Sums, products and negations of contexts and complete lattices

• Autores: Konrad Deiters, Marcel Erné
• Localización: Algebra universalis, ISSN 0002-5240, Vol. 60, Nº. 4, 2009, págs. 469-496
• Idioma: inglés
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• Resumen

  • In Formal Concept Analysis, one associates with every context $${\mathbb{K}}$$ its concept lattice $${\mathcal{B}} {\mathbb{K}}$$, and conversely, with any complete lattice L the standard context $${\mathcal{S}}$$ 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 $${\mathcal{B}}$$ and $${\mathcal{S}}$$ on various (finite or infinite) sum and product constructions. The rules obtained confirm the �exponential� behavior of $${\mathcal{B}}$$ and the �logarithmic� behavior of $${\mathcal{S}}$$ 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.