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Hauptvortrag / Eingeladener Vortrag auf einer Tagung

Higher commutators, nilpotence, and supernilpotence

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Zusammenfassung: The $n$-ary commutator operation of a universal algebra associates a congruence $\beta := [ \alpha_1, \ldots, \alpha_n]$ with every $n$-tuple $(\alpha_1,\ldots, \alpha_n) \in (\mathrm{Con} \mathbf{A})^n$. These commutator operations were introduced by A.\ Bulatov to distinguish between polynomially inequivalent algebras, and their properties in Mal'cev algebras were investigated by N.\ Mudrinski and the speaker. Using commutator operations, a different concept of nilpotence can be defined: an algebra is defined to be \emph{supernilpotent} if for some $n \in \mathbb{N}$, $[1,\ldots,1] = 0$ ($n$ repetitions of $1$). For finite Mal'cev algebras, being supernilpotent is equivalent to $\log (\mathbf{F}_{V(\mathbf{A})} (n))$ being bounded from above by a polynomial in $n$. We will review some basic results on higher commutators and supernilpotent Mal'cev algebras, discuss results by J.\ Berman, W.\ Blok, and K.\ Kearnes that link supernilpotence to nilpotence, provide a generalization of one of these structural results to infinite expanded groups, and use these results to establish that the clone of congruence preserving functions of certain algebras is finitely generated.

Tagungstitel: NSAC 2013 The 4th Novi Sad Algebraic Conference in conjunction with the workshop Semigroups and Applications 2013
Vortragsdatum: 09.06.2013
Web: http://www.dmi.uns.ac.rs/nsac2013/ (Novi Sad Algebraic Conference 2013)
Land: Serbien
Ort: Novi Sad

Beteiligte

ReferentInnen: Assoz.Univprof. DI Dr. Erhard Aichinger

Forschungseinheiten der JKU:

Wissenschaftszweige: 1102 Algebra | 1107 Geometrie | 1119 Zahlentheorie | 1131 Computer Algebra

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