Rank properties of endomorphisms of infinite partially ordered sets

The relative rank rank(S : U) of a subsemigroup U of a semigroup S is the minimum size of a set V subset of S such that U together with V generates the whole of S. As a consequence of a result of Sierpinski, it follows that for U <= T-X, the monoid of all self-maps of an infinite set X, rank(T-X : U) is either 0, 1 or 2, or uncountable. In this paper, the relative ranks rank(T-X : Ox) are considered, where X is a countably infinite partially ordered set and Ox is the endomorphism monoid of X. We show that rank(T-X : O-X) <= 2 if and only if either: there exists at least one element in X which is greater than, or less than, an infinite number of elements of X; or X has vertical bar X vertical bar connected components. Four examples are given of posets where the minimum number of members of T-X that need to be adjoined to O-X to form a generating set is, respectively, 0, 1, 2 and uncountable.