On Minimal Generating Sets of Certain Subsemigroups of Isometries

Let $DP_{n}$ and $ODP_{n}$ be the semigroups of all isometries and of all order-preserving isometries on $X_{n}$, respectively. In this paper we investigate the structure of minimal generating sets of the subsemigroup $DP_{n,r}$= {α ∈ DPn : |im (α)| ≤ r} (similarly of the subsemigroup $ODP_{n,r}$ = {α ∈ ODPn : |im (α)| ≤ r}) for 2 ≤ r ≤ n − 1.  .