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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 C
M 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, C
M is isomorphic to C
e
M, and C
e is adjoint to , where assigns to an object A of IRL0 the product A × A
0 which is an object of NRL. 相似文献
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-autonomous lattices are the algebraic exponentials and without additive constants. In this paper, we investigate the structure theory of this variety and some of its subvarieties, as well as its relationships with other classes of algebras. 相似文献