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Amalgamation through quantifier elimination for varieties of commutative residuated lattices

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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.

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Authors and Affiliations

  1. Artificial Intelligence Research Institute (IIIA), Spanish National Research Council (CSIC), Campus Universitat Autonoma de Barcelona, 08193, Bellaterra, Spain

    Enrico Marchioni

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  1. Enrico Marchioni
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Correspondence to Enrico Marchioni.

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The author would like to thank Vincenzo Marra for some very helpful discussions on the content of this paper. 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).

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Marchioni, E. Amalgamation through quantifier elimination for varieties of commutative residuated lattices. Arch. Math. Logic 51, 15–34 (2012). https://doi.org/10.1007/s00153-011-0251-x

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