On the maximal subsemigroups and relative ranks of certain ideals of finite monotone and order-decreasing partial transformations

Let [Formula: see text] be the semigroup consisting of all monotone and order-decreasing partial transformations, and let [Formula: see text] be the subsemigroup of [Formula: see text] consisting of all order-preserving and order-decreasing partial transformations on the finite chain [Formula: see text]. For [Formula: see text], let [Formula: see text] and [Formula: see text]. In this paper, we determine the maximal subsemigroups of [Formula: see text], and moreover, we show that the smallest number of elements of [Formula: see text] which, together with [Formula: see text], generate [Formula: see text] is [Formula: see text] for every [Formula: see text].