Non-distributive Kalman equivalence for Frobenius-adjoint residuated lattices

Extend the Kalman equivalence from the distributive variety FARDL to the non-distributive variety FARL by determining whether there exists a category of Frobenius-adjoint centered involutive residuated lattices and functors establishing an equivalence with FARL, whose restriction to the distributive subcategories is naturally isomorphic to the equivalence of Theorem 5.15; if no such equivalence exists, characterize the subvarieties of FARL admitting a Kalman-type equivalence and determine whether a largest such subvariety exists.

Background

The paper establishes a Kalman and positive-cone equivalence between the distributive category FARDL of Frobenius-adjoint residuated distributive lattices and the category FA of Frobenius-adjoint c-differential residuated distributive lattices satisfying CK. The corresponding non-distributive algebraic setting is FARL, the variety of Frobenius-adjoint residuated lattices, but the paper does not extend the categorical construction to that setting.

The unresolved issue is whether an appropriate category of Frobenius-adjoint centered involutive residuated lattices can support quasi-inverse Kalman and positive-cone functors for all of FARL. The problem also asks for a classification of the subvarieties where such an equivalence survives and for the existence of a maximal subvariety with this property.