Weak coherence of congruences. (English) Zbl 0796.08003

Chajda, Ivan

Czech. Math. J. 41, No. 1, 149-154 (1991).

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

Keywords:

regularity; permutability; Mal’tsev-type condition; weakly coherent varieties

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