## Weak coherence of congruences.(English)Zbl 0796.08003

An algebra $$A$$ with a nullary operation 0 is weakly coherent if every of its subalgebras containing $$[0]_ \theta$$ for some $$\theta\in\text{Con }A$$ is a union of classes of $$\theta$$. The paper contains a Mal’tsev-type condition characterizing weakly coherent varieties. An algebra $$A$$ with 0 has subalgebras closed under translations of congruence 0-classes, briefly $$A$$ has 0-CUT, if for any subalgebra $$B$$ of $$A$$ and every $$n$$-ary polynomial $$p$$ over $$A$$ and each $$x\in A$$, $$y\in B$$, if $$[0]_ \theta\subseteq B$$ and $$p(0,\dots,0)= y$$, then $$p([0]_ \theta)\subseteq B$$ for $$\theta= \theta(x,y)$$. It is proven that a variety $$\mathcal V$$ with 0 is weakly coherent if and only if $$\mathcal V$$ is 0-regular, permutable and has 0-CUT. These conditions are independent.
Reviewer: I.Chajda (Přerov)

### MSC:

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

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