Some applications of higher commutators in Mal’cev algebras

Aichinger, Erhard; Mudrinski, Nebojša

doi:10.1007/s00012-010-0084-1

Cited by 58 publications

(142 citation statements)

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“…We note that an n-supernilpotent congruence is nilpotent of class n by property (HC8) in [3]. In rings and groups the converse holds as well by Lemmas 3.5 and 3.6.…”

Section: Introductionmentioning

confidence: 79%

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Mal’cev algebras with supernilpotent centralizers

Mayr¹

2011

Algebra Univers.

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Let A be a finite algebra in a congruence permutable variety. We assume that for every subdirectly irreducible homomorphic image of A the centralizer of the monolith is n-supernilpotent. Then the clone of polynomial functions on A is determined by relations of arity |A| n+1 . As consequences we obtain finite implicit descriptions of the polynomial functions on finite local rings with 1 and on finite groups G such that in every subdirectly irreducible quotient of G the centralizer of the monolith is a p-group.

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“…We note that an n-supernilpotent congruence is nilpotent of class n by property (HC8) in [3]. In rings and groups the converse holds as well by Lemmas 3.5 and 3.6.…”

Section: Introductionmentioning

confidence: 79%

“…This definition is equivalent to the one given in [3,Definition 3.2] and in [6,Definition 3]. We use the notation M A (α 1 , .…”

Section: Introductionmentioning

confidence: 99%

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Mal’cev algebras with supernilpotent centralizers

Mayr¹

2011

Algebra Univers.

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“…The following special class of binary polynomials can be used to define commutators in Mal'cev algebras, as it has been done in [3,Lemma 6.9]. Definition 2.2.…”

Section: The Commutator [•mentioning

confidence: 99%

Uniform Mal’cev algebras with small congruence lattices

Mudrinski

2014

Algebra Univers.

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Abstract. A congruence of an algebra is called uniform if all the congruence classes are of the same size. An algebra is called uniform if each of its congruences is uniform. All algebras with a group reduct have this property. We prove that almost every finite uniform Mal'cev algebra with a congruence lattice of height at most two is polynomially equivalent to an expanded group. MotivationIn this note, we investigate finite uniform algebras. We say that a congruence θ of an algebra A is uniform if all its congruence classes have the same cardinality. We say that an algebra A is uniform if all congruences of A are uniform. Uniform congruences have been introduced by W. Taylor in [13] and studied by R. McKenzie in [11]. In these papers, as in [4, p. 93], varieties of uniform algebras are considered rather than single uniform algebras. In 2005, K. Kaarli, [9], proved that finite uniform lattices are congruence permutable.Clearly, all algebras with a group reduct (groups, rings and modules as the most famous examples) are uniform. Similarly, all quasigroups are also uniform and therefore all algebras with a quasigroup reduct (expanded quasigroups) are uniform. Let us recall that a quasigroup is a groupoid (G, ·) such that for all a, b ∈ G there exist unique x and y in G such that ax = b and ya = b. Let a/θ and b/θ be two different classes of a congruence θ of the quasigroup G. We know that there is a c ∈ G such that a · c = b. We take the right translation f c : G → G, defined by f c (x) := x · c for all x ∈ G, and restrict it to a/θ. Then f c (a/θ) ⊆ b/θ because θ is compatible with · . Furthermore, f c is an injective mapping because · is a quasigroup operation. Hence, |a/θ| ≤ |b/θ|. Analogously, we obtain |b/θ| ≤ |a/θ|. If · has a neutral element in G, then we call (G, ·) a loop. An algebra A is called an expanded quasigroup (loop) if it has a quasigroup (loop) operation among its fundamental operations.Presented by E. Kiss.

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“…These proofs require the existence of a Mal'cev term in an essential way, hence the validity of (HC3'), (HC4), (HC7), (HC8) outside Mal'cev algebras still remains to be explored. We note that the properties (HC5) and (HC6) listed in [3] are missing from our list here. The property (HC5) relates higher commutators to a certain centralizing relation, and (HC6) relates the commutator operations of an algebra to the commutator operations of a homomorphic image.…”

Section: Introductionmentioning

confidence: 99%

Sequences of Commutator Operations

Aichinger

Mudrinski

2013

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Given the congruence lattice L of a finite algebra A with a Mal'cev term, we look for those sequences of operations on L that are sequences of higher commutator operations of expansions of A. The properties of higher commutators proved so far delimit the number of such sequences: the number is always at most countably infinite; if it is infinite, then L is the union of two proper subintervals with nonempty intersection.

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Some applications of higher commutators in Mal’cev algebras

Cited by 58 publications

References 15 publications

Mal’cev algebras with supernilpotent centralizers

Mal’cev algebras with supernilpotent centralizers

Uniform Mal’cev algebras with small congruence lattices

Sequences of Commutator Operations

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