Remarks on the sequential effect algebras

TL;DR

This paper proves that sub-sequential effect algebras generated by commutative subsets are also commutative and investigates conditions under which elements are equal when scaled, addressing open problems in effect algebra theory.

Contribution

It affirms an open problem about the commutativity of sub-sequential effect algebras and explores conditions for element equality under scalar multiplication.

Findings

Sub-sequential effect algebras generated by commutative sets are commutative.

If c is sharp and a|b, then a=b when na=nb=c.

Counterexamples show conditions are necessary.

Abstract

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 $(E,0,1, \oplus, \circ)$ is a sequential effect algebra and $A$ is a commutative subset of $E$, then the sub-sequential effect algebra $\bar{A}$ generated by $A$ is also commutative. Next, we also study the following uniqueness problem: If $na=nb=c$ for some positive integer $n\geq 2$, 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.