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The associated order of a preorder

  • John Boris Miller (a1)

Abstract

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′.

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References

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[1]Cameron, N. and Miller, J. B., ‘Topology and axioms of interpolation in partially ordered spaces’, J. Reine Angew. Math. 278/279 (1975), 113.
[2]Loy, R. J. and Miller, J. B., ‘Tight Riesz groups’, J. Austral. Math. 13 (1972), 224240.
[3]Miller, J. B., ‘Quotient groups and realization of tight Riesz groups’, J. Austral. Math. Soc. Ser. A 16 (1973), 416430.
[4]Miller, J. B., ‘Simultaneous lattice and topological completion of topological posets’, Compositio Math. 30 (1975), 6380.
[5]Miller, J. B., ‘The order-dual of a TRL group, I’, J. Austral. Math. Soc. Ser. A 25 (1978), 129141.
[6]Miller, J. B., ‘Local convexity in topological lattices’, Portugal. Math. 38 (1979), 1931.
[7]Miller, J. B., ‘Ramification and deramification of preordered sets’ (Analysis Paper 56, Department of Mathematics, Monash University)
[8]Miller, J. B., ‘Eventual periodicity of the associated sequence’ (Analysis Paper 62, Department of Mathematics, Monash University).
[9]Wirth, A., ‘An order determined multiattribute decision rule’ (Preprint, Graduate School of Management, University of Melbourne, 08 1986).
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