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Semiconic idempotent logic sCI is a common generalization of intuitionisticlogic, semilinear idempotent logic sLI, and in particular relevance logic withmingle. We establish the projective Beth definability property and thedeductive interpolation property for many extensions of sCI, and identifyextensions where these properties fail. We achieve these results by studyingthe (strong) amalgamation property and the epimorphism-surjectivity propertyfor the corresponding algebraic semantics, viz. semiconic idempotent residuatedlattices. Our study is made possible by the structural decomposition of conicidempotent models achieved in the prequel, as well as a detailed analysis ofthe structure of idempotent residuated chains serving as index sets in thisdecomposition. Here we study the latter on two levels: as certain enrichedGalois connections and as enhanced monoidal preorders. Using this, we show thatalthough conic idempotent residuated lattices do not have the amalgamationproperty, the natural class of rigid and conjunctive conic idempotentresiduated lattices has the strong amalgamation property, and thus hassurjective epimorphisms. This extends to the variety generated by rigid andconjunctive conic idempotent residuated lattices, and we establish the (strong)amalgamation and epimorphism-surjectivity properties for several importantsubvarieties. Using the algebraizability of sCI, this yields the deductiveinterpolation property and the projective Beth definability property for thecorresponding substructural logics extending sCI.

Semiconic Idempotent Logic II: Beth Definability and Deductive Interpolation