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Betweenness relations and cycle-free partial orders

Published online by Cambridge University Press:  24 October 2008

J. K. Truss
J. K. Truss
Affiliation:
Department of Pure Mathematics, University of Leeds, LS2 9JT
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Extract

The intuition behind the notion of a cycle-free partial order (CFPO) is that it should be a partial ordering (X, ≤ ) in which for any sequence of points (x0, x1;…, xn–1) with n ≤ 4 such that xi is comparable with xi+1 for each i (indices taken modulo n) there are i and j with j ╪ i, i + 1 such that xj lies between xi and xi+1. As its turn out however this fails to capture the intended class, and a more involved definition, in terms of the ‘Dedekind–MacNeille completion’ of X was given by Warren[5]. An alternative definition involving the idea of a betweenness relation was proposed by P. M. Neumann [1]. It is the purpose of this paper to clarify the connections between these definitions, and indeed between the ideas of semi-linear order (or ‘tree’), CFPO, and the betweenness relations described in [1]. In addition I shall tackle the issue of the axiomatizability of the class of CFPOs.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 4 , May 1996 , pp. 631 - 643
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Adeleke, S. A. and Neumann, P. M.. Relations related to betweenness: their structure and automorphisms, in preparation.Google Scholar
[2]Droste, M.. Structure of partially ordered sets with transitive automorphism groups. Memoirs of the American Math. Soc. 57 (1985), No. 334.Google Scholar
[3]MacNeille, H. M.. Partially ordered sets. Trans. Amer. Math. Soc. 42 (1937), 416–460.CrossRefGoogle Scholar
[4]Rubin, M.. The reconstruction of trees from their automorphism groups. Contemporary Mathematics 151 (American Math. Soc., 1993).CrossRefGoogle Scholar
[5]Warben, R.. The structure of k-CS-transitive cycle-free partial orders, to appear in the Memoirs of the American Math. Soc.Google Scholar