Regularity of transformation semigroups defined by a partition

Abstract

Let X be a nonempty set, and let F = (Yi: i ∈ I) be a family of nonempty subsets of X with the properties that X = ∪i∈I Yi, and Yi ∩ Yj = ∅ for all i, j ∈ I with i ≠ j. Let ∅ ≠ J ⊆ I, and let Tℱ(J) (X) = (α ∈ T(X): ∀i ∈ I∃j ∈ J, Yiα ⊆ Yj). Then Tℱ(J) (X) is a subsemigroup of the semigroup T (X, Y(J)) of functions on X having ranges contained in Y(J), where Y(J)∪i∈J Yi. For each α ∈ Tℱ(J) (X), let χ(α): I → J be defined by iχ(α) = j ⇔ Yiα ⊆ Yj. Next, we define two congruence relations χ and χ on Tℱ(J) (X) as follows: (α,β) ∈ χ ⇔ χ(α) = χ(β) and (α,β) ∈ χ ⇔ χ(α)|J = χ(β)|J. We begin this paper by studying the regularity of the quotient semigroups Tℱ(J)(X)/χ and Tℱ(J) (X)/χ, and the semigroup Tℱ(J) (X). For each α ∈ Tℱ(X) Tℱ(I) (X), we see that the equivalence class [α] of α under χ is a subsemigroup of Tℱ(X) if and only if χ(α) is an idempotent element in the full transformation semigroup T(I). Let Iℱ(X), Sℱ(X) and Bℱ(X) be the sets of functions α in Tℱ(X) such that χ(α) is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups [α], Iℱ(X), Sℱ(X) and Bℱ(X) of Tℱ(X).