Not each sequential effect algebra is sharply dominating

TL;DR

This paper constructs a counterexample showing that not all sequential effect algebras are sharply dominating, addressing an open problem in effect algebra theory.

Contribution

The paper provides a specific example demonstrating that some sequential effect algebras are not sharply dominating, answering an open question posed by Gudder.

Findings

Counterexample disproves the universal sharply dominating property

Not all sequential effect algebras are sharply dominating

Addresses an open problem in effect algebra research

Abstract

Let $E$ be an effect algebra and $E_S$ be the set of all sharp elements of $E$. $E$ is said to be sharply dominating if for each $a\in E$ there exists a smallest element $\widehat{a}\in E_s$ such that $a\leq \widehat{a}$. In 2002, Professors Gudder and Greechie proved that each $\sigma$-sequential effect algebra is sharply dominating. In 2005, Professor Gudder presented 25 open problems in International Journal of Theoretical Physics, Vol. 44, 2199-2205, the 3th problem asked: Is each sequential effect algebra sharply dominating? Now, we construct an example to answer the problem negatively.