Green's relations and unit-regularity for semigroup of transformations whose characters are bijective

2022/12/29 by Mosarof Sarkar, Sarkar, Mosarof, Shubh N. Singh +1
Computer Science · #20B30 #20M17 #20M20 #FOS: Mathematics #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2212.14239

openalex publication_date 2022/12/29 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28

Abstract

Let X be a nonempty set and P=\X_i\colon i∈ I\ be a partition of X. Denote by T(X, P) the semigroup of all transformations of X that preserve P. In this paper, we study the semigroup B(X,P) of all transformations f∈ T(X, P) such that χ^(f)∈ \rm Sym(I), where \rm Sym(I) is the symmetric group on I and χ^(f)\colon I → I is the character (map) of f defined by iχ^(f)=j whenever X_if⊆ X_j. We describe unit-regular elements in B(X,P), and determine when B(X,P) is a unit-regular semigroup. We alternatively prove that B(X,P) is a regular semigroup. We describe Green's relations on B(X,P), and prove that D = J on B(X,P) when P is finite. We also give a necessary and sufficient condition for D = J on B(X,P). We end the paper with a conjecture.

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