On the monoid of partial isometries with fixed set

Abstract

Denote [n] to be a finite n-chain 12n ,and let DPn denote the semigroup of all partial isometries on [n]. For a nonempty subset D of [n], let Fix([n]D) be the set of all maps in DPn which fix all elements of D. Then Fix([n]D) is a subsemigroup of DPn. In this paper, we show that the semigroup Fix([n]D) is a semilattice of idempotents, and further, we obtain the cardinality and the rank of the semigroup.