标度广义效应代数和标度效应代数的结构

Construction of Scale Generalized Effect Algebras and Scale Effect Algebras

研究了标度广义效应代数与标度效应代数的代数结构,给出了比较完整的结果.通过引入全标度广义代数的概念,本文证明了区间[0,1)上的标度广义效应代数和单位区间[0,1]上的标度效应代数完全由单位区间上的阿基米德余模确定,标度广义效应代数恰同构于全标度广义代数的下集.若标度广义代数满足局部有限条件,则它同构于实数加法群的子群代数.满足(S)条件的标度效应代数同构于实数加法群的子群代数和全标度广义代数的字典序乘积的子代数.

The complete constructions of scale generalized effect algebras and scale effect algebras are studied in this paper. We introduce the concept of total scale generalized algebra, then we show that scale generalized effect algebras on the interval [0, 1） and scale effect algebras on the unit interval [0, 1] are completely determined by the Archimedean co-norm on the unit interval [0, 1]. Scale generalized effect algebras are exactly the lower set of total scale generalized algebras. Furthermore, if a scale general- ized effect algebra is locally finite, then it is isomorphic to a sub-algebra of real additive group. Scale effect algebra satisfying （S） condition is isomorphic to the lexicographic product of a sub-algebra of real additive group and a total scale generalized algebra.