Meet-Irreducible Elements in Implicative Lattices

Smith, Dorothy P.

doi:10.2307/2037894

Cited by 5 publications

(5 citation statements)

References 8 publications

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“…Using the well-known characterization of the relative pseudo-complement in terms of ∧irreducible elements [15], we have that given A,…”

Section: Relative Pseudo-complement and Pseudo-differencementioning

confidence: 99%

“…14 All these algorithms have been implemented in LaMa4J (Lattice Manipulation for Java), a free Java library supporting computation in lattices. 15 An interesting open question is that relative to the width of the lattice E n , that is, the cardinality of a maximum antichain. The sequence of the first few widths of E n for n = 0, 1, .…”

Section: An Elementary Characterization Of the Relative Pseudo-comple...mentioning

confidence: 99%

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On the Lattice of Antichains of Finite Intervals

Boldi

Vigna

2016

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Motivated by applications to information retrieval, we study the lattice of antichains of finite intervals of a locally finite, totally ordered set. Intervals are ordered by reverse inclusion; the order between antichains is induced by the lower set they generate. We discuss in general properties of such antichain completions; in particular, their connection with Alexandrov completions. We prove the existence of a unique, irredundant ∧-representation by ∧-irreducible elements, which makes it possible to write the relative pseudo-complement in closed form. We also discuss in detail properties of additional interesting operators used in information retrieval. Finally, we give a formula for the rank of an element and for the height of the lattice.

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“…Using the well-known characterization of the relative pseudo-complement in terms of ∧irreducible elements [15], we have that given A,…”

Section: Relative Pseudo-complement and Pseudo-differencementioning

confidence: 99%

Section: An Elementary Characterization Of the Relative Pseudo-comple...mentioning

confidence: 99%

On the Lattice of Antichains of Finite Intervals

Boldi

Vigna

2016

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“…A meet decomposition (8) is irredundant if x < i∈J m i for every proper subset J I. For distributive relatively pseudocomplemented lattices, the following criterion for meet irreducibility of elements is known [14]. Throughout this section, we assume that L is an arbitrary distributive relatively pseudocomplemented lattice.…”

Section: Irredundant Meet Decompositions In Lattices Of Colour-familiesmentioning

confidence: 99%

Lattices of relative colour-families and antivarieties

Kravchenko

2007

Discuss. Math. - General Algebra and Applications

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We consider general properties of lattices of relative colour-families and antivarieties. Several results generalise the corresponding assertions about colour-families of undirected loopless graphs, see [1]. Conditions are indicated under which relative colour-families form a lattice. We prove that such a lattice is distributive. In the class of lattices of antivarieties of relation structures of finite signature, we distinguish the most complicated (universal) objects. Meet decompositions in lattices of colour-families are considered. A criterion is found for existence of irredundant meet decompositions. A connection is found between meet decompositions and bases for anti-identities.

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“…It is a subsemilattice of L (see [6], [9]). We refer the reader to [4], [6], [7], [8] and [9] for lists of properties satisfied by * in implicative and a-implicative semilattices.…”

Section: S Hoo [Septembermentioning

confidence: 99%

“…Introduction. In [8], D. P. Smith obtained various results regarding atoms and meet-irreducible elements in implicative lattices. She also considered the situation in the MacNeille completion.…”

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confidence: 99%

Atoms, Primes and Implicative Lattices

Hoo¹

1984

Can. math. bull.

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Let L be an a-implicative semilattice. We obtain a characterization of those elements which cover a. This gives a characterization of atoms in pseudocomplemented semilattices, and leads to various results on primes and irreducibles in semilattices. As an application, we prove that in a complete, atomistic lattice L, the following are equivalent (i) L is implicative (ii) L is (2, ∞) meet distributive (iii) each element of L is a meet of primes.

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Meet-Irreducible Elements in Implicative Lattices

Cited by 5 publications

References 8 publications

On the Lattice of Antichains of Finite Intervals

On the Lattice of Antichains of Finite Intervals

Lattices of relative colour-families and antivarieties

Atoms, Primes and Implicative Lattices

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