On unit-regular elements in various monoids of transformations

2021/02/20 by Mosarof Sarkar, Sarkar, Mosarof, Shubh N. Singh +1
Computer Science · Mathematics · #20M15 #20M20 #FOS: Mathematics #Group Theory (math.GR) #Rings, Modules, and Algebras #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2102.10282

openalex publication_date 2021/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be an arbitrary set and let T(X) denote the full transformation monoid on X. We prove that an element of T(X) is unit-regular if and only if it is semi-balanced. For infinite X, we discuss regularity of the submonoid of T(X) consisting of all injective (resp. surjective) transformations. For a partition P of X, we characterize unit-regular elements in the monoid T(X, P), under composition, defined as T(X, P) = \f∈ T(X)| (∀ X_i ∈ P) (∃ X_j ∈ P) X_i f ⊆ X_j\. We also characterize (unit-)regular elements in various known submonoids of T(X, P).

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