J Austral Math Soc Ser A 48 pp1--24, 1990.

The Associated Order of a Preorder

John Boris Miller

Abstract

Any preorder P on a set X has an associated preorder P¢, and hence an associate sequence preorders P, P¢, P¢¢, P¢¢¢, .... The properties of this sequence are studied. When X is finite the sequence is eventually periodic with period p = 1 or p = 2. If p = 1, the eventual constant preorder is full. For 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 be in a heuristic sense be of unbounded complexity. If X is orderdense the periodicity starts at P¢.

1980 AMS Subject Classification (1985 Revision): 06A99

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Authors

John Boris Miller

Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.