Isomorphism Theorems for Variants of Semigroups of Linear Transformations
Ruanglak Jongchotinon, Yupaporn Kemprasit · 2010
If S is a semigroup and a ∈ S, the semigroup (S, ◦) defined by x ◦y = xay for all x,y ∈ S is called a variant of S and (S, ◦) is denoted by (S,a). In 2003-2004, Tsyaputa characterized when two variants of the following transformation semigroups are isomorphic : the symmetric inverse semigroup, the full transformation semigroup and the partial transformation semigroup on a finite nonempty set. In this paper, we consider the semigroups under composition of all linear transformations of a finite-dimensional vector space over a finite field. We determine when its variants are isomorphic. We also obtain as a consequence in the same matter for the full n × n matrix semigroup over a finite field. Mathematics Subject Classification: 20M20, 20M10