Betweenness of partial orders

We construct a monadic second-order sentence that characterizes the ternary relations that are the betweenness relations of finite or infinite partial orders. We prove that no first-order sentence can do that. We characterize the partial orders that can be reconstructed from their betweenness relations. We propose a polynomial time algorithm that tests if a finite relation is the be-tweenness of a partial order

Road Systems and Betweenness

A road system is a collection of subsets of a set—the roads—such that every singleton subset is a road in the system and every doubleton subset is contained in a road. The induced ternary (betweenness) relation is defined by saying that a point c lies between points a and b if c is an element of every road that contains both a and b . Traditionally, betweenness relations have arisen from a plethora of other structures on a given set, reflecting intuitions that range from the order-theoretic to the geometric and topological. In this paper we initiate a study of road systems as a simple mechanism by means of which a large majority of the classical interpretations of betweenness are induced in a uniform way

A THEORY OF QUALITATIVE SIMILARITY

The central result of this paper establishes an isomorphism between two types of mathematical structures: ""ternary preorders"" and ""convex topologies."" The former are characterized by reflexivity, symmetry and transitivity conditions, and can be interpreted geometrically as ordered betweenness relations; the latter are defined as intersection-closed families of sets satisfying an ""abstract convexity"" property. A large range of examples is given. As corollaries of the main result we obtain a version of Birkhoff''s representation theorem for finite distributive lattices, and a qualitative version of the representation of ultrametric distances by indexed taxonomic hierarchies.

Into the Square: On the Complexity of Some Quadratic-time Solvable Problems

International audienceWe analyze several quadratic-time solvable problems, and we show that these problems are not solvable in truly subquadratic time (that is, in time O(n2−ϵ) for some ϵ>0), unless the well known Strong Exponential Time Hypothesis (in short, SETH) is false. In particular, we start from an artificial quadratic-time solvable variation of the k-Sat problem (already introduced and used in the literature) and we will construct a web of Karp reductions, proving that a truly subquadratic-time algorithm for any of the problems in the web falsifies SETH. Some of these results were already known, while others are, as far as we know, new. The new problems considered are: computing the betweenness centrality of a vertex (the same result was proved independently by Abboud et al.), computing the minimum closeness centrality in a graph, computing the hyperbolicity of a graph, and computing the subset graph of a collection of sets. On the other hand, we will show that testing if a directed graph is transitive and testing if a graph is a comparability graph are subquadratic-time solvable (our algorithm is practical, since it is not based on intricate matrix multiplication algorithms)

Finite Sholander Trees, Trees, and their Betweenness

We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof.Comment: 8 page

Betweenness algebras

We introduce and study a class of betweenness algebras-Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators of possibility and of suffciency.Comment: 26 pages, 2 figure

Network structure of the Wisconsin Schizotypy Scales—Short Forms: Examining psychometric filtering approaches

Schizotypy is a multidimensional construct that provides a useful framework for understanding the etiology, development, and risk for schizophrenia-spectrum disorders. Past research has applied traditional methods, such as factor analysis, to uncovering common dimensions of schizotypy. In the present study, we aimed to advance the construct of schizotypy, measured by the Wisconsin Schizotypy Scales–Short Forms (WSS-SF), beyond this general scope by applying two different psychometric network filtering approaches—the state-of-the-art approach (lasso), which has been employed in previous studies, and an alternative approach (information-filtering networks; IFNs). First, we applied both filtering approaches to two large, independent samples of WSS-SF data (ns = 5,831 and 2,171) and assessed each approach’s representation of the WSS-SF’s schizotypy construct. Both filtering approaches produced results similar to those from traditional methods, with the IFN approach producing results more consistent with previous theoretical interpretations of schizotypy. Then we evaluated how well both filtering approaches reproduced the global and local network characteristics of the two samples. We found that the IFN approach produced more consistent results for both global and local network characteristics. Finally, we sought to evaluate the predictability of the network centrality measures for each filtering approach, by determining the core, intermediate, and peripheral items on the WSS-SF and using them to predict interview reports of schizophrenia-spectrum symptoms. We found some similarities and differences in their effectiveness, with the IFN approach’s network structure providing better overall predictive distinctions. We discuss the implications of our findings for schizotypy and for psychometric network analysis more generally

