Transit functions provide a unified approach to study notions of intervals, convexities, and betweenness. Recently, their scope has been extended to certain set systems associated with clustering. We characterize here the class of set systems that correspond to k-ary monotonic transit functions. Convexities form a subclass and are characterized in terms of transit functions by two additional axioms. We then focus on axiom systems associated with weak hierarchies as well as other generalizations of hierarchical set systems. c b This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Set systems identified by transit functions Throughout this contribution, V is a finite, non-empty set. Consider an arbitrary set system X ⊆ 2 V and the function R X (x, y) ={A ∈ X | x, y ∈ A}.(2.1)

k-point crossover operators and their recombination sets are studied from different perspectives. We show that transit functions of k-point crossover generate, for all k > 1, the same convexity as the interval function of the underlying graph. This settles in the negative an open problem by Mulder about whether the geodesic convexity of a connected graph G is uniquely determined by its interval function I. The conjecture of Gitchoff and Wagner that for each transit set R k (x, y) distinct from a hypercube there is a unique pair of parents from which it is generated is settled affirmatively. Along the way we characterize transit functions whose underlying graphs are Hamming graphs, and those with underlying partial cube graphs. For general values of k it is shown that the transit sets of k-point crossover operators are the subsets with maximal Vapnik-Chervonenkis dimension. Moreover, the transit sets of k-point crossover on binary strings form topes of uniform oriented matroid of VC-dimension k + 1. The Topological Representation Theorem for oriented matroids therefore implies that k-point crossover operators can be represented by pseudosphere arrangements. This provides the tools necessary to study the special case k = 2 in detail.

No abstract

No abstract

n-ary transit functions are introduced as a generalization of binary (2-ary) transit functions. We show that they can be associated with convexities in natural way and discuss the Steiner convexity as a natural n-ary generalization of geodesicaly convexity. Furthermore, we generalize the betweenness axioms to n-ary transit functions and discuss the connectivity conditions for underlying hypergraph. Also n-ary all paths transit function is considered.