Notes on congruence implication. (English) Zbl 0760.08002

Summary: Besides the usual implication between lattice identities, the classes \(\text{Con} {\mathcal V}=\{\text{Con} A:A\in{\mathcal V}\}\), with \({\mathcal V}\) being closed with respect to some operators, give rise to some new kind(s) of implication. Other kinds of implication arise when \({\mathcal V}\) has a nullary operation \(e\). Then \(\text{Con} {\mathcal V}\) is said to satisfy a lattice identity \(p(x_ 1,\dots,x_ t)\leq q(x_ 1,\dots,x_ t)\) at \(e\) if the congruence block \([e]p(\alpha_ 1,\dots,\alpha_ t)\) is included in \([e]q(\alpha_ 1,\dots,\alpha_ t)\) for any \(A\in{\mathcal V}\) and arbitrary \(\alpha_ 1,\dots,\alpha_ t\in\text{Con} A\). This paper shows that some classical results on the implication in congruence varieties (the case when \({\mathcal V}\) is closed with respect to \(\mathbb{H},\mathbb{S}\) and \(\mathbb{P})\) can be strengthened by using the above- mentioned kinds of implication. An example shows that this strengthening is not always possible.


08B05 Equational logic, Mal’tsev conditions
08B10 Congruence modularity, congruence distributivity
03C05 Equational classes, universal algebra in model theory
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