Details
| Original language | English |
|---|---|
| Pages (from-to) | 469-496 |
| Number of pages | 28 |
| Journal | Algebra universalis |
| Volume | 60 |
| Issue number | 4 |
| Publication status | Published - 14 Apr 2009 |
Abstract
In Formal Concept Analysis, one associates with every context K its concept lattice BK, and conversely, with any complete lattice L the standard context S L, constituted by the join-irreducible elements as 'objects', the meet-irreducible elements as 'attributes', and the incidence relation induced by the lattice order. We investigate the effect of the operators B and S on various (finite or infinite) sum and product constructions. The rules obtained confirm the 'exponential' behavior of B and the 'logarithmic' behavior of S with respect to cardinal operations but show a 'linear' behavior on ordinal sums. We use these results in order to establish several forms of De Morgan's law for the lattice-theoretical negation operator, associating with any complete lattice the concept lattice of the complementary standard context.
Keywords
- Complete lattice, Concept, Context, De Morgan's law, Direct product, Horizontal sum, Negation, Tensor product, Vertical sum
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Algebra universalis, Vol. 60, No. 4, 14.04.2009, p. 469-496.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Sums, products and negations of contexts and complete lattices
AU - Deiters, Konrad
AU - Erné, Marcel
PY - 2009/4/14
Y1 - 2009/4/14
N2 - In Formal Concept Analysis, one associates with every context K its concept lattice BK, and conversely, with any complete lattice L the standard context S L, constituted by the join-irreducible elements as 'objects', the meet-irreducible elements as 'attributes', and the incidence relation induced by the lattice order. We investigate the effect of the operators B and S on various (finite or infinite) sum and product constructions. The rules obtained confirm the 'exponential' behavior of B and the 'logarithmic' behavior of S with respect to cardinal operations but show a 'linear' behavior on ordinal sums. We use these results in order to establish several forms of De Morgan's law for the lattice-theoretical negation operator, associating with any complete lattice the concept lattice of the complementary standard context.
AB - In Formal Concept Analysis, one associates with every context K its concept lattice BK, and conversely, with any complete lattice L the standard context S L, constituted by the join-irreducible elements as 'objects', the meet-irreducible elements as 'attributes', and the incidence relation induced by the lattice order. We investigate the effect of the operators B and S on various (finite or infinite) sum and product constructions. The rules obtained confirm the 'exponential' behavior of B and the 'logarithmic' behavior of S with respect to cardinal operations but show a 'linear' behavior on ordinal sums. We use these results in order to establish several forms of De Morgan's law for the lattice-theoretical negation operator, associating with any complete lattice the concept lattice of the complementary standard context.
KW - Complete lattice
KW - Concept
KW - Context
KW - De Morgan's law
KW - Direct product
KW - Horizontal sum
KW - Negation
KW - Tensor product
KW - Vertical sum
UR - http://www.scopus.com/inward/record.url?scp=67349264449&partnerID=8YFLogxK
U2 - 10.1007/s00012-009-2141-1
DO - 10.1007/s00012-009-2141-1
M3 - Article
AN - SCOPUS:67349264449
VL - 60
SP - 469
EP - 496
JO - Algebra universalis
JF - Algebra universalis
SN - 0002-5240
IS - 4
ER -