论文标题
与操作员的部分功能的差异代数：离散二元性和完成
Difference-restriction algebras of partial functions with operators: discrete duality and completion
论文作者
论文摘要
我们在部分函数的抽象代数类别与集合商类别之间展示了相关性。代数是那些可代表的原子代数为在相对补体和域限制下关闭的部分函数的集合；形态是完整的同态。这概括了原子布尔代数与集合类别之间的离散邻接。我们定义了代表代数的兼容完成，并表明我们的邻接诱导的单子可产生任何原子代表代数的兼容完成。作为推论，辅助限制在兼容完整的原子代表代数上的双重性，从而概括了完整的原子布尔代数和集合之间的离散二元性。然后，我们将这些辅助，二元性和完成结果扩展到具有任意额外的完全添加剂和兼容性保留运算符的代表代数。
We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative complement and domain restriction; the morphisms are the complete homomorphisms. This generalises the discrete adjunction between the atomic Boolean algebras and the category of sets. We define the compatible completion of a representable algebra, and show that the monad induced by our adjunction yields the compatible completion of any atomic representable algebra. As a corollary, the adjunction restricts to a duality on the compatibly complete atomic representable algebras, generalising the discrete duality between complete atomic Boolean algebras and sets. We then extend these adjunction, duality, and completion results to representable algebras equipped with arbitrary additional completely additive and compatibility preserving operators.
