Identities in congruence lattices of universal algebras

Polin, S. V.

doi:10.1007/bf02412505

Cited by 8 publications

(6 citation statements)

References 3 publications

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“…However, as we mentioned, some identities we have considered are strictly weaker than congruence modularity. Using Polin's variety [47,10], we shall check that the identities (4.4) and (4.6) from Theorem 4.5 do not generally imply congruence modularity. In particular, having a switch or a J-switch level (Definition 9.7) does not imply congruence modularity.…”

Section: Proof (I) Is Witnessed Bymentioning

confidence: 99%

Day's Theorem is sharp for $n$ even

Lipparini¹

2019

Preprint

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Both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms t 0 , . . . , tn witnessing congruence distributivity it is possible to construct terms u 0 , . . . , u 2n−1 witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when n is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, specular, mixed and defective.All the results hold also when restricted to locally finite varieties. We introduce some families of congruence distributive varieties and characterize many congruence identities they satisfy.

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Section: Proof (I) Is Witnessed Bymentioning

confidence: 99%

Day's Theorem is sharp for $n$ even

Lipparini¹

2019

Preprint

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“…McKenzie's Conjecture was refuted by S. V. Polin in his famous paper [74]. Polin constructed a locally finite variety P that is not congruence modular, but satisfies a nontrivial congruence identity.…”

Section: Con(v) = H S P ({Con(a) | a ∈ V})mentioning

confidence: 99%

The shape of congruence lattices

Kearnes

Kiss

2012

Memoirs of the AMS

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Abstract. We develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvability theory for the nonmodular TC commutator. These theories are used to show that each of the following sets of statements are equivalent for a variety V of algebras.(I) (a) V satisfies a nontrivial congruence identity. throughout V. We prove that a residually small variety that satisfies a congruence identity is congruence modular. IntroductionThis monograph is concerned with the relationships between Maltsev conditions, commutator theories and the shapes of congruence lattices in varieties of algebras. Shapes of Congruence LatticesCarl F. Gauss, in [23], introduced the notationwhich is read as "a is congruent to b modulo m", to mean that the integers a and b have the same remainder upon division by the integer modulus m, equivalently that a − b ∈ mZ. As the notation suggests, congruence modulo m is an equivalence relation on Z. It develops that congruence modulo m is compatible with the ring operations of Z, and that the only equivalence relations on Z that are compatible with the ring operations are congruences modulo m for m ∈ Z. Richard Dedekind conceived of a more general notion of "integer", which nowadays we call an ideal in a number ring. Dedekind extended the notation (1.1) towhere a, b ∈ C and µ ⊆ C; (1.2) is defined to hold if a − b ∈ µ. Dedekind called a subset µ ⊆ C a module if it could serve as the modulus of a congruence, i.e., if this relation of congruence modulo µ is an equivalence relation on C. This happens precisely when µ is closed under subtraction. For Dedekind, therefore, a "module" was an additive subgroup of C. The set of Dedekind's modules is closed under the operations of intersection and sum. These two operations make the set of modules into a lattice. Dedekind proposed and investigated the problem of determining the identities of this lattice (the "laws of congruence arithmetic"). In 1900, in [13], he published the discovery that if α, β, γ ⊆ C are modules, then (1.3) α ∩ (β + (α ∩ γ)) = (α ∩ β) + (α ∩ γ).

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“…Indeed, there are locally finite varieties satisfying the conditions of Theorem 1.2 that are not congruence distributive, for example, Polin's variety (see [14]). The fact that this variety has the properties we are claiming for it follows from Exercise 9.20 (6) of [9].…”

Section: Vol 54 2005mentioning

confidence: 99%

The triangular principle is equivalent to the triangular scheme

Kearnes

Kiss

2005

Algebra univers.

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The triangular scheme is a diagrammatic characterization of congruence joinsemidistributivity. The triangular principle is a variant of this condition, where one of the congruences is replaced by a tolerance. This paper contains two proofs showing that the triangular principle and the triangular scheme are equivalent for varieties. The first one is a routine argument using tame congruence theory, and works only for locally finite varieties. The second proof is an elementary, but nontrivial calculation with terms, which works for arbitrary varieties. This yields a stronger Mal'tsev-condition characterizing congruence join-semidistributivity than the one obtained from the triangular scheme.

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Identities in congruence lattices of universal algebras

Cited by 8 publications

References 3 publications

Day's Theorem is sharp for $n$ even

Day's Theorem is sharp for $n$ even

The shape of congruence lattices

The triangular principle is equivalent to the triangular scheme

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