效应代数上态射的注记

A Note on Morphisms on Effect Algebras

研究了效应代数上的态、态射和单同态的一些重要性质和给了它们的刻画;对于可加映射φ:E→F,证明了φ是一个态射当且仅当φ(a′)=(φ(a))′(■a∈E);φ是一个单同态当且仅当φ是反保序的,即φ(a)≤φ(b)■a≤b。最后,证明了:如果(P,≤)是一个偏序集且以0为最小元、1为最大元,φ是从P到[0,1]的一族映射且满足一定条件,则在P上存在部分二元运算⊕,使得(P,0,1,⊕)成为一个效应代数且以φ为一个序决定态系统。

A series of properties of morphisms, monomorphisms and states on an effect algebra are discussed, and characterizations of morphisms and monomorphisms are given. It is proved that if Ф ： E →F is additive, then it is morphism if and only if Ф（a＇） = （Ф（a））＇ （ a ∈ E）; and it is a monomorphism if and only if it is anti-order preserving, i.e., Ф（a） 〈 Ф（b） implies a ≤ b. Finally, it is shown that if （P, ≤） is a partially ordered set with the minimal and maxmual elements 0 and 1, respectively, and if Ф is a family of mappings from P into [0, 1] satisfying some conditions, then there exists a partial binary operation in P such that （P, 0, 1, ） becomes an effect algebra with Ф as an order-determining state system.