On Probability Domains |
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Authors: | Roman?Fri? Martin?Pap?o |
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Institution: | 1.Mathematical Institute,Slovak Academy of Sciences,Ko?ice,Slovak Republic;2.Department of Mathematics, Faculty of Education,Catholic University in Ru?omberok,Ru?omberok,Slovak Republic |
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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 S n 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 (μ 1,ν 1)≤(μ 2,ν 2) whenever μ 1≤μ 2 and ν 2≤ν 1) 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 I X ), has nice properties and basic probabilistic notions and constructions are categorical. For example, states are morphisms. We introduce the category S n D cogenerated by \(S_{n}=\{(x_{1},x_{2},\ldots ,x_{n})\in I^{n};\:\sum_{i=1}^{n}x_{i}\leq 1\}\) carrying the coordinatewise partial order, difference, and sequential convergence and we show how basic probability notions can be defined within S n D. |
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