The clone of polynomial functions on an algebra A is formed by all finitary functions on A which can be obtained by composing the operations of A, the constant functions, and the projections on A. For rings this defines exactly the classical polynomial functions. Algebras whose polynomial clones are equal, such as the Boolean algebra and its corresponding Boolean ring, are said to be polynomial equivalent.
In the 1990's Idziak and McKenzie asked whether there are only countably many finite Mal'cev algebras (algebras that generate a congruence permutable variety) up to polynomial equivalence.
For proving that the answer is yes it would suffice to show that the following question has an affirmative answer:
Is the clone of polynomial functions on a finite Mal'cev algebra determined by finitely many relations?
We discuss invariants that proved useful in describing polynomial clones so far (e.g., congruences, extended types, higher commutators, ...) and present some recent positive results as well as their applications to groups and rings.
Sprache der Kurzfassung:
Eingeladener Vortrag an anderen Institutionen
Details zum Vortragsort:
CAUL, Centro de Álgebra da Universidade de Lisboa, Lissabon