The decidability of the affine completeness generation problem. (Q1771940)

An algebra \(A\) is affine complete if each congruence-compatible operation on \(A\) is a polynomial of \(A\). A variety is affine complete if each of its members is affine complete. The authors give an explicit proof of the theorem: There is an effective procedure for deciding whether or not a given finite algebra of finite type generates an affine complete variety. The proof is based on the fact that if such an algebra \(A\) generates an affine complete variety then \(A\) must have an \((n+1)\)-ary near unanimity term. Then the variety is congruence distributive. On the other hand, there exist congruence non-distributive affine complete algebras, thus the problem of characterizing affine complete algebras remains still open.