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 \(n\leq 8\).


08B05 Equational logic, Mal’tsev conditions


Zbl 0382.08003