Regularity of the semigroup of transformations preserving a length

TL;DR

This paper investigates the regularity of specific subsemigroups of the full transformation semigroup on a finite set, providing necessary and sufficient conditions for regularity and establishing the regularity of another related subsemigroup.

Contribution

It offers a complete characterization of when the subsemigroup T_n(l) is regular and proves that T*_n(l) is always regular, advancing understanding of transformation semigroup structures.

Findings

Necessary and sufficient conditions for T_n(l) to be regular.

Proof that T*_n(l) is a regular semigroup.

Characterization of subsemigroups preserving a fixed length difference.

Abstract

Let $X_n = \{1,2,\dots,n\}$ be a finite set $(n\geq 2)$ and $T_n$ the full transformation semigroup on $X_n$. For a positive integer $l\leq n-1$, we define $$T_n(l) = \{\alpha\in T_n \colon \forall x,y\in X_n,\, |x-y| = l \;\Rightarrow\; |x\alpha - y\alpha| = l\}$$ and $$T^*_n(l) = \{\alpha\in T_n \colon \forall x,y\in X_n,\, |x-y| = l \;\Leftrightarrow\; |x\alpha - y\alpha| = l\}.$$ 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.