On Probability Domains

Abstract

Motivated by IF-probability theory (intuitionistic fuzzy), we study n-component probability domains in which each event represents a body of competing components and the range of a state represents a simplex Sn of n-tuples of possible rewards-the sum of the rewards is a number from [0,1]. For n=1 we get fuzzy events, for example a bold algebra, and the corresponding fuzzy probability theory can be developed within the category ID of D-posets (equivalently effect algebras) of fuzzy sets and sequentially continuous D-homomorphisms. For n=2 we get IF-events, i. e., pairs (μ,ν) of fuzzy sets μ,ν∈[0,1]X such that μ(x)+ν(x)≤1 for all x∈X, but we order our pairs (events) coordinatewise. Hence the structure of IF-events (where (μ1,ν1)≤(μ2,ν2) whenever μ1≤μ2 and ν2≤ν1) is different and, consequently, the resulting IF-probability theory models a different principle. The category ID is cogenerated by I=[0,1] (objects of ID are subobjects of powers IX), has nice properties and basic probabilistic notions and constructions are categorical. For example, states are morphisms. We introduce the category SnD cogenerated by carrying the coordinatewise partial order, difference, and sequential convergence and we show how basic probability notions can be defined within SnD. © 2009 Springer Science+Business Media, LLC.