2024/11/22 by Utsithon Chaichompoo, Chaichompoo, Utsithon, Kritsada Sangkhanan +1
Computer Science · #20M17 #20M19 #20M20 #FOS: Mathematics #Rings and Algebras (math.RA) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2411.15081
openalex publication_date 2024/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a nonempty set and T(X) the full transformation semigroup on X. For any equivalence relation E on X, define a subsemigroup TE^*(X) of T(X) by TE^*(X)=\α∈ T(X):for all x,y∈ X, (x,y)∈ E⇔ (xα,yα)∈ E\. We have the regular part of TE^*(X), denoted by Reg(T), is the largest regular subsemigroup of TE^*(X). Defined the subsemigroup QE^*(X) of TE^*(X) by QE^*(X)=\α∈ TE^*(X):|Aα|=1 and A∩ Xα≠∅ for all A∈ X/E\. Then we can prove that this subsemigroup is the (unique) minimal ideal of Reg(T) which is called the kernel of Reg(T). In this paper, we will compute the rank of QE^*(X) when X is finite and prove an isomorphism theorem. Finally, we describe and count all maximal subsemigroups of QE^*(X) where X is a finite set.