Remarks on the sequential effect algebras

In this paper, first, we answer affirmatively an open problem which was presented in 2005 by professor Gudder on the sub-sequential effect algebras. That is, we prove that if (E, 0, 1, ⊕, o) is a sequential effect algebra and A is a commutative subset of E, then the sub-sequential effect algebra Ā generated by A is also commutative. Next, we also study the following uniqueness problem: If na = nb = c for some positive integer n ≥ 2, then under what conditions a = b hold? We prove that if c is a sharp element of E and a|b, then a = b. We give also two examples to show that neither of the above two conditions can be discarded.