## Mal’tsev functions on small algebras.(English)Zbl 0805.08003

Given a set $$A$$, a function $$p: A\to A^ 3$$ is called a Mal’tsev function if $$p(x,y,y)= p(y,y,x)= x$$ holds for any $$x,y\in A$$. It is well- known that if an algebra $$A$$ has a Mal’tsev function which is compatible with all congruences of $$A$$ then $$A$$ is congruence permutable. However, the converse is not true in general [H. P. Gumm, Algebra Univers. 8, 320-329 (1978; Zbl 0382.08003)]. In the present paper, the following problem is considered. Given an $$n$$-element set $$A$$ and a lattice $$L$$ of permuting equivalences on $$A$$; does there exist a Mal’tsev function on $$A$$ which is compatible with all members of $$L$$? It is shown that the answer is in general negative when $$n\geq 25$$ (even in the case if $$A$$ is an algebra and $$L=\text{Con}(A))$$, and it is affirmative for $$n\leq 8$$.

### MSC:

 08B05 Equational logic, Mal’tsev conditions

Zbl 0382.08003