Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

The relative rank (S : U) of a subsemigroup U of a semigroupS is the minimum size of a set V S such that U together withV generates the whole of S. As a consequence of a result ofSierpiski, it follows that for U T_X, the monoid of all self-mapsof 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 : O_X) are considered,where X is a countably infinite partially ordered set and O_Xis the endomorphism monoid of X. We show that rank(T_X : O_X) 2 if and only if either: there exists at least one elementin X which is greater than, or less than, an infinite numberof elements of X; or X has |X| connected components. Four examplesare given of posets where the minimum number of members of T_Xthat need to be adjoined to O_X to form a generating set is,respectively, 0, 1, 2 and uncountable. 2000 Mathematics SubjectClassification 08A35 (primary), 06A07, 20M20 (secondary).