Conditions for factorable relations. (English) Zbl 0795.08007

Duda, Jaromír

Czech. Math. J. 41, No. 1, 131-134 (1991).

It is shown that a variety \(\mathcal V\) has the Fraser-Horn property for congruences (tolerances) iff the congruence (tolerance) condition \(\langle\langle x,x,x\rangle,\langle y,y,x\rangle\rangle\in \Theta\) implies \(\langle\langle x,x,y\rangle,\langle y,x,y\rangle\rangle\in \Theta\) \((\langle\langle x,x,x\rangle,\langle y,y,x\rangle\rangle,\langle\langle y,y,y\rangle,\langle y,y,x\rangle\rangle\in T\) imply \(\langle\langle x,y,y\rangle,\langle y,y,x\rangle\rangle\in T)\) holds for any congruence \(\Theta\) (tolerance \(T\)) on \(A\times A\times A\), \(x,y\in A\in{\mathcal V}\). The former conditions were written in three (four, respectively) variables.

Reviewer: J.Duda (Brno)

MSC:

08B05

Equational logic, Mal’tsev conditions

Keywords:

tolerances; Fraser-Horn property; congruences

References:

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