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Many-valued complete distributivity

Hongliang Lai·Dexue Zhang
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Abstract

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.

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Hongliang Lai, Dexue Zhang. 2006-05-12. Many-valued complete distributivity. https://arxiv.org/abs/math/0603590

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