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 ⊆ S such that U together with V generates the whole of S. As a consequence of a result by 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, 2 or uncountable. In this paper we consider the relative ranks rank(TX : OX), 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. We give four examples of posets where the respective minimum number of members of TX that need to be adjoined to OX to form a generating set is 0, 1, 2 and uncountable. 1. Background & introduction Let X be a set and let ≤ be a partial order (a reflexive, anti-symmetric and transitive relation) on X. Such a pair (X,≤) is called a partially ordered set, or poset for short. Where there is no possibility of confusion we will write X instead of (X,≤). A mapping α in TX , the monoid of all mappings from X to X, is called order preserving if whenever x ≤ y we have xα ≤ yα. We adhere to the convention of writing mappings on the right and composing from left to right. We may also refer to order preserving mappings as endomorphisms of (X,≤). We denote the set of all order preserving maps by OX and remark that OX is a monoid under composition of mappings. When X is a linearly ordered set with, say, n elements the monoid OX has order ( 2n−1 n−1 ) and the minimum size of a generating set (called the rank) of OX is n, see [6] and [12]. Determining even the simplest properties of OX when X is finite but not linearly ordered proves difficult. A classical topic in the study of monoids of endomorphisms of ‘structures’ is the characterisation of such ‘structures’ whose endomorphism monoid satisfies a particular property. Thus the importance of monoids of order preserving mappings goes beyond the theory of ordered sets. In [7] it was shown that every monoid is isomorphic to the monoid of all maximum and minimum preserving endomorphisms of some lattice. Since OX always contains all the constant mappings on X (i.e. right zeros) it follows that not every monoid is isomorphic to the monoid of endomorphisms of some poset X. In [1] necessary and sufficient conditions on the set X are given for OX to be regular. A characterization of those posets X for which OX is abundant is given in [2]. In this paper we are concerned with determining a rank property, to be defined shortly, of the monoid of endomorphisms of an arbitrary countably infinite poset. 2000 Mathematics Subject Classification 08A35 (primary), 06A07 & 20M20 (secondary). 2 p.m. higgins, j.d. mitchell, m. morayne and n. ruškuc Such properties were considered in [10] for order preserving mappings of linearly ordered sets. The ‘classical’ rank of a semigroup S is the minimum cardinality of any generating set for S. For example, the rank of TX when X is a finite set is 3; see [13, Exercise 1.9.7]. When considering finitely generated semigroups this definition of rank obviously yields some information about that semigroup. However, not all semigroups are finitely generated and, in fact, when a semigroup S is uncountable its rank is |S|, which does not give us any new information. Here we want to measure a subset A of a fixed semigroup S with respect to S. In order to do this we introduce the so-called relative rank of S with respect to A. For a subset A ⊆ S the relative rank of S modulo A is the minimum cardinality of any set B such that 〈A∪B 〉 = S (i.e. the semigroup generated by A∪B is S); we denote this cardinal by rank(S : A). Similar properties were considered in the context of groups in [4], [5] and [14] where subgroups of symmetric groups were studied. Relative ranks of subsemigroups of the full transformation monoid were first considered in [8] and [9]. This study was continued in [11] and was extended to subsemigroups of the monoid of all binary relations and the symmetric inverse monoid (i.e. all injective partial mappings) over infinite sets. Here we are concerned with finding the relative rank of TX modulo OX where X is a countably infinite poset. It is known that rank(TX : OX) is either 0, 1, 2, or uncountable. We find a necessary and sufficient condition on X for rank(TX : OX) ≤ 2 to hold.