Regularity of the Transformation Semigroups Restricted by an Equivalence Relation with a Restricted Range - Pokpong Srimora; Chinram, Ronnason; Yonthanthum, Winita (Science & Technology Asia, 2025)
Let 𝑋 be a nonempty set. Denote by 𝑃(𝑋) and 𝑇(𝑋) the partial transformation semigroup and the full transformation semigroup on 𝑋, respectively. For a nonempty subset 𝑌 of 𝑋 and an equivalence relation 𝜌 on 𝑋, let 𝑃(𝑋,𝑌, 𝜌) = 𝛼 ∈ 𝑃(𝑋) | ran𝛼 ⊆ 𝑌 and ∀𝑥,𝑦 ∈ dom𝛼,(𝑥,𝑦) ∈ 𝜌 ⇒ 𝑥𝛼 = 𝑦𝛼, 𝑇(𝑋,𝑌, 𝜌) = 𝛼 ∈ 𝑇(𝑋) | ran𝛼 ⊆ 𝑌 and ∀𝑥,𝑦 ∈ 𝑋,(𝑥,𝑦) ∈ 𝜌 ⇒ 𝑥𝛼 = 𝑦𝛼. Then 𝑃(𝑋,𝑌𝜌) and 𝑇(𝑋,𝑌, 𝜌) are subsemigroups of 𝑃(𝑋) and 𝑇(𝑋), respectively. In this paper, we characterize the regular elements, left regular elements and right regular elements of both 𝑃(𝑋,𝑌, 𝜌) and 𝑇(𝑋,𝑌, 𝜌). In addition, the regularity, left regularity and right regularity of 𝑃(𝑋,𝑌, 𝜌) and 𝑇(𝑋,𝑌, 𝜌) are determined.