Amalgamation through quantifier elimination for varieties of commutative residuated lattices

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 T ∀ has the amalgamation property. Let Th(K) be the theory of an elementary subclass K of the linearly ordered members of a variety V of semilinear commutative residuated lattices. We show that whenever Th(K) has elimination of quantifiers, and every linearly ordered structure in V is a model of Th ∀(K), then 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. © 2011 Springer-Verlag.Moreover, Marchioni acknowledges partial support from the Spanish projects TASSAT (TIN2010-20967-C04-01), Agreement Technologies (CONSOLIDER CSD2007-0022, INGENIO 2010), the Generalitat de Catalunya grant 2009-SGR-1434, and Juan de la Cierva Program of the Spanish MICINN, as well as the ESF Eurocores-LogICCC/MICINN project (FFI2008-03126-E/FILO), and the Marie Curie IRSES Project (FP7-PEOPLE-2009).Peer Reviewe