Rank and idempotent rank of finite full transformation semigroups with restricted range

Abstract

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. In 2011, Sanwong studied the regular part F(X,Y) = {α ∈ T(X,Y):Xα ⊆ Y α}, of T(X,Y) and described its Green's relations and ideals. In this paper, we compute the rank of F(X,Y) when X is a finite set. Moreover, we obtain the rank and idempotent rank of its ideals. © 2013 Springer Science+Business Media New York.