Regularity of the semigroup of transformations preserving a length

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.