On the semigroup of order-preserving partial isometries of a finite chain with restricted range

Abstract

Let In be the symmetric inverse semigroup on Xn= { 1 , 2 , ⋯ , n} , and let ODPn be the subsemigroup of In consisting of all order-preserving partial isometries of Xn. Let Y= { 1 , 2 , ⋯ , r} be a non-empty subset of Xn. Define ODPIn,r={α∈ODPn:imα⊆Y}.In this paper, we give a necessary and sufficient condition for ODPIn,r to be an inverse semigroup and characterize its Green relations. Moreover, the cardinality and the rank of ODPIn,r are investigated.