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Zbl 0777.08005
Duda, Jaromír
Coherence and weak coherence in the square of algebras. (English)
[J] Czech. Math. J. 42, No.4, 613-618 (1992). ISSN 0011-4642; ISSN 1572-9141/e

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.
[J.Duda (Brno)]

MSC 2000:: *08A30 Subalgebras of general algebraic systems
08A05 Structure theory of general algebraic systems

Keywords: coherence; tolerance; factorable congruences; variety

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