The Cartesian closedness of the category Fuzz and function spaces on topological fuzzes

Abstract

Abstract A fuzz is a pair (F,′) consisting of a completely distributive lattice F and an order reversing involution ′:F →F. A mapping ƒ : (F,′)→ (G, ′ ) preserving arbitrary joins is called an order homomorphism if f−1: (G, ′)→ (F, ′) preserves ′, where ƒ −1 (b) = ∨ {a ϵ F: ƒ(a) ⩽b} for every b ϵ G.The category Fuzz consists of all fuzzes and all order homomorphisms. In a fuzz a generalized topology has been defined and discussed. In this paper, we define function spaces in topological fuzzes. For this purpose, we have to consider on which fuzz the function space is set up. That is, for every pair of (topological) fuzzes F and G, it is necessary that to set up a fuzz GF such that it is ‘very like’ to the set of all mappings between two sets. More exactly, we hope to construct the exponents in the category Fuzz or to prove the Cartesian closedness of the category. By introducing a concept of parallelism of two mappings, the exponents in the category are obtained and, moreover, on the exponents pointwise convergence and uniformly convergence topologies are defined and proved with properties analogous to general topology. In particular, the former is proper and the latter is admissible. The limit of a net consisting of continuous order homomorphisms with respect to the uniformly convergence topology is continuous.