Regularity of the semigroup of transformations preserving a length

2024/06/03 by Worachead Sommanee, Sommanee, Worachead
Computer Science · Mathematics · #20M20 #FOS: Mathematics #Group Theory (math.GR) #Mathematical Control Systems and Analysis #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2406.01015

openalex publication_date 2024/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X_n = \1,2,…,n\ be a finite set (n≥ 2) and T_n the full transformation semigroup on X_n. For a positive integer l≤ n-1, we define T_n(l) = \α∈ T_n \colon ∀ x,y∈ X_n, |x-y| = l ⇒ |xα- yα| = l\ and T^*_n(l) = \α∈ T_n \colon ∀ x,y∈ X_n, |x-y| = l ⇔ |xα- yα| = l\. Then T_n(l) and T^*_n(l) are subsemigroups of T_n. In this paper, we give a necessary and sufficient condition for T_n(l) to be regular. Moreover, we prove that T^*_n(l) is a regular semigroup.

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