Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 538-580 |
| Seitenumfang | 43 |
| Fachzeitschrift | Algebra universalis |
| Jahrgang | 30 |
| Ausgabenummer | 4 |
| Publikationsstatus | Veröffentlicht - Dez. 1993 |
Abstract
We study several kinds of distributivity for concept lattices of contexts. In particular, we find necessary and sufficient conditions for a concept lattice to be (1) distributive, (2) a frame (locale, complete Heyting algebra), (3) isomorphic to a topology, (4) completely distributive, (5) superalgebraic (i.e., algebraic and completely distributive). In cases (2), (4) and (5), our criteria are first order statements on objects and attributes of the given context. Several applications are obtained by considering the completion by cuts and the completion by lower ends of a quasiordered set as special types of concept lattices. Various degrees of distributivity for concept lattices are expressed by certain "separation axioms" for the underlying contexts. Passing to complementary contexts makes some statements and proofs more elegant. For example, it leads to a one-to-one correspondence between completely distributive lattices and so-called Cantor lattices, and it establishes an equivalence between partially ordered sets and doubly founded reduced contexts with distributive concept lattices.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Algebra und Zahlentheorie
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in: Algebra universalis, Jahrgang 30, Nr. 4, 12.1993, S. 538-580.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Distributive laws for concept lattices
AU - Erné, Marcel
PY - 1993/12
Y1 - 1993/12
N2 - We study several kinds of distributivity for concept lattices of contexts. In particular, we find necessary and sufficient conditions for a concept lattice to be (1) distributive, (2) a frame (locale, complete Heyting algebra), (3) isomorphic to a topology, (4) completely distributive, (5) superalgebraic (i.e., algebraic and completely distributive). In cases (2), (4) and (5), our criteria are first order statements on objects and attributes of the given context. Several applications are obtained by considering the completion by cuts and the completion by lower ends of a quasiordered set as special types of concept lattices. Various degrees of distributivity for concept lattices are expressed by certain "separation axioms" for the underlying contexts. Passing to complementary contexts makes some statements and proofs more elegant. For example, it leads to a one-to-one correspondence between completely distributive lattices and so-called Cantor lattices, and it establishes an equivalence between partially ordered sets and doubly founded reduced contexts with distributive concept lattices.
AB - We study several kinds of distributivity for concept lattices of contexts. In particular, we find necessary and sufficient conditions for a concept lattice to be (1) distributive, (2) a frame (locale, complete Heyting algebra), (3) isomorphic to a topology, (4) completely distributive, (5) superalgebraic (i.e., algebraic and completely distributive). In cases (2), (4) and (5), our criteria are first order statements on objects and attributes of the given context. Several applications are obtained by considering the completion by cuts and the completion by lower ends of a quasiordered set as special types of concept lattices. Various degrees of distributivity for concept lattices are expressed by certain "separation axioms" for the underlying contexts. Passing to complementary contexts makes some statements and proofs more elegant. For example, it leads to a one-to-one correspondence between completely distributive lattices and so-called Cantor lattices, and it establishes an equivalence between partially ordered sets and doubly founded reduced contexts with distributive concept lattices.
KW - (complete) lattice
KW - (completely) distributive
KW - (doubly) founded
KW - AMS Mathematics Subject Classification 1991: 06A15, 06A23, 06D05, 06D10
KW - closure system
KW - completion by cuts
KW - concept
KW - Context
KW - frame
KW - join- and meet-dense
KW - join- and meet-prime
UR - http://www.scopus.com/inward/record.url?scp=0042714737&partnerID=8YFLogxK
U2 - 10.1007/BF01195382
DO - 10.1007/BF01195382
M3 - Article
AN - SCOPUS:0042714737
VL - 30
SP - 538
EP - 580
JO - Algebra universalis
JF - Algebra universalis
SN - 0002-5240
IS - 4
ER -