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 × A 0 which is an object of NRL.

In this article, we introduce and study a family of compatible functions in Hilbert algebras which in the case of Heyting algebras agree with the frontal operators given by Esakia (2006, J. Appl. 16,[349][350][351][352][353][354][355][356][357][358][359][360][361][362][363][364][365][366]. Moreover, we give a representation theory, based on previous works by Cabrer, Celani and Montangie, for Hilbert algebras with a frontal operator and for Hilbert algebras with some particular frontal operators.

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In this paper we expand previous results obtained in [2] about the study of categorical equivalence between the category IRL0 of integral residuated lattices with bottom, which generalize MV -algebras and a category whose objects are called c-differential residuated lattices. The equivalence is given by a functor K • , motivated by an old construction due to J. Kalman, which was studied by Cignoli in [3] in the context of Heyting and Nelson algebras. These results are then specialized to the case of MV -algebras and the corresponding category MV • of monadic MV -algebras induced by "Kalman's functor" K • . Moreover, we extend the construction to `-groups introducing the new category of monadic `-groups together with a functor Γ ] , that is "parallel" to the well known functor Γ between `-groups and MV -algebras.

The (dual) Dold-Kan correspondence says that there is an equivalence of categories K : Ch ¿0 → Ab between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR * → Rings , although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR * and Rings are Quillen closed model categories and the total left derived functor of K is an equivalence:LK : Ho DGR * ∼ → Ho Rings :The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through di erent methods. Our proof is based on a functor Q : DGR * → Rings , naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-Kostant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module [n] → n R S that arises from a homomorphism R → S of not necessarily commutative rings, using the coproduct R of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S ⊗ n R originally deÿned by Nuss K-theory 12 (1997) 23, using braids.