Standard completions for quasiordered sets |
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Authors: | Marcel Erné Gerhard Wilke |
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Institution: | 1. Institut für Mathematik, Universit?t Hannover, Welfengarten 1, 3000, Hannover 1, West Germany
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Abstract: | A standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators: - V-distributive completions,
- Completely distributive completions,
- A-completions (i.e. standard completions which are completely distributive algebraic lattices),
- Boolean completions.
Moreover, completely distributive completions are described by certain idempotent relations, and the A-completions are shown to be in one-to-one correspondence with the join-dense subsets of Q. If a pseudocomplemented meet-semilattice Q has a Boolean completion ?, then Q must be a Boolean lattice and ? its MacNeille completion. |
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