Semiconic idempotent logic is a common generalization of intuitionistic logic, relevance logic with mingle, and semilinear idempotent logic. It is an algebraizable logic and it admits a cut-free hypersequent calculus. We give a structural decomposition of its characteristic algebraic semantics, conic idempotent residuated lattices, showing that each of these is an ordinal sum of simpler partially ordered structures. This ordinal sum is indexed by a totally ordered residuated lattice, which serves as its skeleton and is both a subalgebra and nuclear image. We equationally characterize the totally ordered residuated lattices appearing as such skeletons. Further, we describe both congruence and subalgebra generation in conic idempotent residuated lattices, proving that every variety generated by these enjoys the congruence extension property. In particular, this holds for semilinear idempotent residuated lattices. Based on this analysis, we obtain a local deduction theorem for semiconic idempotent logic, which also specializes to semilinear idempotent logic.

翻译：半圆锥幂等逻辑是直觉主义逻辑、带混合的相关逻辑及半线性幂等逻辑的共同推广。它是一种可代数化的逻辑，并允许无切割的超序列演算。我们给出了其特征代数语义——圆锥幂等剩余格——的结构分解，表明每个这样的结构均为更简单偏序结构的序和。该序和由一个全序剩余格索引，此全序剩余格作为其骨架，同时是子代数和核像。我们从方程角度刻画了作为此类骨架的全序剩余格。进一步，我们描述了圆锥幂等剩余格中的同余与子代数生成，证明由这些结构生成的每个簇都满足同余扩展性质。特别地，这一性质适用于半线性幂等剩余格。基于此分析，我们获得了半圆锥幂等逻辑的局部演绎定理，该定理同样可特化为半线性幂等逻辑的情形。