Algebraic structure theory and interpolation failures in semilinear logics

Abstract

In this work we study integral residuated chains, and we solve some open problems related to the study of the amalgamation property in varieties of residuated lattices, or equivalently, about the deductive interpolation property in substructural logics. More precisely, we find a V-formation consisting of 2-potent finite commutative integral chains that does not have an amalgam, nor a one-amalgam, in residuated chains; as most relevant consequences, this entails that the following varieties do not have the amalgamation property: semilinear commutative (integral) residuated lattices, MTL-algebras, involutive and pseudocomplemented MTL-algebras, and all of their n-potent subvarieties for n strictly larger than 1. These results entail the failure of the deductive interpolation property for the corresponding substructural logics.