Quasi-idempotent ranks of the proper ideals in finite symmetric inverse semigroups

Abstract

Let In and Sn be the symmetric inverse semigroup and the symmetric group on a finite chain Xn = {1, . . . , n}, respectively. Also, let In,r = {α ∈ In : |im(α)| ≤ r} for 1 ≤ r ≤ n − 1. For any α ∈ In , if α ̸= α 2 = α 4 then α is called a quasi-idempotent. In this paper, we show that the quasi-idempotent rank of In,r (both as a semigroup and as an inverse semigroup) is (n 2 ) if r = 2, and (n r ) + 1 if r ≥ 3. The quasi-idempotent rank of In,1 is n (as a semigroup) and n − 1 (as an inverse semigroup).