Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range

W. Sommanee

doi:10.37863/umzh.v73i12.1289

W. Sommanee Chiang Mai Rajabhat Univ., Thailand https://orcid.org/0000-0001-7239-4705

DOI: https://doi.org/10.37863/umzh.v73i12.1289

Keywords: linear transformation; restricted range; embedding; maximal subsemigroup

Abstract

UDC 512.64

Let $V$ be a vector space and let $T(V)$ denote the semigroup (under composition) of all linear transformations from $V$ into $V$. For a fixed subspace $W$ of $V$, let $T(V,W)$ be the semigroup consisting of all linear transformations from $V$ into $W$. It is known that \[ F(V,W) =\{\alpha\in T(V,W): V\alpha\subseteq W\alpha\} \] is the largest regular subsemigroup of $T(V,W)$. In this paper, we prove that any regular semigroup $S$ can be embedded in $F(V,W)$ with $\dim(V) = |S^1|$ and $\dim(W) = |S|$, and determine all the maximal subsemigroups of $F(V,W)$ when $W$ is a finite dimensional subspace of $V$ over a finite field.

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