# Cycle Orders

DOI: 10.1023/A:1006381208272

- Cite this article as:
- Fishburn, P.C. & Woodall, D.R. Order (1999) 16: 149. doi:10.1023/A:1006381208272

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## Abstract

Let *X*, *T* and *C* be, respectively, a finite set with at least three points, a set of ordered triples of distinct points from *X*, and a cyclic ordering of the points in *X*. Define *T*\( \subseteq \)*C* to mean that, for every *a b c* ∈ *T*, the elements *a*, *b*, *c* occur in that cyclic order in *C*, and let C(*T*) denote the set of cyclic orderings of *X* for which *T*\( \subseteq \)*C*. We say that *T* is *noncyclic* if C(*T*) is empty, *cyclic* if C(*T*) is nonempty, *uniquely cyclic* if | C(*T*) | = 1, a *partial cycle order* if it is cyclic and *T* ={*a b c* :{*a b c*} \( \subseteq \)*C* for all *C* ∈ C(*T*)}, and a *total cycle order* if it is a uniquely cyclic partial cycle order. Many years ago E. V. Huntington axiomatized total cycle orders by independent necessary and sufficient conditions on *T*. The present paper studies the more relaxed structures of cyclic *T* sets and partial cycle orders. We focus on conditions for cyclicity, a theory of cycle dimension of partial cycle orders, and extremal problems that address combinatorial structures of *T* sets.

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