Not each sequential effect algebra is sharply dominating

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.