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Journal of Logic and Computation Advance Access published online on March 13, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp014
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© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

States on Bold Algebras: Categorical Aspects

Roman Fric

Mathematical Institute, Slovak Academy of Sciences, Gresákova 6, 040 01 Kosice, Slovak Republic and Catholic University in Ruzomberok, Pedagogical Faculty, Department of Mathematics, Nám.A. Hlinku 60, 034 01 Ruzomberok, Slovak Republic.
E-mail: fric{at}saske.sk

Received 2 October 2008.


  Abstract

We study bold algebras and states on bold algebras in the context of transition from classical probability theory to fuzzy probability theory. Our aim is to point out the role of bold algebras and states on bold algebras in a categorical approach to probability theory. In particular, we formulate several fundamental questions related to basic probability notions and constructions and provide possible answers in terms of bold algebras and states on bold algebras. We show that the category ID of D-posets of fuzzy sets and sequentially continuous difference homomorphisms can serve as a base category in which both classical and fuzzy probability theory can be developed and generalized. Classical and fuzzy random events such as fields of sets and measurable real-valued functions into the interval [0,1], considered as bold algebras, become special objects. Observables, considered as morphisms between objects, become dual to generalized random variables. States become morphisms into [0,1], considered as an object of ID. Properties of objects of ID follow from classical theorems of analysis such as the Lebesgue Dominated Convergence Theorem (states are sequentially continuous) and categorical constructions such as the product (the structure of a probability domain is completely determined by the states as the initial structure). We prove that each generated Lukasiewicz tribe is the epireflection of its underlying Butnariu–Klement {sigma}-field of sets. This helps to understand the transition from classical crisp random events to fuzzy random events. Indeed, the corresponding fuzzification is necessary to cover generalized random variables having a quantum character, i.e. fuzzy random variables in the Gudder–Bugajski sense sending a classical elementary event (point measure) to a non-trivial probability measure.

Keywords: Bold algebra; state; duality; D-poset of fuzzy sets; categorical probability theory


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