E-Inversive Elements in Some Semigroups of Transformations that Preserve Equivalence

Nares Sawatraksa, Punyapat Kammoo, Chaiwat Namnak

Abstract

Let X be a nonempty set and T(X) the full transformation semigroup
on a set X. For an equivalence relation E on X and a cross-section R of the
partition X/E induced by E, let
TE∗ (X) = {α ∈ T(X) : ∀x, y ∈ X,(x, y) ∈ E ⇔ (xα, yα) ∈ E} and
TE(X, R) = {α ∈ T(X) : Rα = R and ∀x, y ∈ X,(x, y) ∈ E ⇒ (xα, yα) ∈ E}.
Then TE∗ (X) and TE(X, R) are subsemigroups of T(X). In this paper, we describe
the E-inversive elements of TE∗ (X) and TE(X, R). We also show that TE∗ (X) and
TE(X, R) are E-inversive semigroups in terms of the cardinality of X/E and R,
respectively.