Semiconic idempotent logic sCI is a common generalization of intuitionistic logic, semilinear idempotent logic sLI, and in particular relevance logic with mingle. We establish the projective Beth definability property and the deductive interpolation property for many extensions of sCI, and identify extensions where these properties fail. We achieve these results by studying the (strong) amalgamation property and the epimorphism-surjectivity property for the corresponding algebraic semantics, viz. semiconic idempotent residuated lattices. Our study is made possible by the structural decomposition of conic idempotent models achieved in the prequel, as well as a detailed analysis of the structure of idempotent residuated chains serving as index sets in this decomposition. Here we study the latter on two levels: as certain enriched Galois connections and as enhanced monoidal preorders. Using this, we show that although conic idempotent residuated lattices do not have the amalgamation property, the natural class of rigid and conjunctive conic idempotent residuated lattices has the strong amalgamation property, and thus has surjective epimorphisms. This extends to the variety generated by rigid and conjunctive conic idempotent residuated lattices, and we establish the (strong) amalgamation and epimorphism-surjectivity properties for several important subvarieties. Using the algebraizability of sCI, this yields the deductive interpolation property and the projective Beth definability property for the corresponding substructural logics extending sCI.

翻译：半锥幂等逻辑sCI是直觉主义逻辑、半线性幂等逻辑sLI，特别是包含混合的相关性逻辑的共同推广。我们为sCI的许多扩展建立了投影Beth可定义性质和演绎插值性质，并指出了这些性质失效的扩展。我们通过研究相应代数语义（即半锥幂等剩余格）的（强）合并性质和满同态-满射性质来获得这些结果。我们的研究得益于前序工作中实现的锥幂等模型的结构分解，以及对该分解中作为索引集的幂等剩余链结构的详细分析。这里我们在两个层面上研究后者：作为特定增强的Galois连接和作为增强的幺半群预序。利用这一点，我们证明尽管锥幂等剩余格不具有合并性质，但刚性且合取的锥幂等剩余格的自然类具有强合并性质，从而具有满射满同态。这推广到由刚性且合取的锥幂等剩余格生成的簇，并且我们为几个重要的子簇建立了（强）合并和满同态-满射性质。利用sCI的可代数化性，这为扩展sCI的相应子结构逻辑推导出了演绎插值性质和投影Beth可定义性质。