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Research articles

ScienceAsia 44 (2018): 288-291 |doi: 10.2306/scienceasia1513-1874.2018.44.288

On a generalization of transformation semigroups that preserve equivalences

Nares Sawatraksa*, Chaiwat Namnak

ABSTRACT:     Let T(X) be the full transformation semigroup on a nonempty set X. For an equivalence relation σ on X, Pei introduced and studied the subsemigroup of T(X) defined by T(X,σ) = {α ∈ T(X) : ∀x, y ∈ X,(x, y) ∈ σ implies (xα, yα) ∈ σ}, which is called a transformation semigroup preserving the equivalence σ. In this paper, for two equivalence relations σ, ρ with ρ ⊆ σ on a nonempty set X, we introduce the subsemigroup T(X,σ,ρ) = {α ∈ T(X) : ∀x, y ∈ X,(x, y) ∈ σ implies (xα, yα) ∈ ρg of T(X) which generalizes the notation of the subsemigroup T(X,σ) of T(X). A necessary and sufficient condition under which T(X,σ,ρ) is a BQ-semigroup (a semigroup whose biideals and quasi-ideals coincide) is given. We also prove that T(X,σ) of T(X) can be embedded into a semigroup of T(Y, Z) = {α ∈ T(Y ) : Y α ⊆ Z} for some sets Y and Z with Z ⊆ Y .

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a Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000 Thailand

* Corresponding author, E-mail: naress58@nu.ac.th

Received 25 Jan 2017, Accepted 22 Jul 2018