The n-th root of sequential effect algebras

Abstract

Sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Professor Gudder presented 25 open problems to motivate its study. The 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique ? We can strengthen the problem as following: For each given positive integer $n>1$, is there a sequential effect algebra such that the n-th root of its some element $c$ is not unique and the n-th root of $c$ is not the k-th root of $c$ ($k<n$) ? Recently, we answered the strengthened problem affirmatively.