The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
-
José Gil-Férez
,
Antonio Ledda
Abstract
The amalgamation property (AP) is of particular interest in the study of residuated lattices due to its relationship with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the AP. The variety SemRL of semilinear residuated lattices is a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the AP are semilinear. However, we prove that this is not the case, and in the process we establish that the same is true for the variety SemCanRL of semilinear cancellative residuated lattices. In addition, we prove that the variety whose members have a distributive lattice reduct and satisfy the identity x(y ∧ z)w ≈ xyw ∧ xzw also fails the AP.
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© 2015
Articles in the same Issue
- Editorial. Many-Valued Logic ’12
- Antonio Di Nola
- On the Modal μ-Calculus Over Finite Symmetric Graphs
- A Note on Saturated Models for Many-Valued Logics
- ℍ-Perfect Pseudo MV-Algebras and Their Representations
- Some Notes on Elimination Properties for The Theory of Riesz MV-Chains
- The Riesz Hull of a Semisimple MV-Algebra
- The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
- The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups
- Algebraically Closed Abelian l-Groups
- Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P
- Functional Many-Valued Relations
Articles in the same Issue
- Editorial. Many-Valued Logic ’12
- Antonio Di Nola
- On the Modal μ-Calculus Over Finite Symmetric Graphs
- A Note on Saturated Models for Many-Valued Logics
- ℍ-Perfect Pseudo MV-Algebras and Their Representations
- Some Notes on Elimination Properties for The Theory of Riesz MV-Chains
- The Riesz Hull of a Semisimple MV-Algebra
- The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
- The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups
- Algebraically Closed Abelian l-Groups
- Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P
- Functional Many-Valued Relations
