States as Morphisms

Domains of generalized probability have been introduced in order to provide a general construction of random events, observables and states. It is based on the notion of a cogenerator and the properties of product. We continue our previous study and show how some other quantum structures fit our categorical approach. We discuss how various epireflections implicitly used in the classical probability theory are related to the transition to fuzzy probability theory and describe the latter probability theory as a genuine categorical extension of the former. We show that the IF-probability can be studied via the fuzzy probability theory. We outline a "tensor modification" of the fuzzy probability theory.

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“…An interested reader can find additional information on D-posets and related quantum structures, e.g. in [4,5,7,23,29,31,32]. In FPT (cf.…”

Section: From Classical To Fuzzymentioning

confidence: 99%

On Probability Domains III

Frič

Papčo

2015

Int J Theor Phys

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“…They generalize ( [4]) various structures, e.g. D-lattices, orthoalgebras, Boolean algebras, MV-algebras and provide a category in which states and observables become morphisms [3]. Recall that a D-poset is a partially order set X with the least element 0 X , the greatest element 1 X , and a partial binary operation called difference, such that a ⊖ b is defined iff b ≤ a, and the following axioms are assumed:…”

Section: Domains Of Probabilitymentioning

confidence: 99%

Fuzzification of probabilistic objects

Papčo¹

2013

Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology

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A categorical approach to probability allows to put basic notions of probability into a broader mathematical perspective, to evaluate their roles, and mutual relationships. Classical probability theory and fuzzy probability theory lead to two particular categories and their relationship (in categorical terms) enable us to understand and explicitly formulate the difference between them. Using our previous results, we show that the category ID of D-posets of fuzzy sets provides a framework in which the transition from classical to fuzzy probability theory is the consequence of some natural assumptions imposed on classical notions. Probability domains are constructed via suitable cogenerators and we study the transition in terms of the fuzzification of classical Boolean cogenerator. We introduce two categories CP and FP of probability spaces and observables corresponding to the classical probability theory and the fuzzy probability theory, respectively. We show that CP is isomorphic to a subcategory of FP.

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“…For example, using particular results from [12], [6], [10], [13], [18], it has been proved in [14] that the fuzzy probability theory developed by S. Gudder (cf. [15]) and S. Bugajski (cf.…”

Section: Introductionmentioning

confidence: 99%

Simplex-valued probability

Frič¹

2010

Mathematica Slovaca

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ABSTRACT. We continue our study of generalized probability from the viewpoint of category theory. Assuming that each generalized probability measure is a morphism, we model basic probabilistic notions within the category cogenerated by its range. It is known that the closed unit interval I = [0, 1], carrying a suitable difference structure, cogenerates the category ID in which the classical and fuzzy probability theories can be modeled. We study generalized probability theories modeled within two different categories cogenerated by a simplexx i ≤ 1 , carrying a suitable difference structure; since I and S 1 coincide, for n = 1 we get the fuzzy probability theory as a special case. In the first case, when the morphisms preserve the so-called pure elements, the resulting category S n D, n > 1, and ID are isomorphic and the generalized probability theories modeled in ID and S n D are "the same". In the second case, when the morphisms map each maximal element to a maximal element, the resulting categories W S n D, n > 1, lead to different models of generalized probability theories. We define basic notions of the corresponding simplex-valued probability theories and mention some applications.

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States as Morphisms

Cited by 12 publications

References 17 publications

On Probability Domains III

On Probability Domains III

Fuzzification of probabilistic objects

Simplex-valued probability

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