Notes on congruence implication. (English) Zbl 0760.08002

Czédli, Gábor

Arch. Math., Brno 27b, 149-153 (1991).

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

Keywords:

Mal’tsev condition; locally equational class; lattice identities; implication in congruence varieties