Orthomodular Posets Research Articles

Łukasiewicz Operations in Fuzzy Set and Many-Valued Representations of Quantum Logics

It, is shown that Birkhoff –von Neumann quantum logic (i.e., an orthomodular lattice or poset) possessing an ordering set of probability measures S can be isomorphically represented as a family of fuzzy subsets of S or, equivalently, as a family of propositional functions with arguments ranging over S and belonging to the domain of infinite-valued Łukasiewicz logic. This representation endows BvN quantum logic with a new pair of partially defined binary operations, different from the order-theoretic ones: Łukasiewicz intersection and union of fuzzy sets in the first case and Łukasiewicz conjunction and disjunction in the second. Relations between old and new operations are studied and it is shown that although they coincide whenever new operations are defined, they are not identical in general. The hypothesis that quantum-logical conjunction and disjunction should be represented by Łukasiewicz operations, not by order-theoretic join and meet is formulated and some of its possible consequences are considered.

Direct Limits of Effect Algebras

In this paper, we prove that direct limits exist in the category of effect algebrasand effect algebra-morphisms. Then, as a consequence, we obtain similar knownresults for the categories of orthomodular posets and orthomodular lattices.

Concrete Quantum Logics

Concrete quantum logics are quantum logics which allow for a set representation.They seem to be of significant conceptual value within quantum axiomatics andthey play an important role in the theory of orthomodular structures asset-representable orthomodular posets or lattices and they also sometimes constitutea “domain” for investigations in “noncommutative.” measure theory. This paperpresents a survey of recent results on this class of logics. Stress is put on thealgebraic and measure-theoretic aspects. Several open questions relevant tothe logicoalgebraic foundation of quantum theories are posed.

Disjunctivity and Alternativity in Projection Logics

We study the notions of disjunctivity and alternativity of orthomodular posets inthe context of orthoprojections or skew projections in C *-algebras.

Fuzzy set representations of some quantum structures

Quantum structures are algebraic structures like quantum logics (orthomodular posets and lattices), orthoalgebras, effect algebras (D-posets) which model event structures of quantum mechanics. The elements of orthomodular posets can be represented by question observables having their spectrum in {0, 1}, so they are considered as two-valued physical quantities. We show that if such a structure possesses an order determining system of states (=probability measures), it can be represented by a system of fuzzy sets and conversely.

An Overview of the 1997 IQSA Meeting in Atlanta

The International Quantum Structures Association (IQSA) met in Atlanta the week of October 12, 1997, principally to plan its 1998 biannual meeting. This was a historic meeting as it was the first to be held in the United States. As is its custom whenever its members assemble, scientific ideas were exchanged, not only informally, but also in the form of research papers presented formally to the group. Many of these papers have been written up for publication. Some of them have been published elsewhere; others are being modified and will eventually appear in an evolved form, but a representative portion of them were submitted to this journal, have survived the refereeing process, and are assembled here to be published together. The papers are fairly representative of IQSA’ s activities. They are loosely assembled into three clusters. The first consists of those papers which focus on the so-called orthomodular structures or closely related structures. These are the (sometimes partial) algebraic structures (in order of increasing generality): Boolean algebras, orthomodular lattices, orthomodular posets, orthoalgebras, and effect algebras. Cattaneo, Dalla Chiara, and Giuntini, working in the setting of effect algebras or their lexicomorphic cousins quantum-MV-algebras, delineate six notions of a sharp physical property.o They study the relationships among these notions and show, among other things, that these six notions collapse into one in an MV-algebra. Navara discusses a theorem that insures the existence of an orthomodular lattice (OML) with certain prespecified state space properties given that a certain quite weak structure exists with the same property. These weak structures are, in fact, weaker than all the orthomodular structures listed above. He therefore provides us with a simple means of showing that OMLs exist with certain state space properties.

Axioms of an Experimental System

Inthispaper a system of axioms ispresented todefine the notion of an experimental system. The primary feature of these axioms is that they are based solely on the mathematical notion of a direct product decomposition of a set. Properties of experimental systems are then developed. This includes defining negation, implication, conjunction, and disjunction on the set 4 of all binary experiments of the system and showing that the resulting structure is a regular orthomodular poset. The theory of observables of experimental systems is also developed. Finally, the usual models of experiments from classical as well as quantum physics are shown to satisfy the axioms of an experimental system, and a mechanism to create new models of the axioms is given.

Probability Weights and Measures on Finite Effect Algebras

We study probability weights and measures onfinite effect algebras, thus generalizing the existingtheory for orthomodular posets and orthoalgebras. Ourdevelopment proceeds somewhat more generally in that we study weights and measures associatedwith an antichain in the positive cone of a euclideanvector space with the standard partialordering.

