Listing the subsets of Xn to build a (minimal) quasi-idempotent generating set of In,r

Let [Formula: see text] be the symmetric inverse semigroup on the finite chain [Formula: see text] and let [Formula: see text] for [Formula: see text]. A quasi-idempotent element is an element [Formula: see text] with the property that [Formula: see text]. In this paper, we obtain a useful method by listing the subsets of [Formula: see text] to build a (minimal) quasi-idempotent generating set of [Formula: see text] both as a semigroup and also as an inverse semigroup for [Formula: see text] and [Formula: see text].