On a generalization of transformation semigroups that preserve equivalences
Nares Sawatraksa, Chaiwat Namnak · ScienceAsia · 2018
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) ∈ ρ} 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 .