbb{F})\) by defining \(\varphi(A)=0\) for all singular matrices \(A\). These semigroup homomorphisms are called degenerate and are known. The author determines all nondegenerate semigroup homomorphisms \(\varphi\) from \({\mathcal M}_2(\mathbb{F})\) to \({\mathcal M}_3(\mathbb{F})\) with the property that \(\varphi(0)=0\) and \(\varphi(I)=I\). To be somewhat more specific, the author shows that either \(\varphi(M)=SPS^{-1}\) where \(S\in{\mathcal M}_3(\mathbb{F})\) is invertible and \(P\) is a \(3\times 3\) matrix involving a field endomorphism of \(\mathbb{F}\) and a semigroup homomorphism of \(\mathbb{F}\) or \(\varphi(M)=SQS^{-1}\) where, again, \(S\in{\mathcal M}_3(\mathbb{F})\) is invertible and \(Q\) is a \(3\times 3\) matrix involving a field endomorphism of \(\mathbb{F}\). There are more possibilities for the matrix \(Q\) if \(\text{char }\mathbb{F}=2\).
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1371434
zbMATH Keywords: semigroups of matrices
zbMATH Keywords: group homomorphisms
zbMATH Keywords: semigroup homomorphisms
zbMATH Keywords: field endomorphisms
author: Damjana Kokol Bukovšek
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
Wikidata QID: Q126582973
description: scientific article; zbMATH DE number 1371434