A Characterization of Identities Implying Congruence Modularity I

Day, Alan; Freese, Ralph

doi:10.4153/cjm-1980-087-6

Cited by 43 publications

(30 citation statements)

References 8 publications

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“…The proof that the resulting equational class satisfies some nontrivial congruence identity is practically the same as that presented by Day and Freese for Polin's variety in [1]. The tame congruence theoretic properties of the class referred to earlier can easily be established.…”

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A modification of Polin's variety

Kearnes

Valeriote

1999

Algebra Universalis

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In this note we settle a question posed by Hobby and McKenzie in [2] on the nature of locally finite equational classes which satisfy some nontrivial congruence identity.We settle problem #14 from [2] by exhibiting a locally finite equational class V which omits types 1 and 5 and which satisfies some non-trivial congruence identity, but which contains a finite algebra having a type 4 minimal set with a nonempty tail. For background to this problem, the reader is encouraged to consult Chapter 9 of [2] and the paper [1] by Day and Freese on Polin's Variety.Loosely stated, Polin's variety consists of algebras created by replacing the points of an (external) boolean algebra B by a family of (internal) boolean algebras {B a : a ∈ B} so that whenever a ≥ b in B, there is a homomorphism ξ Our example is a simple modification of this idea wherein we replace the internal boolean structures by distributive lattices having a distinguished largest element 1. The proof that the resulting equational class satisfies some nontrivial congruence identity is practically the same as that presented by Day and Freese for Polin's variety in [1]. The tame congruence theoretic properties of the class referred to earlier can easily be established. * 1991 Mathematical Subject Classification. Primary 08B05; Secondary 08A30.† Support of NSERC is gratefully acknowledged 1

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“…The corresponding results for Polin's variety are established in sections 2 and 7 of [1]. The proofs presented there can be used, almost without change, to prove our theorem.…”

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A modification of Polin's variety

Kearnes

Valeriote

1999

Algebra Universalis

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“…Figure 9 diagrams the congruence lattice of the free algebra on one generator in Polin's variety. This diagram played a critical role in the characterization of varieties with modular congruence lattice of [5].…”

Section: Some Examplesmentioning

confidence: 99%

Automated Lattice Drawing

Freese

2004

Concept Lattices

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Abstract. Lattice diagrams, known as Hasse diagrams, have played an ever increasing role in lattice theory and fields that use lattices as a tool. Initially regarded with suspicion, they now play an important role in both pure lattice theory and in data representation. Now that lattices can be created by software, it is important to have software that can automatically draw them. This paper covers:-The role and history of the diagram.-What constitutes a good diagram.-Algorithms to produce good diagrams. Recent work on software incorporating these algorithms into a drawing program will also be covered.An ordered set P = (P, ≤) consists of a set P and a partial order relation ≤ on P . That is, the relation ≤ is reflexive (x ≤ x), transitive (x ≤ y and y ≤ z imply x ≤ z) and antisymmetric (x ≤ y and y ≤ x imply x = y). If P is finite there is a unique smallest relation ≺, known as the cover or neighbor relation, whose transitive, reflexive closure is ≤. (Graph theorists call this the transitive reduct of ≤.) A Hasse diagram of P is a diagram of the acyclic graph (P, ≺) where the edges are straight line segments and, if a < b in P, then the vertical coordinate for a is less than the one for b. Because of this second condition arrows are omitted from the edges in the diagram.A lattice is an ordered set in which every pair of elements a and b has a least upper bound, a ∨ b, and a greatest lower bound, a ∧ b, and so also has a Hasse diagram.These Hasse diagrams 1 are an important tool for researchers in lattice theory and ordered set theory and are now used to visualize data. This paper deals the special issues involved in such diagrams. It gives several approaches that have been used to automatically draw such diagrams concentrating on a three dimension force algorithm especially adapted for ordered sets that does particularly well.We begin with some examples.1 In the second edition of his famous book on lattice theory [3] Birkhoff says these diagrams are called Hasse diagrams because of Hasse's effective use of them but that they go back at least to H. Vogt, Résolution algébrique deséquation, Paris, 1895.

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“…Many important splitting lattices have singular covers for their critical quotients. For example, the congruence lattices of the finitely generated free algebras in Polin's variety have this property [4]. Another such class of lattices is investigated in [3].…”

Section: Introductionmentioning

confidence: 99%