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首页> 外文期刊>Journal of logic and computation >States on Bold Algebras: Categorical Aspects
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States on Bold Algebras: Categorical Aspects

机译：大胆代数上的状态：分类方面

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摘要

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 a-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.
机译：在从经典概率论到模糊概率论的过渡中，我们研究了大胆的代数和大胆的代数上的状态。我们的目的是指出在概率论的一种分类方法中，粗体代数和状态在粗体代数上的作用。特别是，我们提出了一些与基本概率概念和构造有关的基本问题，并提供了关于粗体代数和粗体代数上的状态的可能答案。我们表明，模糊集和连续连续差分同态的D姿态的类别ID可以作为基础类别，经典和模糊概率理论都可以得到发展和推广。经典和模糊随机事件（例如集合的字段和进入区间[0,1]的可测量实值函数）（被视为粗体代数）成为特殊对象。被视为对象之间的态射的可观变量变成广义随机变量的对偶。状态变成[0,1]的态射，被视为ID的对象。 ID对象的属性来自经典的分析定理，例如Lebesgue支配的收敛定理（状态是连续连续的）和分类构造（例如乘积）（概率域的结构完全由状态确定为初始结构）。我们证明，每个产生的卢卡西维奇部落都是其底层Butnariu-Klement a域集合的外向反射。这有助于理解从经典的清晰随机事件到模糊随机事件的过渡。实际上，相应的模糊化对于覆盖具有量子特征的广义随机变量是必要的，即，在Gudder-Bugajski意义上的模糊随机变量将经典基本事件（点测度）发送到非平凡概率测度。

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