## Every at most four element algebra has a Mal’cev theory for permutability.(English)Zbl 0779.08001

For any algebra $$A$$ with at most four elements, it is proved that $$A$$ has permutable congruences if and only if there exists a ternary Mal’tsev function on $$A$$ which is compatible with all congruences of $$A$$.

### MSC:

 08A30 Subalgebras, congruence relations 08B05 Equational logic, Mal’tsev conditions
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### References:

 [1] GUMM H.-P.: Is there a Maľcev theory for single algebras?. Algebra univ., 8, 1978, 320-329. · Zbl 0382.08003 [2] KOREC I.: A ternary function for distributivity and permutability of an equivalence lattice. Proc. Amer. Math. Soc., 69, 1978, 8-10. · Zbl 0382.08004 [3] MAĽCEV A. I.: On the general theory of algebraic systems. Mat. Sborník., 35, 1954, 3-20. [4] PIXLEY A. F.: Distributivity and permutability of congruence relations in equational classes of algebras. Proc. Amer. Math. Soc., 14, 1963, 105-109. · Zbl 0113.24804 [5] PIXLEY A. F.: Local Maľcev conditions. Canad. Math. Bull., 15, 1972, 559-568. · Zbl 0254.08009
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