Higher commutators, nilpotence, and supernilpotence
Sprache des Vortragstitels:
Englisch
Original Tagungtitel:
NSAC 2013 The 4th Novi Sad Algebraic Conference in conjunction with the workshop Semigroups and Applications 2013
Sprache des Tagungstitel:
Englisch
Original Kurzfassung:
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.
Sprache der Kurzfassung:
Englisch
Vortragstyp:
Hauptvortrag / Eingeladener Vortrag auf einer Tagung
Forschungs-