Leírás
Equivalence relations appear everywhere in mathematics, in particular, if they are compatible with some mathematical structure (and therefore allow factor constructions). They satisfy a strong property: An n-ary operation f preserves an equivalence relation ρ if and only if each unary polynomial function which can be obtained from f by substituting constants at all but one argument preserves ρ (such unary functions are called basic translations of f). This preservation criterion can be generalized to a condition that ranges over those n-ary polynomial functions that are obtained from f by substituting constants at all but n many arguments. In fact, there are many examples of such relations for each positive n which play an important role in higher commutator theory.
Jakubıkova-Studenovska, Poeschel, and Radeleczki provide an essentially complete characterization of all finitary relations ρ with the property that f preserves if and only if all basic translations of f preserve ρ. In this talk we will explain how to generalize this characterization to relations with the analogous ‘higher dimensional’ compatibility criterion with respect to n-ary polynomial functions for some positive n. These considerations will lead us to a finiteness condition for polynomial clones which is distinct from both 'finitely generated' and 'finitely related'.
This is joint work with Reinhard Poeschel.