Effect test spaces and effect algebras

Stanley Gudder

Sylvia Pulmannová

Compatibility relations in effect algebras and their connections with refinements of the orthogonal partitions of unity are studied. Properties of blocks as maximal sets of compatible elements are discussed. Some special kinds of effect algebras are characterized using properties of compatibility. Using refinements, an additional structure on the effect test spaces is introduced and used to a characterization of different types of effect algebras.

W. A. Majewski, Tomasz I. Tylec

Sander Uijlen

This thesis presents two new approaches to two topics in quantum foundations: Part I: The study of causal structures and higher order processes, i.e., transformations of processes.Part II: The study of non-locality and contextuality using effect algebras.embed in a category of presheaves and this allows us to make a connection between the two approaches using adjunctions.In particular, we are able to relate our effect algebraic framework to the established presheaf approach of Abramsky and Brandenburger.In a similar fashion we relate effect algebras to test spaces.We describe explicitly the examples of Bell, Kochen-Specker, Hardy and GHZ.

Stanley Gudder

Aneesh Raghavan, John S. Baras

In multi-agent systems, agents observe data, and use them to make inferences and take actions. As a result sensing and control naturally interfere, more so from a real-time perspective. A natural consequence is that in multi-agent systems there are propositions based on the set of observed events that might not be simultaneously verifiable, which leads to the need for probability structures that allow such \textit{incompatible events}. We revisit the structure of events in a multi-agent system and we introduce the necessary new models that incorporate such incompatible events in the formalism. These models are essential for building non-commutative probability models, which are different than the classical models based on the Kolmogorov construction. From this perspective, we revisit the concepts of \textit{event-state-operation structure} and the needed \textit{relationship of incompatibility} from the literature and use them as a tool to study the needed new algebraic structure of the set of events. We present an example from multi-agent hypothesis testing where the set of events does not form a Boolean algebra, but forms an ortholattice. A possible construction of a `noncommutative probability space', accounting for \textit{incompatible events} is discussed. We formulate and solve the binary hypothesis testing problem in the noncommutative probability space. We illustrate the occurrence of `order effects' in the multi-agent hypothesis testing problem by computing the minimum probability of error that can be achieved with different orders of measurements.