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.
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机译：由J. Kalman提出的将分布晶格和居中Kleene代数相关联的旧结构所激发，我们定义了函子K？。将带有0（IRL0）的积分剩余格与某些渐开线剩余格相关联。我们的工作还基于Cignoli获得的关于Heyting和Nelson代数之间的附加项的结果，该结果丰富了晶格和Kleene代数之间的基本附加项。将函子提升到剩余格的类别，使我们研究了其他附加条件和等价形式。例如，我们处理其函数域为cuRL的函子C，这是对合残差格矩阵M的类别，其单位由对合固定并具有布尔补码c（CM的基础集合是大于或等于c的元素集合） 。如果我们将具有零幂的那些对象限制为cuRL的完整子类别NRL，则C是等价的。实际上，C M与C e M同构，并且C e与C e M同构，其中，将IRL0的对象A分配为NRL的对象的乘积A×A 0。
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