广义博斯巴赫态
Generalized Bosbach States
摘要：
Bosbach状态表示一种概率地评估来自各种（交换或非交换）多值逻辑的公式的方式。它们被定义在对应于这些逻辑的代数上，值在$[0,1]$中。从观察到Bosbach态的定义中有$[0,1]$的标准MV-代数结构介入出发，本文引入了定义在剩余格上的Bosbach态，其值在剩余格上.我们被引导到两种类型的广义Bosbach状态，具有不同的行为。本文所证明的广义Bosbach态的性质，可以作为发展某些概率多值逻辑的代数基础。{ \bf Keywords}：Bosbach态，剩余格，MV-代数，$s$ -柯西完备化，度量完备化. { \bf MSC 2010}：主要06 F35。中学06 D35。
英文摘要：
Bosbach states represent a way of probabilisticly evaluating the formulas from various (commutative or non-commutative) many-valued logics. They are defined on the algebras corresponding to these logics with values in $[0,1]$. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of $[0,1]$, in this paper we introduce Bosbach states defined on residuated lattices with values in residuated lattices. We are led to two types of generalized Bosbach states, with distinct behaviours. The properties of generalized Bosbach states, proven in the paper, may serve as an algebraic foundation for developping some probabilistic many-valued logics. {\bf Keywords}: Bosbach states, residuated lattices, MV-algebras, $s$-Cauchy completion, metric completion. {\bf MSC 2010}: Primary 06F35. Secondary 06D35.
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