Categories with Fuzzy Sets and Relations
Categories with Fuzzy Sets and Relations John Harding, Carol Walker, Elbert Walker Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, USA jhardingfhardy,elbertg@nmsu.edu Abstract We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categories. A discussion is made of extending these results to the setting of type-2 fuzzy sets. 1 Introduction A fuzzy set is a map A : X ! I from a set X to the unit interval I. Several authors [2, 6, 7, 20, 22] have considered fuzzy sets as a category, which we will call FSet, where a morphism from A : X ! I to B : Y ! I is a function f : X ! Y that satisfies A(x) ≤ (B◦f)(x) for each x 2 X. Here we continue this path of investigation. The underlying idea is to lift additional structure from the unit interval, such as t-norms, t-conorms, and negations, to provide additional structure on the category. Our eventual aim is to provide a setting where processes used in fuzzy control can be abstractly studied, much in the spirit of recent categorical approaches to processes used in quantum computation [1]. Order preserving structure on I, such as t-norms and conorms, lifts to provide additional covariant structure on FSet. In fact, each t-norm T lifts to provide a symmetric monoidal tensor ⊗T on FSet.
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