Ivan Rival

A structure theory for ordered sets

The theory of ordered sets lies at the confluence of several branches of mathematics including set theory, lattice theory, combinatorial theory, and even aspects of modern operations research. While ordered sets are often peripheral to the mainstream of any of these theories there arise, from time to time, problems which are order-theoretic in substance. The aim of this work is to fashion a classification scheme for ordered sets which aimed at providing a unified vantage point for some of the problems encountered with ordered sets. This classification scheme is based on a structure theory much akin to the familiar subdirect representation theory so useful in general algebra. The novelry of the structure theory lies in the importance that we attach to, and the widespread use that we make of, the concept of retract. At present, some vindication for our classification scheme can be found by examining its effectiveness for totally ordered sets (e.g., well-ordered sets, the rationals, reals, etc.) as well as for those finite ordered sets that arise commonly in combinatorial investigations (e.g., crowns and fences).

Fixed-edge theorem for graphs with loops

Let G be an undirected graph without multiple edges and with a loop at every vertex—the set of edges of G corresponds to a reflexive and symmetric binary relation on its set of vertices. Then every edge-preserving map of the set of vertices of G to itself fixes an edge [{f(a), f(b)} = {a, b} for some edge (a, b) of G] if and only if (i) G is connected, (ii) G contains no cycles, and (iii) G contains no infinte paths. The proof is conerned with those subgraphs H of a graph G for which there is an edge-preserving map f of the set of vertices of G onto the set of vertices of H and satisfying f(a) = a for each vertex a of H.

Bipartite graphs, upward drawings, and planarity

Abstract We show that a bipartite digraph admits an upward drawing, i.e., a planar drawing with the additional constraint that all the edges flow in the same direction if and only if it is planar. This result finds applications both in the field of automatic graph layout and in the field of ordered sets.

The smallest graph variety containing all paths

This paper is inspired by two problems. Chatuctetize the retpacts of a graph. In this respect this paper continues the investigations of [2-6]. Ck&fi gruphs u.ccor&ng to their retracts. In this respect this paper begins the classification theory for graphs, initiated by D. Duffus and I. Eva1 in [l] for ordered sets. For a graph G let V(G) denote its vertex set and E(G) E V(G) x V(G) its edge set. A graph H is a retract of the graph G if there are edge-preserving maps, f of V(H) to V(G).. and g of V(G) to V(H), satisfying g 0 f(u) = 27 for each v E V(H). Our interest in this paper is with the retracts of a reflexive graph-an undirected graph without any multiple edges but with a loop at every vertex (cf. Fig. l(a)). In [2] we have shown, for instance, that in a reflexive graph every cycle of minimum order and e:uery isometric tree is a retract. For graphs G and H the direct product G x H is the graph with vertex set V(G) x V(H) and edge set consisting of all pairs ((a, x), (6, y)) where (a, 6) E E(G) and (x, y)~ .E(H) (cf. Fig. l(b)). A representation of a reflexive graph G is a family (Gi 1 i E I) of reflexive graphs such that each Gi is a retract of G and G is itself a retract of the direct product ni,, Gi. (See Fig. 2.) G is imeducible if, for every representation (Gi 1 i E I) of G, G is a retract of Gi for some i E I; otherwise, G is reducible. A path P is a graph whose vertex set consists of a sequence uo, u1, u2, ’ , %, l ’ of distinct vertices and (a,, &) E E(P), for each i = 1,2,... ; the length I(P) of a finite path P = (uO, al, u2, . . . , a,,:) is a, where n 3 1.

Chains, antichains, and fibres

Abstract Call a subset of an ordered set a fibre if it meets every maximal antichain. We prove several instances of the conjecture that, in an ordered set P without splitting elements, there is a subset F such that both F and P − F are fibres. For example, this holds in every ordered set without splitting points and in which each chain has at most four elements. As it turns out several of our results can be cast more generally in the language of graphs from which we may derive “complementary” results about cutsets of ordered sets, that is, subsets which meet every maximal chain. One example is this: In a finite graph G every minimal transversal is independent if and only if G contains no path of length three.

The rank of a distributive lattice

Abstract The rank of a partial ordering P is the maximum size of an irredundant family of linear extensions of P whose intersection is P . A simple relationship is established between the rank of a finite distributive lattice and its subset of join irreducible elements.

Representing orders on the plane by translating convex figures

Given a finite collection of disjoint, convex figures on the plane, is it possible to assign to each a single direction of motion so that this collection of figures may be separated, through an arbitrary large distance, by translating each figure one at a time, along its assigned direction? We present a computational model for this separability problem based on the theory of ordered sets.

Graphical Data Structures for Ordered Sets

Ordered sets occur widely in computation, in scheduling, in sorting, in social choice, and even in geography. For some years research on these themes has focussed first on combinatorial optimization and then on “algorithmics”. Important advances have been made both at practical and, at theoretical levels. There is little doubt that the modern mathematical theory of ordered sets owes much of its vitality to these recent developments. While some of the problems remain exceedingly difficult, such as the “three-machine scheduling problem”, attention is shifting from the usual optimization themes to data structures; indeed, there is emerging a need for efficient data structures to code and store ordered sets. Among these data structures, graphical ones are coming to play a decisive role, for instance, in problems in which decisions must be made from among alternatives ranked according to precedence or preference relations.

Reading, Drawing, and Order

The modern theoretical computer science literature is preoccupied with efficient data structures to code and store ordered sets. Among these data structures, graphical ones play a decisive role especially in decision-making problems. Choices must be made, from among alternatives ranked hierarchically according to precedence or preference. Loosely speaking, graphical data structures must be drawn in order that they may be easily read.

Spanning retracts of a partially ordered set

Two general kinds of subsets of a partially ordered set P are always retracts of P: (1) every maximal chain of P is a retract; (2) in P, every isometric, spanning subset of length one with no crowns is a retract. It follows that in a partially ordered set P with the fixed point property, every maximal chain of P is complete and every isometric, spanning fence of P is finite.