## Conditions for factorable relations.(English)Zbl 0795.08007

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
Full Text:

### References:

 [1] Duda J.: On two schemes applied to Maľcev type theorems. Ann. Univ. Sci. Budapest, Sectio Math. 26 (1983), 39-45. · Zbl 0518.08002 [2] Duda J.: Varieties having directly decomposable congruence classes. Čas. Pěst. Mat. 111 (1986), 394-403. · Zbl 0606.08001 [3] Duda J.: Maľcev conditions for directly decomposable compatible relations. Czech. Math. J. 39 (1989), 674-680. · Zbl 0704.08002 [4] Duda J.: Fraser-Horn identities can be written in two variables. Algebra Univ. 26 (1989), 178-180. · Zbl 0669.08003 [5] Fraser G. A., Horn A.: Congruence relations in direct products. Proc. Amer. Math. Soc. 26 (1970), 390-394. · Zbl 0241.08004 [6] Hagemann J.: Congruences on products and subdirect products of algebras. Preprint Nr. 219. TH-Darmstadt 1975. [7] Niederle J.: Decomposability conditions for compatible relations. Czech. Math. J. 33 (1983), 522-524. · Zbl 0538.08007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.