Defining relations of semigroups of all directed transformations of an ordered finite set

The semigroup $\mathfrak A$ of all transformations $X$ of a finite (partially) ordered set $\Omega$, such that $\alpha\le X\alpha$ for all $\alpha\in\Omega$, is considered. All possible generating sets of a $\Omega$ are elucidated. Only one of those sets is irreducible. A system of defining relations is found for that generating set.