Amalgamation through quantifier elimination for varieties of commutative residuated lattices

Abstract

This work presents a model-theoretic approach to the study of the amalgamation property for varieties of semilinear commutative residuated lattices. It is well-known that if a first-order theory T enjoys quantifier elimination in some language L, the class of models of the set of its universal consequences $${\rm T_\forall}$$ has the amalgamation property. Let $${{\rm Th}(\mathbb{K})}$$ be the theory of an elementary subclass $${\mathbb{K}}$$ of the linearly ordered members of a variety $${\mathbb{V}}$$ of semilinear commutative residuated lattices. We show that whenever $${{\rm Th}(\mathbb{K})}$$ has elimination of quantifiers, and every linearly ordered structure in $${\mathbb{V}}$$ is a model of $${{\rm Th}_\forall(\mathbb{K})}$$ , then $${\mathbb{V}}$$ has the amalgamation property. We exploit this fact to provide a purely model-theoretic proof of amalgamation for particular varieties of semilinear commutative residuated lattices.