Maximal sublattices of finite distributive lattices

Adams, Michael E.; Dwinger, Philip; Schmid, Jürg

doi:10.1007/bf01233919

Cited by 8 publications

(15 citation statements)

References 11 publications

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“…It is clear that this reduction requires only log-space. To complete the proof we must check that (1) A, X, a ∈ Gen-SubAlg ⇐⇒ B, X ∈ GenSet.…”

Section: Problem: Gensetmentioning

confidence: 99%

“…Numerous papers have discussed Frattini sublattices. For a sample, see [1,2,9,27]. There are also papers studying the concept in the context of Moufang loops [17], closure algebras [30], Stone algebras [8], semilattices [22] and Lie algebras [3].…”

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confidence: 99%

“…Numerous papers have discussed Frattini sublattices. For a sample, see [1,2,9,27]. There are 2000 Mathematics Subject Classification.…”

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confidence: 99%

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Computational Complexity of Generators and Nongenerators in Algebra

Bergman

Slutzki

2002

Int. J. Algebra Comput.

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Abstract. We discuss the computational complexity of several problems concerning subsets of an algebraic structure that generate the structure. We show that the problem of determining whether a given subset X generates an algebra A is P-complete, while determining the size of the smallest generating set is NP-complete. We also consider several questions related to the Frattini subuniverse, Φ(A), of an algebra A. We show that the membership problem for Φ(A) is co-NP-complete, while the membership problems for Φ(Φ(A)), Φ(Φ(Φ(A))),... all lie in the class P (NP).In the analysis of any algebraic structure, determining those subsets that generate the structure frequently plays a key role. This is evident in linear algebra for example, where the discussion of bases and spanning sets forms a central element of the subject. The same is true in other branches of algebra such as group and lattice theory.Knowledge of the generating subsets of an algebra gives us information on its subalgebras, homomorphic images, automorphism group etc. As computer algebra systems become commonplace in the toolboxes of scientists (see for example GAP [11], and the "algebra calculator" [21]), questions of efficiency in the determination of generating subsets arise. In this paper we address this fundamental issue by providing completeness results for several variants of the basic question: does the subset X generate the algebra A? In particular, we consider the question of minimal generating sets and the existence of a generating set of a given cardinality.In addition to questions such as these, it is sometimes of interest to ask about the role an individual element can play in the generation of an algebra. Schmid [28] suggests classifying elements as generators, nongenerators and irreducibles. To explain these, we need the notion of the Frattini subuniverse of an algebra.The Frattini subgroup has played an important role in group theory since it was first considered by Giovanni Frattini in 1885 [10]. It was observed quite some time ago that most of the basic properties of the Frattini subgroup hold more generally in any algebraic structure. Numerous papers have discussed Frattini sublattices. For a sample, see [1,2,9,27]. There are 2000 Mathematics Subject Classification. 68Q17, 08A30, 20D25.

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“…It is clear that this reduction requires only log-space. To complete the proof we must check that (1) A, X, a ∈ Gen-SubAlg ⇐⇒ B, X ∈ GenSet.…”

Section: Problem: Gensetmentioning

confidence: 99%

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confidence: 99%

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Computational Complexity of Generators and Nongenerators in Algebra

Bergman

Slutzki

2002

Int. J. Algebra Comput.

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“…During the 1970's the work of Adams [2], Chen, Koh and Tan [8], and Rival [18,19], carried these investigations forward. In recent years, Abad and Adams [1], Adams, Dwinger and Schmid [3], Ryter and Schmid [20], and Vogt [22] have added considerably to our understanding of maximal sublattices of finite distributive lattices. In addition, the recent paper of Adams, Freese, Nation and Schmid [4] examines maximal sublattices of finite bounded lattices, a class which includes the finite distributive lattices.…”

Section: Introductionmentioning

confidence: 99%

“…Three lattices witnessing the ambiguity of Theorem 1 terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700035928 [3] Subinverses of finite distributive lattices 335…”

Section: Introductionmentioning

confidence: 99%

Covering in the lattice of subuniverses of a finite distributive lattice

Lengvárszky

McNulty

1998

J Aust Math Soc A

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The covering relation in the lattice of subuniverses of a finite distributive lattice is characterized in terms of how new elements in a covering sublattice fit with the sublattice covered. In general, although the lattice of subuniverses of a finite distributive lattice will not be modular, nevertheless we are able to show that certain instances of Dedekind's Transposition Principle still hold. Weakly independent maps play a key role in our arguments.1991 Mathematics subject classification {Amer. Math. Soc): primary 06B05.

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Many varieties of lattices with nonsurjective epimorphisms

Wasserman

2006

Algebra univers.

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We use dominions to show that many varieties of lattices have nonsurjective epimorphisms. The variety D of distributive lattices is treated in detail. We show that the dominion in D of a sublattice M ⊆ L is the closure of M under relative complementation in L. This dominion is also the largest sublattice of L in which M is epimorphically embedded. In any variety of lattices larger than D, the dominion of M in L is just M .

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Maximal sublattices of finite distributive lattices

Cited by 8 publications

References 11 publications

Computational Complexity of Generators and Nongenerators in Algebra

Computational Complexity of Generators and Nongenerators in Algebra

Covering in the lattice of subuniverses of a finite distributive lattice

Many varieties of lattices with nonsurjective epimorphisms

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