The ranks of certain semigroups of partial isometries

Abstract

Let $$I_{n}$$ be the symmetric inverse semigroup on $$X_{n}=\{1,\ldots ,n\}$$ , and let $$DP_{n}$$ and $$ODP_{n}$$ be its subsemigroups of partial isometries and of order-preserving partial isometries on $$X_{n}$$ under its natural order, respectively. In this paper we find the ranks of the subsemigroups $$DP_{n,r}=\{ \alpha \in DP_{n}:|\mathrm {im\, }(\alpha )|\le r\}$$ and $$ODP_{n,r}=\{ \alpha \in ODP_{n}: |\mathrm {im\, }(\alpha )|\le r\}$$ for $$2\le r\le n-1$$ .