2-affine complete algebras need not be affine complete
Sprache des Vortragstitels:
Universal Algebra and Lattice Theory (Dedicated to the 70th birthday of B. Csakany)
Sprache des Tagungstitel:
Given a finite algebra A, we study the clone Comp (A) of all congruence preserving functions on A, and the clone Pol (A) of all polynomial functions on A. We call an algebra n-affine complete iff every n-ary congruence preserving function is polynomial.
For each k in N, we exhibit an algebra that is k-affine complete, but not (k+1)-affine complete.
However, as a consequence of a Theorem by J. Hagemann and Chr. Herrmann, we know that if each homomorphic image of a finite algebra A in a congruence permutable variety is 2-affine complete, then A is k-affine complete for all k in N.