Finite Distributive Concept Algebras

Ganter, Bernhard; Kwuida, Léonard

doi:10.1007/s11083-006-9045-x

Cited by 4 publications

(5 citation statements)

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“…Even if they are not always isomorphic to concept algebras (Theorem 4.1), they embed into concept algebras (Theorem 4.4). Finite distributive weakly dicomplemented lattices are isomorphic to concept algebras [GK07]. Extending results to finite weakly dicomplemented lattices in one sense and to distributive weakly dicomplemented lattices in the other are the next steps towards the representation of weakly dicomplemented lattices.…”

Section: Discussionmentioning

confidence: 92%

“…We proved (see [15] or [9]) that finite distributive weakly dicomplemented lattices are isomorphic to concept algebras. However we cannot expect all weakly dicomplemented lattices to be isomorphic to concept algebras, since concept algebras are necessary complete lattices.…”

Section: Proposition 2 Let H Be a Closure Operator On X And K A Kernementioning

confidence: 99%

“…CEP Concrete embedding problem: given a weakly dicomplemented lattice L, is there a context K △ ▽ (L) such that L embeds into the concept algebra of K △ ▽ (L)? We proved (see [Kw04] or [GK07]) that finite distributive weakly dicomplemented lattices are isomorphic to concept algebras. However we cannot expect all weakly dicomplemented lattices to be isomorphic to concept algebras, since concept algebras are necessary complete lattices.…”

Section: Weak Dicomplementationmentioning

confidence: 99%

“…The main problem we address in this paper is when a weakly dicomplemented lattice is isomorphic to a concept algebra. Characterizing concept algebras remains an open problem, but substantial results are obtained, especially in the finite case [Kw04,GK07]. The rest of this contribution is divided as follows: in Section 2 we introduce some formal definitions, give a characterization of weakly dicomplemented lattices and present several constructions of weakly dicomplemented lattices.…”

Section: Introductionmentioning

confidence: 99%

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On the isomorphism problem of concept algebras

Kwuida

Machida

2010

Ann Math Artif Intell

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Section: Discussionmentioning

confidence: 92%

Section: Proposition 2 Let H Be a Closure Operator On X And K A Kernementioning

confidence: 99%

Section: Weak Dicomplementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the isomorphism problem of concept algebras

Kwuida

Machida

2010

Ann Math Artif Intell

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“…Example 5.3 By [7,15], the only weak complementations on the direct product of chains C 2 × C 3 , with the elements denoted as in the leftmost Hasse diagram below, are:…”

Section: Remark 53 For Any Coatomic Bounded Lattice L Any J ⊆ L and Anymentioning

confidence: 99%

On Nontrivial Weak Dicomplementations and the Lattice Congruences that Preserve Them

Kwuida

Mureșan

2022

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We study the existence of nontrivial and of representable (dual) weak complementations, along with the lattice congruences that preserve them, in different constructions of bounded lattices, then use this study to determine the finite (dual) weakly complemented lattices with the largest numbers of congruences, along with the structures of their congruence lattices. It turns out that, if $$n\ge 7$$ n ≥ 7 is a natural number, then the four largest numbers of congruences of the n–element (dual) weakly complemented lattices are: $$2^{n-2}+1$$ 2 n - 2 + 1 , $$2^{n-3}+1$$ 2 n - 3 + 1 , $$5\cdot 2^{n-6}+1$$ 5 · 2 n - 6 + 1 and $$2^{n-4}+1$$ 2 n - 4 + 1 , which yields the fact that, for any $$n\ge 5$$ n ≥ 5 , the largest and second largest numbers of congruences of the n–element weakly dicomplemented lattices are $$2^{n-3}+1$$ 2 n - 3 + 1 and $$2^{n-4}+1$$ 2 n - 4 + 1 . For smaller numbers of elements, several intermediate numbers of congruences appear between the elements of these sequences.

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