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The associated order of a preorder

Part of: Ordered sets

Published online by Cambridge University Press:  09 April 2009

John Boris Miller
John Boris Miller
Affiliation:
Department of Mathematics, Monash UniversityClayton, Victoria 3168, Australia
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Abstract

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Any preorder P on a set X has an associated preorder P′, P″, P‴, … The proerties of this sequence are studied. When X is finite the sequence is eventually periodic with period P = 1 or p = 1, the eventual constant preorder is full p = 2 the possible forms which the eventual alternating order can take are examined: first, the possible combinations of components are enumerated; second, the notion of ramification at a caste is used to show that X may in a heuristic sense be of unbounded complexity. If X is orderdense the periodicity starts at P′.

Keywords

preorderpartial orderassociated ordercasteramification.

MSC classification

Secondary: 06A99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 1 , February 1990 , pp. 1 - 24
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Cameron, N. and Miller, J. B., ‘Topology and axioms of interpolation in partially ordered spaces’, J. Reine Angew. Math. 278/279 (1975), 1–13.Google Scholar
[2]Loy, R. J. and Miller, J. B., ‘Tight Riesz groups’, J. Austral. Math. 13 (1972), 224–240.CrossRefGoogle Scholar
[3]Miller, J. B., ‘Quotient groups and realization of tight Riesz groups’, J. Austral. Math. Soc. Ser. A 16 (1973), 416–430.CrossRefGoogle Scholar
[4]Miller, J. B., ‘Simultaneous lattice and topological completion of topological posets’, Compositio Math. 30 (1975), 63–80.Google Scholar
[5]Miller, J. B., ‘The order-dual of a TRL group, I’, J. Austral. Math. Soc. Ser. A 25 (1978), 129–141.CrossRefGoogle Scholar
[6]Miller, J. B., ‘Local convexity in topological lattices’, Portugal. Math. 38 (1979), 19–31.Google Scholar
[7]Miller, J. B., ‘Ramification and deramification of preordered sets’ (Analysis Paper 56, Department of Mathematics, Monash University)Google Scholar
[8]Miller, J. B., ‘Eventual periodicity of the associated sequence’ (Analysis Paper 62, Department of Mathematics, Monash University).Google Scholar
[9]Wirth, A., ‘An order determined multiattribute decision rule’ (Preprint, Graduate School of Management, University of Melbourne, 08 1986).Google Scholar