Details
| Original language | English |
|---|---|
| Pages (from-to) | 351-376 |
| Number of pages | 26 |
| Journal | SEMIGROUP FORUM |
| Volume | 27 |
| Issue number | 1 |
| Publication status | Published - Dec 1983 |
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.
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: SEMIGROUP FORUM, Vol. 27, No. 1, 12.1983, p. 351-376.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Standard completions for quasiordered sets
AU - Erné, Marcel
AU - Wilke, Gerhard
PY - 1983/12
Y1 - 1983/12
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=51249182054&partnerID=8YFLogxK
U2 - 10.1007/BF02572747
DO - 10.1007/BF02572747
M3 - Article
AN - SCOPUS:51249182054
VL - 27
SP - 351
EP - 376
JO - SEMIGROUP FORUM
JF - SEMIGROUP FORUM
SN - 0037-1912
IS - 1
ER -