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

• Date: 2021
• Main Author: Sommanee, W.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2021
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/1289
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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.