Extensions of witness mappings

TL;DR

This paper investigates the extension of witness mappings in interval effect algebras to determine conditions under which a witness mapping for a subset can be extended to include an additional element, enhancing understanding of coexistence in these structures.

Contribution

It provides necessary and sufficient conditions for extending witness mappings in interval effect algebras, advancing the theoretical framework for coexistence analysis.

Findings

Extension exists iff a specific mapping e_t satisfies certain conditions

Main result applied to various relationships between t and S

Enhances understanding of coexistence in effect algebras

Abstract

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 $\beta$ for $S$, and an element $t\in E\setminus S$. We study the question whether there is a witness mapping $\beta_t$ for $S\cup\{t\}$ such that $\beta_t$ is an extension of $\beta$. 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 $\beta$, then $\beta_t$ exists".