关于模糊结构推广的概率论中的奇异演绎系统
Fantastic deductive systems in probability theory on generalizations of fuzzy structures
摘要：
本文的目的是介绍关于模糊结构推广的奇异演绎系统的概念，并强调它们在这些代数上的概率论中的作用。给出了交换伪BE代数的一个刻划，并将一个由四个恒等式组成的公理系统推广到交换伪BE代数的情形。我们定义了伪BE代数的奇异演绎系统，并研究了它们的性质。证明了如果伪BE（A）代数$A$是交换的，则它的所有演绎系统$A$都是奇异的.此外，我们还将测度、状态测度和测度态射的概念推广到伪BE代数的情形，并证明了有界伪BE代数上的Bosbach态集与其状态测度集之间存在一一对应关系。将伪BCK代数上的内部状态和状态态射算子的概念推广到伪BE代数上，证明了伪BE代数上的任何II型状态算子都是伪BE代数上的状态态射算子，定义并研究了伪BE代数上的伪赋值和交换伪赋值的概念.对于交换伪BE代数，我们证明了这两类伪赋值是一致的。给出了伪赋值和交换伪赋值的刻画。我们表明，内核的Bosbach状态（状态态射，措施，II型状态算子，伪估值）是一个梦幻般的演绎系统。
英文摘要：
The aim of this paper is to introduce the notion of fantastic deductive systems on generalizations of fuzzy structures, and to emphasize their role in the probability theory on these algebras. We give a characterization of commutative pseudo-BE algebras and we generalize an axiom system consisting of four identities to the case of commutative pseudo-BE algebras. We define the fantastic deductive systems of pseudo-BE algebras and we investigate their properties. It is proved that, if a pseudo-BE(A) algebra $A$ is commutative, then all deductive systems of $A$ are fantastic. Moreover, we generalize the notions of measures, state-measures and measure-morphisms to the case of pseudo-BE algebras and we also prove that there is a one-to-one correspondence between the set of all Bosbach states on a bounded pseudo-BE algebra and the set of its state-measures. The notions of internal states and state-morphism operators on pseudo-BCK algebras are extended to the case of pseudo-BE algebras and we also prove that any type II state operator on a pseudo-BE algebra is a state-morphism operator on it. The notions of pseudo-valuation and commutative pseudo-valuation on pseudo-BE algebras are defined and investigated. For the case of commutative pseudo-BE algebras we prove that the two kind of pseudo-valuations coincide. Characterizations of pseudo-valuations and commutative pseudo-valuations are given. We show that the kernel of a Bosbach state (state-morphism, measure, type II state operator, pseudo-valuation) is a fantastic deductive system.
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