Paper detail

Extensions of witness mappings

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".