Representations of effect algebras
ZhengLi CHEN, QiaoWei ZHANG, Huaixin Cao, Zhihua Guo · Scientia Sinica Mathematica · 2011
This note is devoted to introducing and discussing the representations of abstract effect algebras. An abstract effect algebra (E,⊕, 0, 1) is said to be representable if there exists a Hilbert space H and a monomorphism φ:E →ε(H), where ε(H) denotes the effect algebra consisting of all positive contractions on H. Such a pair (φ,H) is called a representation of (E,⊕, 0, 1). Many examples of effect algebras are given, some of which are representable and the others are not. It is also proved that every fuzzy set system F and Boolean algebra BX on a nonempty set X are representable effect algebras.