The average value inequality in sequential effect algebras

A sequential effect algebra (E, 0, 1, ., .) is an effect algebra on which a sequential product . with certain physics properties is defined; in particular, sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Gudder asked the following problem: If a, b ., (E, 0, 1, ., .) and a.b and a . b.a . b, is it the case that 2(a . b) . a 2 . b 2? In this paper, we construct an example to answer the problem negatively.