A two-dimensional observable is a special kind of a σ-homomorphism defined on the Borel σ-algebra of the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra. A two-dimensional spectral resolution is a mapping defined on the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra which has properties similar to a two-dimensional distribution function in probability theory. We show that there is a one-to-one correspondence between two-dimensional observables and two-dimensional spectral resolutions defined on a σ-complete MV-algebras as well as on the monotone σ-complete effect algebras with the Riesz decomposition property. The result is applied to the existence of a joint two-dimensional observable of two one-dimensional observables.

中文翻译：

---

二维观测值和光谱分辨率

二维可观测量是一种特殊的 σ-同态，定义在实平面的 Borel σ-代数上，其值在 σ-完全 MV-代数或单调 σ-完全效应代数中。二维光谱分辨率是在实平面上定义的映射，其值位于 σ-完全 MV-代数或单调 σ-完全效应代数中，其性质类似于概率论中的二维分布函数。我们表明，在 σ 完全 MV 代数以及具有 Riesz 分解性质的单调 σ 完全效应代数上定义的二维可观测量和二维光谱分辨率之间存在一一对应的关系。结果适用于两个一维可观察量的联合二维可观察量的存在性。