见证映射的扩张
Extensions of witness mappings
摘要：
我们使用见证映射的概念来处理区间效应代数中的共存问题。设给定区间效应代数$E$、$E$的共存子集$S$、$S$的见证映射$β$和元素$t\in E\setminus S$ .我们研究了是否存在$S\cup\{t\}$的见证映射$β_t$，使得$β_t$是$β$的扩展.在主要结果中，我们证明了这样一个扩张存在的充要条件是存在一个从$S$的有限子集到$E$的映射$e_t$满足一定条件。然后，主要结果被多次应用，以证明“如果$t$与$S$和$β$具有某某关系，则$β_t$存在”类型的主张。
英文摘要：
We deal with the problem of coexistence in interval effect algebras using the notion of a witness mapping. Suppose that we are given an interval effect algebra $E$, a coexistent subset $S$ of $E$, a witness mapping $β$ for $S$, and an element $t\in E\setminus S$. We study the question whether there is a witness mapping $β_t$ for $S\cup\{t\}$ such that $β_t$ is an extension of $β$. In the main result, we prove that such an extension exists if and only if there is a mapping $e_t$ from finite subsets of $S$ to $E$ satisfying certain conditions. The main result is then applied several times to prove claims of the type "If $t$ has a such-and-such relationship to $S$ and $β$, then $β_t$ exists".
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