THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE
Worachead Sommanee, Kritsada Sangkhanan · Journal of the Australian Mathematical Society · 2017
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 $W$ . In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that $$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle onumber\end{eqnarray}$$ is the largest regular subsemigroup of $T(V,W)$ and characterized Green’s relations on $T(V,W)$ . In this paper, we determine all the maximal regular subsemigroups of $Q$ when $W$ is a finite-dimensional subspace of $V$ over a finite field. Moreover, we compute the rank and idempotent rank of $Q$ when $W$ is an $n$ -dimensional subspace of an $m$ -dimensional vector space $V$ over a finite field $F$ .