The ranks of certain semigroups of partial isometries

Let In be the symmetric inverse semigroup on Xn= { 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={??DPn:|im(?)|?r} and ODPn,r={??ODPn:|im(?)|?r} for 2 ? r? n- 1. © 2018, Springer Science+Business Media, LLC, part of Springer Nature