On Some Categories of Involutive Centered Residuated Lattices

Abstract

Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K• relating integral residuated lattices with 0 (IRL0) with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the category of residuated lattices leads us to study other adjunctions and equivalences. For example, we treat the functor C whose domain is cuRL, the category of involutive residuated lattices M whose unit is fixed by the involution and has a Boolean complement c (the underlying set of CM is the set of elements greater or equal than c). If we restrict to the full subcategory NRL of cuRL of those objects that have a nilpotent c, then C is an equivalence. In fact, CM is isomorphic to CeM, and Ce is adjoint to (_), where (_) assigns to an object A of IRL0 the product A × A0 which is an object of NRL.