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Craig interpolation for semilinear substructural logics
Author(s) -
Marchioni Enrico,
Metcalfe George
Publication year - 2012
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201200004
Subject(s) - mathematics , idempotence , axiom , pure mathematics , property (philosophy) , algebraic semantics , interpolation (computer graphics) , commutative property , algebra over a field , characterization (materials science) , algebraic number , discrete mathematics , mathematical analysis , computer science , image (mathematics) , artificial intelligence , philosophy , materials science , geometry , epistemology , nanotechnology
The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R‐mingle with unit” logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model‐theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property.