×

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
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] Abbot J. C.: Semi-boolean algebras. Matem. Vestnik 4 (1967), 117-198.
[2] Chajda I.: Coherence, regularity and permutability of congruences. Algebra Univ. 17 (1983), 170-173. · Zbl 0537.08006
[3] Csákány B.: Characterizations of regular varieties. Acta Sci. Math. (Szeged), 31 (1970), 187-189. · Zbl 0216.03302
[4] Duda J.: Coherence in varieties of algebras. Czech. Math. J. 39 (1989), 711 - 716. · Zbl 0704.08003
[5] Geiger D.: Coherent algebras. Notices AMS 21 (1974), A-436.
[6] Taylor W.: Uniformity of congruences. Algebra Univ. 4 (1974), 342-360. · Zbl 0313.08001
[7] Werner H.: A Maľcev condition on admissible relations. Algebra Univ. 3 (1973), 263. · Zbl 0276.08004
[8] Prodan N. I.: On varieties of CHQ-algebras. (Russian), Algebra i logika 20 (1980), 92-100.
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.