Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 197-221 |
| Seitenumfang | 25 |
| Fachzeitschrift | ORDER |
| Jahrgang | 8 |
| Ausgabenummer | 2 |
| Publikationsstatus | Veröffentlicht - Juni 1991 |
Abstract
By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called L-generators resp. C-subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) L-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable L, the complete lattices with a smallest join-dense L-subbasis consisting of L-primes are the L-completions of principally separated posets.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Algebra und Zahlentheorie
- Mathematik (insg.)
- Geometrie und Topologie
- Informatik (insg.)
- Theoretische Informatik und Mathematik
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in: ORDER, Jahrgang 8, Nr. 2, 06.1991, S. 197-221.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Bigeneration in complete lattices and principal separation in ordered sets
AU - Erné, Marcel
PY - 1991/6
Y1 - 1991/6
N2 - By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called L-generators resp. C-subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) L-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable L, the complete lattices with a smallest join-dense L-subbasis consisting of L-primes are the L-completions of principally separated posets.
AB - By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called L-generators resp. C-subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) L-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable L, the complete lattices with a smallest join-dense L-subbasis consisting of L-primes are the L-completions of principally separated posets.
KW - (Complete) lattice
KW - AMS subject classifications (1991): 06A23, 06B15, 06D05, 54D15
KW - join-(meet-)dense
KW - join-(meet-)irreducible
KW - join-(meet-)prime
KW - L-completion
KW - L-generator
KW - L-subbasis
KW - principal separation
KW - subset selection
UR - http://www.scopus.com/inward/record.url?scp=0007544829&partnerID=8YFLogxK
U2 - 10.1007/BF00383404
DO - 10.1007/BF00383404
M3 - Article
AN - SCOPUS:0007544829
VL - 8
SP - 197
EP - 221
JO - ORDER
JF - ORDER
SN - 0167-8094
IS - 2
ER -