Amalgamation through quantifier elimination for varieties of commutative residuated lattices (Q661293)

The author aims to investigate the amalgamation property (AP) for varieties of semilinear commutative residuated lattices using model-theoretic methods. It is well known that if a first-order theory \(T(K)\) of some class \(K\) of structures admits quantifier elimination, then the class of models of \(T_\forall(K)\), the set of universal consequences of \(T(K)\), has the AP. By an unpublished result of Metcalfe, Montagna and Tsinakis (which can be derived from the paper [\textit{F. Montagna}, ``Interpolation and Beth's property in propositional many-valued logics: a semantic investigation'', Ann. Pure Appl. Logic 141, No. 1--2, 148--179 (2006; Zbl 1094.03011)]), if \(K\) is an elementary subclass of linearly ordered members of a variety \(V\) of semilinear commutative residuated lattices, then the whole \(V\) has the AP. This general observation is applied in the paper to the varieties of MV-algebras, product algebras, Gödel algebras (these varieties are already known to have AP), nilpotent minimum algebras, involutive uniform mingle logic algebras, and representable uniform algebras.