Conic Idempotent Residuated Lattices
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We give a structural decomposition of conic idempotent residuated lattices, showing that each of them is an ordinal sum of certain 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, and we equationally characterize which totally ordered residuated lattices appear as such skeletons. Using the two inverse operations induced by the residuals, we further characterize both congruence and subalgebra generation in conic idempotent residuated lattices. We show that every variety generated by conic idempotent residuated lattices enjoys the congruence extension property. In particular, this holds for semilinear idempotent residuated lattices. Moreover, we provide a detailed analysis of the structure of idempotent residuated chains serving as index sets 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 enjoy the amalgamation property, the natural class of rigid and conjunctive conic idempotent residuated lattices has the strong amalgamation property, and consequently has surjective epimorphisms. We extend this result to the variety generated by rigid and conjunctive conic idempotent residuated lattices, and establish the amalgamation, strong amalgamation, and epimorphism-surjectivity properties for several important subvarieties. Based on this algebraic work, we obtain local deduction theorems, the deductive interpolation property, and the projective Beth definability property for the corresponding substructural logics.
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