Results on the Generalised Shift Graph

In the paper ‘On Chromatic Number of Infinite Graphs’ (1968), Erdős and Hajnal defined the Shift Graph to be the graph whose vertices are the n-element subsets of some totally ordered set S, regarded as increasing n-tuples, such that A = (a1, ..., an) and B = (b1, ..., bn) are neighbours iff a1 < b1 = a2 <b2 = a3 < ... < bn−1 = an < bn or the other way round. In the paper ‘On Generalised Shift Graphs’ (2014), Avart, Łuczac and Rödl extend this definition to include all possible arrangements of the ais and bis, known as ‘types’. In this thesis, we will consider a selection of these types and study the corresponding graphs. All the types we consider will be written as 1^k3^m2^k, where k + m = n, which means that the final m entries of (a1, ..., an) are identified with the first m entries of (b1, ..., bn). Such a graph with totally ordered set S and type 1^k3^m2^k is denoted G(S,1^k3^m2^k). There are two related questions here. One is when the (undirected) graphs G(S,1^k3^m2^k) and G(S',1^k3^m2^k) are distinct (non-isomorphic) for distinct linear orderings S, S'. The other is to what extent we can recognise S inside the graph (called ‘reconstruction’). A positive solution to the latter also yields one for the former, since if we can recognise S in its graph, and S′ in its graph, and they are distinct, then so must the graphs be. We focus on these main cases: S is finite, S is an ordinal, S is a more general totally ordered set. The tools available for reconstruction depend on whether S is a total ordering, a dense total ordering, or an ordinal. There are additional technical complications in the case where S has endpoints, and similarly for S containing relatively small finite segments. Since these graphs are undirected, we expect in general only to recover a linear ordering up to order reversal. The natural notion here is of ‘linear betweenness’, and we spend some time studying linear betweenness relations in their own right, also considering the induced relations on n-tuples. Betweenness relations on n-tuples correspond to shift graphs of the special form G(S,1^n2^n) (i.e. in which no identifications are made). The main contribution of the thesis is to show how it is possible in many instances to reconstruct the underlying linear order (often just up to order-reversal) from the generalized shift graph. A typical example of this is Theorem 4.4. The techniques are to employ graph-theoretical features of the relevant shift graph, such as co-cliques or pairs of co-cliques fulfilling various conditions to ‘recognize’ points and relations of the underlying linear order. There are many variants depending on the precise circumstances (dense or not, with or without endpoints, well-ordered, only partially ordered). We show that for ordinals α and β, if G(α,1^k3^m2^k) is isomorphic to G(β,1^k3^m2^k) then α = β. Note that the fact that (in the infinite case) α is not isomorphic to its reversed ordering means that the betweenness relation is enough to give us the ordering. This result does not necessarily extend to all total orderings in full generality, but we obtain many results. A suite of techniques is used, which may be adapted suitably depending on circumstances, endpoints or not, density, or finiteness. In a more open-ended chapter, we generalise as much of the material for total orders to partial orders, the easiest case being that of trees. Work by Rubin [15] considers reconstruction in a slightly different sense: that a structure can be reconstructed from its automorphism group. So we have two ‘levels’ of reconstruction: of the graph from its automorphism group, and then if possible of the underlying total order from the graph. With this in mind, we study the automorphism groups of many of the graphs arising, managing in several cases to give quite explicit descriptions, so answering Rubin’s reconstruction question - i.e. whether or not a structure can be ‘re- constructed’ from its automorphism group (as in for example [17]) - where possible. For instance, we show that it is possible to determine S from Aut(G(S,132)) if and only if G(S, 132) contains no two points sharing exactly the same neighbour sets. Finally we return to colouring questions as in the original paper of Erdős and Hajnal, and show that the chromatic number of G(κ, 132) is equal to κ for any strong limit cardinal κ