Green's Relations on a Kind of Semigroups of Linear Transformations

Abstract:

      Let $V$ be a linear space over a field $F$ with finite dimension, $L(V)$ the semigroup, under composition, of all linear transformations from $V$ into itself. Suppose that $V=V_1 \oplus V_2 \oplus\cdots\oplus V_m$ is a direct sum decomposition of $V$, where $V_1,V_2,\ldots,V_m$ are subspaces of $V$ with the same dimension. A linear transformation $f\in L(V)$ is said to be sum-preserving, if for each $i\ (1\leq i\leq m)$, there exists some $j\ (1\leq j\leq m)$ such that $f(V_i)\subseteq V_j$. It is easy to verify that all sum-preserving linear transformations form a subsemigroup of $L(V)$ which is denoted by $L^{\oplus}(V)$. In this paper, we first describe Green's relations on the semigroup $L^{\oplus}(V)$. Then we consider the regularity of elements and give a condition for an element in $L^{\oplus}(V)$ to be regular. Finally, Green's equivalences for regular elements are also characterized.