## Mal’cev functions on smalgebras.(English)Zbl 0993.08007

Let $$A$$ be a nonvoid set. A function $$p:A\times A\times A\to A$$ is called a Mal’tsev function on $$A$$ whenever $$p(x,y,y)= p(y,y,x)=x$$ holds for all $$x,y\in A$$. The authors show that for $$|A|=9$$ and a lattice $$L$$ of permuting equivalences on $$A$$ there is a Mal’tsev function on $$A$$ that preserves all members of $$L$$. The same statement for single algebras with a limited number of elements (= smalgebras) was previously known to hold for $$|A|\leq 8$$ and to fail for $$|A|\geq 25$$. The problem remains open for $$10\leq |A|\leq 24$$.

### MSC:

 08B05 Equational logic, Mal’tsev conditions

### Keywords:

Mal’tsev function; smalgebras
Full Text:

### References:

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