The nth root of sequential effect algebras

Abstract

In 2005, Gudder [Int. J. Theor. Phys. 44, 2219 (2005)] presented 25 problems of sequential effect algebras, the 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique? In this paper, we show that for each given positive integer n>1, there is a sequential effect algebra such that the nth root of its some element c is not unique, and the nth root of c is not the kth root of c (k<n). Thus, we answer the problem negatively.