Regularity of a semigroup of transformations with restricted range that preserves an equivalence relation and a cross-section

Abstract

© 2020 by TJM. All rights reserved. For a fixed nonempty subset Y of X, let T (X, Y) be the semigroup consisting of all transformations from X into Y. Let ρ be an equivalence relation on X, ˆρ the restriction of ρ on Y and R a cross-section of the partition Y/ρ. We define T (X, Y, ρ, R) = {α ∈ T (X, Y): Rα ⊆ R and (a, b) ∈ ρ ⇒ (aα, bα) ∈ ρ}. Then T (X,Y, ρ,R) is a subsemigroup of T (X,Y). In this paper, we describe regular elements in T (X,Y, ρ,R), characterize when T (X, Y, ρ, R) is a regular semigroup and investigate some classes of T (X, Y, ρ, R) such as completely regular and inverse from which the results on T (X, ρ, R) and T (X, Y) can be recaptured easily when taking Y = X and ρ to be the identity relation, respectively. Moreover, the description of unit-regularity on T (X, ρ, R) is obtained.