Standard completions for quasiordered sets

authored by

Marcel Erné, Gerhard Wilke

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: (i) V-distributive completions, (ii) Completely distributive completions, (iii) A-completions (i.e. standard completions which are completely distributive algebraic lattices), (iv) 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 C{black-letter}, then Q must be a Boolean lattice and C{black-letter} its MacNeille completion.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

SEMIGROUP FORUM

Volume

27

Pages

351-376

No. of pages

26

ISSN

0037-1912

Publication date

12.1983

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Electronic version(s)

https://doi.org/10.1007/BF02572747 (Access: Closed)