Model-completion of varieties of co-Heyting algebras

Luck Darnière (1) , Markus Junker (2)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 Mathematisches Institut, Abteilung für Mathematische Logik

It is known that exactly eight varieties of Heyting algebras have a model-completion, but no concrete axiomatisation 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 have a certain embedding property. For four of these six varieties (those which are locally finite) this actually provides a new proof of the existence of a model-completion, this time with an explicit and finite axiomatisation.

Mots clés: Heyting algebras; logic; model-completion; quantifier elimination

Domaines: Logique [math.LO]

Luck Darnière, Markus Junker. Model-completion of varieties of co-Heyting algebras. Houston Journal of Mathematics, 2018, 44 (1), pp.49-82. ⟨hal-00445886v2⟩