On the semigroups of partial one-to-one order-decreasing finite transformations

Let In be the symmetric inverse semigroup on X_n = {1,…,n}, let SI_n be the subsemigroup of strictly partial one-to-one self-maps of X_n and let [formula-omitted] be the semigroup of all partial one-to-one decreasing maps including the empty or zero map of X_n. In this paper it is shown that In is an (irregular, for n ≥ 2) type A semigroup with n D-classes and d=g Further, it is shown that In- is generated by the n(n + 1)/2 quasi-idempotents in J_n-1.