Matrix semigroup homomorphisms from dimension two to three
\({\mathcal M}_n(\mathbb{F})\) 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 \(\text{char }\mathbb{F}=2\).
scientific article; zbMATH DE number 1371434