The Relationship Between Two Commutators

Kearnes, Keith A.; Szendrei, Ágnes

doi:10.1142/s0218196798000247

Cited by 51 publications

(67 citation statements)

References 21 publications

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“…There do exist expressions W (p, q, r) in the symbols ∨, ∧ and • such that the congruence inclusion α ∩ (β • γ) ⊆ W (α, β, γ) is nontrivial but does not imply congruence meet semidistributivity for a variety. (E.g., Theorem 8.13 (2) of this monograph, or Theorem 4.8 (2) and (3) of [52]. )…”

Section: Congruence Identities From Maltsev Conditionsmentioning

confidence: 87%

“…As such it encodes the existence of diagonal skew congruences. A useful and completely general representation theorem for abelian algebras and congruences is not likely to exist, but Keith Kearnes andÁgnes Szendrei extended Herrmann's representation theorem for congruence modular varieties to any variety satisfying some nontrivial idempotent Maltsev condition in [52].…”

Section: Commutator Theoriesmentioning

confidence: 99%

“…Since A also satisfies the nontrivial idempotent Maltsev conditions satisfied by V, it follows from Lemma 2.5 of [47] that A must satisfy an idempotent Maltsev condition that fails in the variety of semilattices. From Theorem 4.10 of [52], it follows that the variety A is affine, and in particular congruence permutable. If d(x, y, z) is a Maltsev operation for A, then it will be a weak difference term for V, as is explained in the proof of Theorem 4.8 of [52].…”

Section: Varieties With a Weak Difference Termmentioning

confidence: 99%

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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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Section: Congruence Identities From Maltsev Conditionsmentioning

confidence: 87%

Section: Commutator Theoriesmentioning

confidence: 99%

Section: Varieties With a Weak Difference Termmentioning

confidence: 99%

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The shape of congruence lattices

Kearnes

Kiss

2012

Memoirs of the AMS

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“…Later, it was shown by Kearnes andÁ. Szendrei [7] and P. Lipparini [8] that C 2 already characterizes congruence meet-semidistributivity. More recently, using the Mal'cev condition directly derivable from C 2 , Willard proved the following.…”

Section: Sequence Lemmasmentioning

confidence: 98%

Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound

Willard

Kearnes

1999

Proc. Amer. Math. Soc.

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“…K. Kearnes andÁ. Szendrei [11] proved that abelian algebras with a Taylor term are always quasi-affine, and they are affine under a somewhat stronger assumption that they satisfy an idempotent Mal'tsev condition that fails in semilattices. On the other hand, virtually nothing is known on quasi-affine representations of abelian non-Taylor algebras.…”

Section: 2mentioning

confidence: 99%

Mal’tsev conditions, lack of absorption, and solvability

2015

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Abstract. We provide a new characterization of several Mal'tsev conditions for locally finite varieties using hereditary term properties. We show a particular example how lack of absorption causes collapse in the Mal'tsev hierarchy, and point out a connection between solvability and lack of absorption. As a consequence, we provide a new and conceptually simple proof of a result of Hobby and McKenzie, saying that locally finite varieties with a Taylor term possess a term which is Mal'tsev on blocks of every solvable congruence in every finite algebra in the variety.

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The Relationship Between Two Commutators

Cited by 51 publications

References 21 publications

The shape of congruence lattices

The shape of congruence lattices

Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound

Mal’tsev conditions, lack of absorption, and solvability

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