Varieties of Lattices

Jipsen, Peter; Rose, H. J.

doi:10.1007/bfb0090224

Cited by 48 publications

(54 citation statements)

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“…According to the Mace4 manual, the results of the isomorphism checker should be taken with a grain of salt. However, our numbers for < 7 elements are confirmed by Jipsen's computations [13].…”

Section: Iteration Algebrassupporting

confidence: 83%

Automated Reasoning in Kleene Algebra

Höfner

Struth

Automated Deduction – CADE-21

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Abstract. It has often been claimed that model checking, special purpose automated deduction or interactive theorem proving are needed for formal program development. We demonstrate that off-the-shelf automated proof and counterexample search is an interesting alternative if combined with the right domain model. We implement variants of Kleene algebras axiomatically in Prover9/Mace4 and perform proof experiments about Hoare, dynamic, temporal logics, concurrency control and termination analysis. They confirm that a simple automated analysis of some important program properties is possible. Particular benefits of this novel approach include "soft" model checking in a first-order setting, crosstheory reasoning between standard formalisms and full automation of some (co)inductive arguments. Kleene algebras might therefore provide light-weight formal methods with heavy-weight automation.

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Section: Iteration Algebrassupporting

confidence: 83%

Automated Reasoning in Kleene Algebra

Höfner

Struth

Automated Deduction – CADE-21

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“…We follow the notation from [14,11,12]. Note that M 3 , L 1 , L 2 , L 3 , L 4 , L 5 , L 13 , L 14 , and L 15 are subdirectly irreducible lattices and each L i is the order-skeleton of the corresponding lattice L i for i = 6, 7, 8 as found in these references.…”

Section: Lemma 34 {M 3 } Is An Exclusion System For D ⊂ Mmentioning

confidence: 99%

Distributivity conditions and the order-skeleton of a lattice

Feng

Greechie

2011

Algebra Univers.

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We introduce "π-versions" of five familiar conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D 0π if a∧(b∨c) (a∧b)∨c for all 3-element antichains {a, b, c}. We consider a congruence relation ∼ whose blocks are the maximal autonomous chains and define the order-skeleton of a lattice L to be e L := L/∼. We prove that the following are equivalent for a lattice L: (i) L satisfies D 0π , (ii) e L satisfies any of the five π-versions of distributivity, (iii) the order-skeleton e L is distributive.

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“…Since every almost distributive lattice is semidistributive (see [13,Theorem 4.2]), there is no non-distributive simple lattice in AD. Further, as shown by Lee [16], there is a family (D i : i < 2 ω ) of distinct subvarieties of AD (which, with no loss in generality, we assume does not include D and the variety N 5 generated by N 5 ).…”

Section: Q-universal Varieties Of Bounded Lattices 339mentioning

confidence: 99%

Q -universal varieties of bounded lattices

Adams¹,

Dziobiak²

2002

Algebra Universalis

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A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A i : i < 2 ω ) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal.

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Varieties of Lattices

Cited by 48 publications

References 0 publications

Automated Reasoning in Kleene Algebra

Automated Reasoning in Kleene Algebra

Distributivity conditions and the order-skeleton of a lattice

Q -universal varieties of bounded lattices

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