Quantifier elimination and other model-theoretic properties of BL-algebras
type: Item
instance of: scholarly article
title: Quantifier elimination and other model-theoretic properties of BL-algebras
zbMATH Open document ID: 1247.03137
DOI: 10.1215/00294527-1499336
author: Tommaso Cortonesi
author: Enrico Marchioni
author: Franco Montagna
published in: Notre Dame Journal of Formal Logic
publication date: 2011-12-21T00:00:00Z
review text: BL is short for Hájek Basic Logic, the logic of continuous t-norms and their adjoint implications. BL-algebras are the Lindenbaum algebras of theories in BL-logic, as described in \textit{P. Hájek}'s book [Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]. When the underlying order of a BL-algebra \(A\) is total, \(A\) is said to be a BL-chain. It is easy to see that the theory of all BL-algebras does not have quantifier elimination. The present paper features a model-theoretic analysis of the first-order theory of classes of BL-chains, with particular reference to classes generating the whole variety and whose first-order theories have quantifier elimination. Applications are given to the amalgamation problem for various classes of BL-algebras. Ultrahomogeneous models and the joint embedding properties are also considered, along with decidability and completeness. Finally, the authors prove a sort of BL-algebraic variant of the MV-algebraic Di Nola theorem. This sort of model-theoretical analysis for MV-algebras (the algebras of Łukasiewicz logic) was carried out by Lacava and Marchioni. Caicedo proved quantifier elimination for the chains arising from rational Pavelka logic. Quantifier elimination results for Gödel and product algebras were proved by Baaz and Veith. In an earlier paper, Marchioni established quantifier for certain theories of totally ordered commutative residuated lattices and obtained the amalgamation property of various classes of chains by purely model-theoretic means.
reviewed by: Daniele Mundici
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 06D35
zbMATH DE Number: 5991004
zbMATH Keywords: BL-algebra
zbMATH Keywords: quantifier elimination
zbMATH Keywords: model completeness
zbMATH Keywords: amalgamation
zbMATH Keywords: MV-algebra
zbMATH Keywords: Hájek Basic Logic
zbMATH Keywords: BL-chain
zbMATH Keywords: decidability
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00294527-1499336
OpenAlex ID: W1977812651
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description: scientific article; zbMATH DE number 5991004