On the monoid of all order-decreasing partial transformations

A partial transformation ? on an n-element set n = {1,..., n} is called order-decreasing if x? ? x for all x ? dom(?). The set of all partial order-decreasing transformations on n forms a monoid PDn. In this paper, we determine the maximal subsemigroups as well as the maximal idempotent generated subsemigroups of PDn. Furthermore, we investigate the abundance of the ideals of PDn, and characterize the structure of the left (right) abundant principal ideal of PDn.