A Study on the Subsemigroup of the Finite Full Transformation Semigroup Restricted by Subsets

Abstract

Let T_n be the semigroup of finite full transformation on a finite set X_n, and Y be any

subset of X_n. We considered a collection:

TG_Y(X) = {α ∈ T_n : α|_Y ∈G(Y),(X \ Y )α ⊆ X \ Y },

of maps in T_n whose restrictions to Y and X \ Y is a permutation on Y , and a transformation on X \ Y respectively. By the instrumentality of existing results in the semigroup

theory, we demonstrated that TG_Y(X) is the subsemigroup of the finite full transformation

semigroups. We explored its regularity, and identified the nature of its idempotent elements.

Furthermore we characterized the size of the semigroup