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

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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Published
17.12.2021
How to Cite
Sommanee, W. “Embedding Theorems and Maximal Subsemigroups of Some Linear Transformation Semigroups With Restricted Range”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 12, Dec. 2021, pp. 1714 -22, doi:10.37863/umzh.v73i12.1289.
Section
Research articles