Green’s relations and natural partial order on the regular subsemigroup of transformations preserving an equivalence relation and fixed a cross-section

Abstract

© 2019 by the Mathematical Association of Thailand. All rights reserved. Let X be an arbitrary nonempty set and T(X) the full transformation semigroup on X. For an equivalence relation E on X and a cross-section R of the partition X=E induced by E, let TE(X, R) = {α ∈ T(X): Rα = R and ∀x, y ∈, (x, y) ∈ E ⇒ (xα, yα) ∈ E}. Then the set Reg(TE(X, R)) of all regular elements of TE(X, R) is a regular sub- semigroup of T(X). In this paper, we describe Green’s relations for elements of the semigroup Reg(TE(X, R)). Also, we discuss the natural partial order on this semigroup and characterize when two elements in Reg(TE(X, R)) are related under this order.