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Coherence and weak coherence in the square of algebras. (English) Zbl 0777.08005
A subalgebra $$B$$ of an algebra $$A$$ is said to be coherent with a congruence $$\Theta$$ on $$A$$ whenever the assumption $$[b]\Theta\subseteq B$$ for some $$b\in B$$ implies $$[x]\Theta\subseteq B$$ for every $$x\in B$$. If $$A$$ is an algebra with a nullary operation 0 we say that a subalgebra $$B$$ of $$A$$ is weakly coherent with a congruence $$\Theta$$ on $$A$$ whenever the assumption $$[0]\Theta\subseteq B$$ implies $$[x]\Theta\subseteq B$$ for every $$x\in B$$. It is shown that a variety $$V$$ is permutable and regular iff any tolerance on $$A$$ is coherent with factorable congruences on $$A\times A$$, $$A\in V$$. Further a variety $$V$$ is regular iff any congruence on $$A$$ is coherent with factorable congruences on $$A\times A$$, $$A\in V$$. Analogous results hold for weak coherence, permutability and weak regularity.
Reviewer: J.Duda (Brno)
##### MSC:
 08A30 Subalgebras, congruence relations 08A05 Structure theory of algebraic structures
##### Keywords:
coherence; tolerance; factorable congruences; variety
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##### References:
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