Details
| Original language | English |
|---|---|
| Pages (from-to) | 91-111 |
| Number of pages | 21 |
| Journal | Discrete mathematics |
| Volume | 181 |
| Issue number | 1-3 |
| Publication status | Published - 15 Feb 1998 |
Abstract
We introduce the negation CL of a complete lattice L as the concept lattice of the complementary context (JL,ML, ≰), formed by the join-irreducible elements as objects and the meet-irreducible elements as attributes. We show that the double negation CCL is always order-embeddable in L, and that for finite lattices, the sequence (CnL)n∈ω runs into a 'flip-flop' (i.e., CnL ≃ Cn+2L for some n). Using vertical sums, we provide constructions of lattices which are isomorphic or dually isomorphic to their own negation. The only finite distributive examples among such 'self-negative' or 'self-contrapositive' lattices are vertical sums of four-element Boolean lattices. Explicitly, we determine all self-negative and all self-contrapositive lattices with less than 11 points.
Keywords
- Complement, Concept, Context, Irreducible element, Lattice, Negation
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Mathematics(all)
- Discrete Mathematics and Combinatorics
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In: Discrete mathematics, Vol. 181, No. 1-3, 15.02.1998, p. 91-111.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Negations and contrapositions of complete lattices
AU - Deiters, K.
AU - Erné, M.
PY - 1998/2/15
Y1 - 1998/2/15
N2 - We introduce the negation CL of a complete lattice L as the concept lattice of the complementary context (JL,ML, ≰), formed by the join-irreducible elements as objects and the meet-irreducible elements as attributes. We show that the double negation CCL is always order-embeddable in L, and that for finite lattices, the sequence (CnL)n∈ω runs into a 'flip-flop' (i.e., CnL ≃ Cn+2L for some n). Using vertical sums, we provide constructions of lattices which are isomorphic or dually isomorphic to their own negation. The only finite distributive examples among such 'self-negative' or 'self-contrapositive' lattices are vertical sums of four-element Boolean lattices. Explicitly, we determine all self-negative and all self-contrapositive lattices with less than 11 points.
AB - We introduce the negation CL of a complete lattice L as the concept lattice of the complementary context (JL,ML, ≰), formed by the join-irreducible elements as objects and the meet-irreducible elements as attributes. We show that the double negation CCL is always order-embeddable in L, and that for finite lattices, the sequence (CnL)n∈ω runs into a 'flip-flop' (i.e., CnL ≃ Cn+2L for some n). Using vertical sums, we provide constructions of lattices which are isomorphic or dually isomorphic to their own negation. The only finite distributive examples among such 'self-negative' or 'self-contrapositive' lattices are vertical sums of four-element Boolean lattices. Explicitly, we determine all self-negative and all self-contrapositive lattices with less than 11 points.
KW - Complement
KW - Concept
KW - Context
KW - Irreducible element
KW - Lattice
KW - Negation
UR - http://www.scopus.com/inward/record.url?scp=0041856407&partnerID=8YFLogxK
U2 - 10.1016/S0012-365X(97)00047-2
DO - 10.1016/S0012-365X(97)00047-2
M3 - Article
AN - SCOPUS:0041856407
VL - 181
SP - 91
EP - 111
JO - Discrete mathematics
JF - Discrete mathematics
SN - 0012-365X
IS - 1-3
ER -