RANK PROPERTIES OF ENDOMORPHISMS OF INFINITE PARTIALLY ORDERED SETS
Peter M. Higgins, James D. Mitchell, Michał Morayne, Nik Ruškuc · Bulletin of the London Mathematical Society · 2006
The relative rank (S : U) of a subsemigroup U of a semigroup S is the minimum size of a set V ⊆ S such that U together with V generates the whole of S. As a consequence of a result of Sierpiński, it follows that for U ⩽ TX, the monoid of all self-maps of an infinite set X, rank(TX : U) is either 0, 1 or 2, or uncountable. In this paper, the relative ranks rank(TX : OX) are considered, where X is a countably infinite partially ordered set and OX is the endomorphism monoid of X. We show that rank(TX : OX) ⩽ 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 |X| connected components. Four examples are given of posets where the minimum number of members of TX that need to be adjoined to OX to form a generating set is, respectively, 0, 1, 2 and uncountable. 2000 Mathematics Subject Classification 08A35 (primary), 06A07, 20M20 (secondary).