SEMIGROUP FORUM, vol.97, no.2, pp.214-222, 2018 (SCI-Expanded)
Let In be the symmetric inverse semigroup on X-n = {1,..., n}, and let DPn and ODPn be its subsemigroups of partial isometries and of order-preserving partial isometries on Xn under its natural order, respectively. In this paper we find the ranks of the subsemigroups DPn, r = {alpha is an element of DPn : | im (alpha)vertical bar = r} and ODPn, r = {a. ODPn : vertical bar im (a)vertical bar <= r} for 2 <= r <= n - 1.