Regularity of the semigroup of transformations preserving a length

Abstract

Let $X_n=\{1,2,\dots ,n\}$ be a finite set $(n\ge 2)$ and $T_n$ the full transformation semigroup on $X_n$. For a positive integer $l\le n-1$, we define $$T_n(l)=\{\alpha \in T_n:\forall x,y\in X_n,|x-y|=l\Rightarrow |x\alpha -y\alpha |=l\}$$ and $$T_n^{\ast }(l)=\{\alpha \in T_n:\forall x,y\in X_n,|x-y|=l\iff |x\alpha -y\alpha |=l\}.$$ Then $T_n(l)$ and $T_n^{\ast }(l)$ 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 $T_n^{\ast }(l)$ is a regular semigroup.

arXiv comment14 pages