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The Csákány theory of regularity for finite algebras. (English) Zbl 0844.08005
An algebra \(A\) of which any two congruences coincide whenever they have a common class is called regular. The following theorem is proved:
Theorem. Let \(A\) be a finite algebra with \(\text{card} (A)\leq 5\). The following conditions are equivalent: (1) \(A\) is regular; (2) there exists a ternary function \(p: A^3\to A\) compatible with every congruence of \(A\) such that \(p(x, y, z)= z\Leftrightarrow x=y\).
This is not true for six-element algebras.
08A30 Subalgebras, congruence relations
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[1] CHAJDA I.: Every at most four element algebra has a Mal’cev theory for permutability. Math. Slovaca 41 (1991), 35-39. · Zbl 0779.08001
[2] CHAJDA I.-CZÉDLI G.: Mal’cev functions on small algebras. Studia Sci. Math. Hungar. 28 (1993), 339-348. · Zbl 0805.08003
[3] CSÁKÁNY B.: Characterizations of regular varieties. Acta Sci. Math. (Szeged) 31 (1970), 187-189. · Zbl 0216.03302
[4] GUMM H.-P.: Is there a Mal’cev theory for single algebras?. Algebra Universalis 8 (1978), 320-329. · Zbl 0382.08003
[5] PIXLEY A.: Completeness in arithmetical algebras. Algebra Universalis 2 (1972), 179-196. · Zbl 0254.08010
[6] TAYLOR W.: Characterizing Mal’cev conditions. Algebra Universalis 3 (1973), 351-397. · Zbl 0304.08003
[7] WILLE R.: Kongruenzklassengeometrien. Lectures Notes in Math. 113, Springer-Verlag, Berlin-New York, 1970. · Zbl 0191.51403
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