On the Semigroup of Partial Isometries of a Finite Chain

Let ℐ_n be the symmetric inverse semigroup on X_n = {1, 2,…, n}, and let 𝒟𝒫_n and 𝒪𝒟𝒫_n be its subsemigroups of partial isometries and of order-preserving partial isometries of X_n, respectively. In this article, we investigate the cycle structure of a partial isometry and characterize the Green's relations on 𝒟𝒫_n and 𝒪𝒟𝒫_n. We show that 𝒪𝒟𝒫_n is a 0-E-unitary inverse semigroup. We also investigate the ranks of 𝒟𝒫_n and 𝒪𝒟𝒫_n.