The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

Effect algebras and unsharp quantum logics

First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four “sine quibus non” conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications. Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one. Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications. Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.

http://sahmir.persiangig.com/document/Volume%2094,%20Number%202%20%20March,%2020/A%20Categorical%20Approach%20to%20Probability%20Theory.pdf

A Categorical Approach to Probability Theory

Preface 1. Planck's radiation law and the Einstein coefficients 2. Quantum mechanics of the atom-radiation interaction 3. Classical theory of optical fluctuations and coherence 4. Quantization of the radiation field 5. Single-mode quantum optics 6. Multimode and continuous-mode quantum optics 7. Optical generation, attenuation and amplification 8. Resonance fluorescence and light scattering 9. Nonlinear quantum optics Index

The quantum theory of light.

As a noncommutative generalization of effect algebras, we introduce pseudoeffect algebras and list some of their basic properties. For the purpose of a structure theory, we further define several kinds of Riesz-like properties for pseudoeffect algebras and show how they are interrelated.

Pseudoeffect Algebras. I. Basic Properties

A physically natural generalization of the notion of observable that encompasses both the classical and the quantum ones is derived. Based on it, the idea of the classical extension of a theory is developed; the states of the extended theory being the probability measures on the pure states of the original one. It is shown that quantum theory admits such a classical extension and that the qualifying features of quantum observables are preserved in the extended model.

https://iopscience.iop.org/article/10.1088/0305-4470/28/12/007/pdf

A classical extension of quantum mechanics

Interval and Scale Effect Algebras

The notion of a Sasaki projectionon an orthomodular lattice is generalized to a mapping Φ: E × E → E, where E is an effect algebra. If E is lattice ordered and Φ is symmetric, then E is called a Φ-symmetric effect algebra.This paper launches a study of such effect algebras. In particular, it is shown that every interval effect algebra with a lattice-ordered ambient group is Φ-symmetric, and its group is the one constructed by Ravindran in his proof that every effect algebra that has the Riesz decomposition property is an interval algebra. It is shown that the doubling construction introduced in the paper is connected to the conditional event algebrasof Goodman, Nguyen, and Walker.

Phi-symmetric effect algebras

The authors investigate the lattice Co(P) of convex subsets of a general partially ordered set P. In particular, they determine the conditions under which Co(P) and Co(Q) are isomorphic; and give necessary and sufficient conditions on a lattice L so that L is isomorphic to Co(P) for some P.

The convexity lattice of a poset

We define a tensor product via a universal mapping property on the class oforthoalgebras, which are both partial algebras and orthocomplemented posets. We show how to construct such a tensor product forunital orthoalgebras, and use the Fano plane to show that tensor products do not always exist.

M. K. Bennett

Papers

Effect algebras and unsharp quantum logics

Interval and Scale Effect Algebras

Phi-symmetric effect algebras

The convexity lattice of a poset

Tensor products of orthoalgebras