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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
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