## Fraser-Horn identities can be written in two variables.(English)Zbl 0669.08003

A Mal’cev condition for varieties with directly decomposable congruences (DDC) was given by G. A. Fraser and A. Horn [Proc. Am. Math. Soc. 26, 390-394 (1970; Zbl 0241.08004)]. The identities they used involved three variables, being built up from binary, ternary, and $$m+1$$- ary terms. This paper provides a system of identities involving only two variables, being built up from binary and $$m+2$$-ary terms. As a consequence of this we have the result that a variety $${\mathcal V}$$ has DDC if and only if $$F_{{\mathcal V}}(2)\times F_{{\mathcal V}}(2)$$ has DDC.
Reviewer: S.Oates-Williams

### MSC:

 08B05 Equational logic, Mal’tsev conditions

Zbl 0241.08004
Full Text:

### References:

 [1] J. Duda,On two schemes applied to Mal’cev type theorems, Ann. Univ. Sci. Budapest, Sectio Math.26 (1983), 39-45. · Zbl 0518.08002 [2] J. Duda,Varieties having directly decomposable congruence classes, ?asopis p?st, matem.111 (1986), 394-403. · Zbl 0606.08001 [3] J. Duda,Congruences on products in varieties satisfying the CEP, Math. Slovaca36 (1986), 171-177. · Zbl 0598.08005 [4] G. A. Fraser andA. Horn,Congruence relations in direct products, Proc. Amer. Math. Soc.26 (1970), 390-394. · Zbl 0241.08004 [5] G. Gr?tzer,Universal Algebra,Second Expanded Edition, Springer Verlag, Berlin, Heidelberg and New York, 1979. [6] J.Hagemann,Congruences on products and subdirect products of algebras, Preprint Nr. 219, TH-Darmstadt, 1975.
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