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.