Many-valued complete distributivity
Many-valued complete distributivity
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发表时间：
2006-03
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通讯作者：
Hongliang Lai;Dexue Zhang
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作者：
Hongliang Lai;Dexue Zhang
在可交换单位量子上丰富的范畴可以被研究为广义的或多值的有序结构。由于格论中的许多概念（例如完全分配性）可以通过某些附加物的存在来表征，因此可以根据分类假设在多值环境中重新表述它们。因此，借助分类机器，建立多值完全格、多值完全分配格等理论是可能的。本文对多值完全分配性进行了系统的研究，包括以下主题：（1）多值完全分配格的子代数和商代数； (2)（左伴随）函子的类别； (3)多值完全分配性与真值量子性质的关系。结果表明，丰富范畴论在研究与阶相关的数学实体的多值版本时是一个非常有用的工具。
Categories enriched over a commutative unital quantale can be studied as generalized, or many-valued, ordered structures. Because many concepts, such as complete distributivity, in lattice theory can be characterized by existence of certain adjunctions, they can be reformulated in the many-valued setting in terms of categorical postulations. So, it is possible, by aid of categorical machineries, to establish theories of many-valued complete lattices, many-valued completely distributive lattices, and so on. This paper presents a systematical investigation of many-valued complete distributivity, including the topics: (1) subalgebras and quotient algebras of many-valued completely distributive lattices; (2) categories of (left adjoint) functors; and (3) the relationship between many-valued complete distributivity and properties of the quantale of truth values. The results show that enriched category theory is a very useful tool in the study of many-valued versions of order-related mathematical entities.