效应代数的表示

Representations of effect algebras

本文讨论了抽象效应代数的表示问题.对于一个抽象效应代数(E,⊕,0,1),如果存在一个Hilbert空间H和一个单态射φ:E→E(H),那么称E为可表示的且称(φ,H)是E的一个表示,其中ε(H)表示H上所有正压缩算子构成的效应代数.给出了一些可表示的和不可表示的效应代数的例子,证明了非空集X上的任一模糊集系统F和Boolean代数BX都是可表示的效应代数.

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 5r and Boolean algebra Bx on a nonempty set X are representable effect algebras.