Uniqueness and order in sequential effect algebras

A sequential effect algebra (SEA) is an effect algebra on which a sequential product is defined. We present examples of effect algebras that admit a unique, many and no sequential product. Some general theorems concerning unique sequential products are proved. We discuss sequentially ordered SEA's in which the order is completely determined by the sequential product. It is demonstrated that intervals in a sequential ordered SEA admit a sequential product

On realization of generalized effect algebras

A well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice $L({\mathcal H})$ of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra ${\mathcal G}_D({\mathcal H})$ of effects of a complex Hilbert space ${\mathcal H}$ iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Rie\v{c}anov\'a and Zajac. Further, any operator generalized effect algebra ${\mathcal G}_D({\mathcal H})$ possesses an order determining set of generalized states

Compression Bases in Unital Groups

We study unital groups with a distinguished family of compressions called a compression base. A motivating example is the partially ordered additive group of a von Neumann algebra with all Naimark compressions as the compression base.Comment: 8 page

Detection and prevention of financial abuse against elders

This article is made available through the Brunel Open Access Publishing Fund. Copyright @ The Authors. This article is published under the Creative Commons Attribution (CC BY 3.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/ by/3.0/legalcode.Purpose – This paper reports on banking and finance professionals' decision making in the context of elder financial abuse. The aim was to identify the case features that influence when abuse is identified and when action is taken. Design/methodology/approach – Banking and finance professionals (n=70) were shown 35 financial abuse case scenarios and were asked to judge how certain they were that the older person was being abused and the likelihood of taking action. Findings – Three case features significantly influenced certainty of financial abuse: the nature of the financial problem presented, the older person's level of mental capacity and who was in charge of the client's money. In cases where the older person was more confused and forgetful, there was increased suspicion that financial abuse was taking place. Finance professionals were less certain that financial abuse was occurring if the older person was in charge of his or her own finances. Originality/value – The research findings have been used to develop freely available online training resources to promote professionals' decision making capacity (www.elderfinancialabuse.co.uk). The resources have been advocated for use by Building Societies Association as well as CIFAS, the UK's Fraud Prevention Service.The research reported here was funded by the UK cross council New Dynamicsof Ageing Programme, ESRC Reference No. RES-352-25-0026, with Mary L.M. Gilhooly asPrincipal Investigator. Web-based training tools, developed from the research findings, weresubsequently funded by the ESRC follow-on fund ES/J001155/1 with Priscilla A. Harries asPrincipal Investigator

An Intrisic Topology for Orthomodular Lattices

We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in the case of a boolean algebra, the obtained topology is the discrete one. Thus, our construction provides a general tool for studying orthomodular lattices but also a way to distinguish classical and quantum logics.Comment: Under submission to the International Journal of Theoretical Physic

Scopes and Limits of Modality in Quantum Mechanics

We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems.Comment: 9 pages, no figure

Kochen-Specker Sets and Generalized Orthoarguesian Equations

Every set (finite or infinite) of quantum vectors (states) satisfies generalized orthoarguesian equations ($n$OA). We consider two 3-dim Kochen-Specker (KS) sets of vectors and show how each of them should be represented by means of a Hasse diagram---a lattice, an algebra of subspaces of a Hilbert space--that contains rays and planes determined by the vectors so as to satisfy $n$OA. That also shows why they cannot be represented by a special kind of Hasse diagram called a Greechie diagram, as has been erroneously done in the literature. One of the KS sets (Peres') is an example of a lattice in which 6OA pass and 7OA fails, and that closes an open question of whether the 7oa class of lattices properly contains the 6oa class. This result is important because it provides additional evidence that our previously given proof of noa =< (n+1)oa can be extended to proper inclusion noa < (n+1)oa and that nOA form an infinite sequence of successively stronger equations.Comment: 16 pages and 5 figure

Information-theoretic principle entails orthomodularity of a lattice

Quantum logical axiomatic systems for quantum theory usually include a postulate that a lattice under consideration is orthomodular. We propose a derivation of orthomodularity from an information-theoretic axiom. This provides conceptual clarity and removes a long-standing puzzle about the meaning of orthomodularity.Comment: Version prior to published, with slight modification

On the lattice structure of probability spaces in quantum mechanics

Let C be the set of all possible quantum states. We study the convex subsets of C with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of quantum mechanics, including entanglement and Jaynes' Max-Ent principle. We also encounter links with entanglement witnesses, which leads to a new separability criteria expressed in lattice language. We also provide an extension of a separability criteria based on convex polytopes to the infinite dimensional case and show that it reveals interesting facets concerning the geometrical structure of the convex subsets. It is seen that the above mentioned framework is also capable of generalization to any statistical theory via the so-called convex operational models' approach. In particular, we show how to extend the geometrical structure underlying entanglement to any statistical model, an extension which may be useful for studying correlations in different generalizations of quantum mechanics.Comment: arXiv admin note: substantial text overlap with arXiv:1008.416