Regularity of the semigroup of transformations preserving a length

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