2006/03/25 by Hongliang Lai, Lai, Hongliang, Dexue Zhang +1
Mathematics · #03G25 #06D10 #06F07 #18B35 #18D20 #68Q55 #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO) #math.CT #math.LO #msc:03G25 #msc:06D10 #msc:06F07 #msc:18B35 #msc:18D20 #msc:68Q55
paper · pdf · doi:10.48550/arxiv.math/0603590
35 pages
arxiv created 2006/05/12 · arxiv updated 2009/12/01
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.