Michał Morayne

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 Sierpinski 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. ruskuc 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.

Generating continuous mappings with Lipschitz mappings

If X is a metric space, then C X and £ X denote the semigroups of continuous and Lipschitz mappings, respectively, from X to itself. The relative rank of C X modulo £ X is the least cardinality of any set U £ X where U generates C X . For a large class of separable metric spaces X we prove that the relative rank of C X modulo £ X is uncountable. When X is the Baire space N N , this rank is 8 1 . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

Topological graph inverse semigroups

Abstract To every directed graph E one can associate a graph inverse semigroup G ( E ) , where elements roughly correspond to possible paths in E. These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C ⁎ -algebras, and Toeplitz C ⁎ -algebras. We investigate topologies that turn G ( E ) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G ( E ) ∖ { 0 } must be discrete for any directed graph E. On the other hand, G ( E ) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E, G ( E ) admits a T 1 semigroup topology in which G ( E ) ∖ { 0 } is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G ( E ) in larger topological semigroups.

How to Choose the Best Twins

We consider a version of the secretary problem where each candidate has an identical twin. The aim, as in the classical problem, is to choose with the largest possible probability a top candidate, i.e., one of the best twins. We find an optimal stopping time for such a choice, the probability of success the optimal stopping time yields, and their asymptotic behavior.

A Secretary Problem with Many Lives

We consider a secretary type problem where an administrator who has only one on-line choice in m consecutive searches has to choose the best candidate in one of them.

A Ratio Inequality for Binary Trees and the Best Secretary

Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = m{S ⊆ Tn : 1n ∈ S, S ≅ T}m, and B(n; T) = m{S ⊆ Tn : 1n ∉ S, S ≅ T}m. In this note we prove that ***** insert equation here ***** for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Abstract Let Ω Ω be the semigroup of all mappings of a countably infinite set Ω . If U and V are subsemigroups of Ω Ω , then we write U ≈ V if there exists a finite subset F of Ω Ω such that the subsemigroup generated by U and F equals that generated by V and F . The relative rank of U in Ω Ω is the least cardinality of a subset A of Ω Ω such that the union of U and A generates Ω Ω . In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω . The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈ . Moreover such semigroups have relative rank 0 , 1 , 2 , or d in Ω Ω where d is the minimum cardinality of a dominating family for N N . We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in Ω Ω are 0 , 1 , 2 , and d . We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2 ℵ 0 .

Repeated Degrees in Random Uniform Hypergraphs

We prove that in a random

An Asymptotic Ratio in the Complete Binary Tree

3

Inform friends, do not inform enemies

-uniform or