## 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.

### MSC:

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