The purpose of this paper is to sketch an attack on the general problem of representing a composite physical system in terms of its constituent parts. For quantum-mechanical systems, this is traditionally accomplished by forming either direct sums or tensor products of the Hilbert spaces corresponding to the component systems. Here, a more general mathematical construction is given which includes the standard quantum-mechanical formalism as a special case.

We show how effect algebras arise in physics and how they can be used to tie together the observables, states and symmetries employed in the study of physical systems. We introduce and study the unifying notion of an effect-observable-state-symmetry-system (EOSS-system) and give both classical and quantum-mechanical examples of EOSS-systems.

We introduce and study the D-model, which reflects the simplest situation in which one wants to calibrate an observable. We discuss the question of representing the statistics of the D-model in the context of an effect algebra.

We relate so-called spin factors and generalized Hermitian (GH-) algebras, both of which are partially ordered special Jordan algebras. Our main theorem states that positive-definite spin factors of dimension greater than one are mathematically equivalent to generalized Hermitian algebras of rank two.

A synaptic algebra is a generalization of the self-adjoint part of a von Neumann algebra. In this article we extend to synaptic algebras the type-I/II/III decomposition of von Neumann algebras, AW∗-algebras, and JW-algebras.

Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras.We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a (...) direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice. (shrink)

A Heyting effect algebra is a lattice-ordered effect algebra that is at the same time a Heyting algebra and for which the Heyting center coincides with the effect-algebra center. Every HEA is both an MV-algebra and a Stone-Heyting algebra and is realized as the unit interval in its own universal group. We show that a necessary and sufficient condition that an effect algebra is an HEA is that its universal group has the central comparability and central Rickart properties.

We propose an alternative resolution of Simpson's paradox in multiple classification experiments, using a different maximum likelihood estimator. In the center of our analysis is a formal representation of free choice and randomization that is based on the notion of incompatible measurements.We first introduce a representation of incompatible measurements as a collection of sets of outcomes. This leads to a natural generalization of Kolmogoroff's axioms of probability. We then discuss the existence and uniqueness of the maximum likelihood estimator for a (...) probability weight on such a generalized sample space, given absolute frequency data.As a first example, we discuss an estimation problem with censured data that classically admits only biased ad hoc estimators.Next, we derive an explicit solution of the maximum likelihood estimation problem for a large class of experiments that arise from various kids of compositions of sample spaces. We identify the (categorical) direct sum of sample spaces as a representation of “free choice,” and the (categorical) direct product as a representation of “randomization.”Finally, we apply the foregoing discussion to the case of multiple classification experiments in order to show that there is no Simpson's paradox if the difference between free choice and randomization is recognized in the structure of the experiment.A comparison between our new estimator and the “usual” calculation can be summarized as follows: Pooling the data over one classification factor in the “usual” way in fact destroys or ignores the information contained in it, whereas our proposed maximum likelihood estimator is a proper marginal over this factor that “averages out” the information contained in it. The estimators agree with each other in the case of proportional sample sizes. (shrink)