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Model completion of varieties of co-Heyting algebras. (English) Zbl 1434.06001

Summary: It is known that exactly eight varieties of Heyting algebras have a model-completion. However no concrete axiomatization of these model-completions were known by now except for the trivial variety (reduced to the one-point algebra) and the variety of Boolean algebras. For each of the six remaining varieties we introduce two axioms and show that 1) these axioms are satisfied by all the algebras in the model-completion, and 2) all the algebras in this variety satisfying these two axioms satisfy a certain remarkable embedding theorem. For four of these six varieties (those which are locally finite) these two results provide a new proof of the existence of a model-completion with, in addition, an explicit and finite axiomatization.

MSC:

06D20 Heyting algebras (lattice-theoretic aspects)
03C10 Quantifier elimination, model completeness, and related topics
06B20 Varieties of lattices
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