Universal state space embeddability of Jordan-Banach algebras

We study extensions of states between projection structures of JB algebras and generalized orthomodular posets. It is shown that projection orthoposet of a JB algebra A admits the universal extension property if and only if the Gleason theorem is valid for A. As a consequence we get that any positive Stone algebra-valued measure on projection lattice of a quotient of a JBW algebra without type I2 direct summand extends to a positive measure on an arbitrary larger generalized orthomodular lattice. The classical extension theorem of Horn and Tarski [5] says that any probability measure on a Boolean algebra B extends to a probability measure on any larger Boolean algebra B' containing B. The first measure theoretic generalization of this result was given by P. Ptak [10] who showed that B' can be replaced by any quantum logic L with reasonably rich state space. We say in this case that Boolean algebras have the universal state extension property. In further development it has been proved in [6] that the universal state extension property for projection structures of unital C*-algebras is equivalent with the existence of non-commutative integral (so called Gleason property). As a consequence we get that projection lattices of von Neumann algebras without type I2 direct summand enjoy the universal state extension property. The aim of this note is to extend the above stated results for (not necessarily unital) Jordan Banach algebras and corresponding orthomodular structures not containing a largest element. This generalization is based on extension technique using determinacy of pure states on Jordan algebras. Moreover, general extension theorem for complete vector space valued measures on projections will be proved. First we recall a few notions and fix the notation. (Our standard references for Jordan algebras, ordered vector spaces, and quantum logics are [7, 1, 11], correspondingly.) * By a Jordan algebra we shall mean a real vector space A equipped with the product (i.e. bilinear form) written as (a, b) -? a o b and satisfying the following properties for all a, b E A: a o b = b o a, a o (bo a 2) = (aob) oa 2. A is said to be urnital if it admits a unit with respect to the Jordan product. In addition, if A is a Banach algebra with respect to the product o and if the norm I I satisfies the following conditions for all a,b E A: IIa211 < Ila2 + b211, Received by the editors May 1, 1997. 1991 Mathematics Subject Classification. Primary 46L70, 46L50, 28B15, 81P10.

Regularity in Quantum Logic

In 1996, Harding showed that the binarydecompositions of any algebraic, relational, ortopological structure X form an orthomodular poset FactX. Here, we begin an investigation of the structuralproperties of such orthomodular posets of decompositions.We show that a finite set S of binary decompositions inFact X is compatible if and only if all the binarydecompositions in S can be built from a common n-arydecomposition of X. This characterization ofcompatibility is used to show that for any algebraic,relational, or topological structure X, the orthomodularposet Fact X is regular. Special cases of this result include the known facts that theorthomodular posets of splitting subspaces of an innerproduct space are regular, and that the orthomodularposets constructed from the idempotents of a ring are regular. This result also establishes theregularity of the orthomodular posets that Mushtariconstructs from bounded modular lattices, theorthomodular posets one constructs from the subgroups ofa group, and the orthomodular posets oneconstructs from a normed group with operators. Moreover,all these orthomodular posets are regular for the samereason. The characterization of compatibility is also used to show that for any structure X, thefinite Boolean subalgebras of Fact X correspond tofinitary direct product decompositions of the structureX. For algebraic and relational structures X, this result is extended to show that the Booleansubalgebras of Fact X correspond to representations ofthe structure X as the global sections of a sheaf ofstructures over a Boolean space. The above results can be given a physical interpretation as well.Assume that the true or false questions $$Q$$ of a quantum mechanical system correspond tobinary direct product decompositions of the state spaceof the system, as is the case with the usual von Neumanninterpretation of quantum mechanics. Suppose S is asubset of $$Q$$ . Then a necessary andsufficient condition that all questions in S can beanswered simultaneously is that any two questions in S can be answeredsimultaneously. Thus, regularity in quantum mechanicsfollows from the assumption that questions correspond todecompositions.

Decomposition theorems in orthomodular posets: The problem of uniqueness

We provide some conditions which are equivalent to the uniqueness of Hewitt-Yosida-type decomposition and Lebesgue-type decomposition for real valued additive functions defined on orthomodular posets.

Abelian Extensions of Quantum Logics

All known a quantum logicsoÐ orthomodular lattices and posets, orthoalgebras, effect algebras, etc.Ðmay be regarded as cancelative, unital partial abelian semigroups. [See Wilce (1995a, b) for pertinent definitions.] An abelian group may also be regarded as a cancelative, unital PASÐ one in which every element is a unit. As noted in Wilce (1995b), every cancelative unital PAS L has a canonical ideal A (L) which is an abelian group, and a canonical quotient by A (L) which is an effect algebra. Thus, we may regard every cancellative, unital PAS as an extension of an effect algebra by an abelian group. The present paper begins a study of such extensions.

Gleason-Type Theorem for Linear Spaces over the Field of Four Elements

We prove an analog of the famous Gleason theoremfor additive functions on the orthomodular poset of allprojections defined on an n-dimensional linear spaceover the field consisting of four elements. An essential part of the proof consists in acomputer calculation.

Effect algebras and statistical physical theories

The dichotomic physical quantities of a physical system can be naturally hosted in a mathematical structure, called effect algebra, of which orthomodular posets and Boolean algebras are particular examples. We examine how effect algebras arise inside statistical physical theories and, conversely, we study to what extent an effect algebra can be taken as a primitive structure on which a satisfactory statistical physical model equipped with a convex set of states can be constructed.

Decompositions in Quantum Logic

We present a method of constructing an orthomodular poset from a relation algebra. This technique is used to show that the decompositions of any algebraic, topological, or relational structure naturally form an orthomodular poset, thereby explaining the source of orthomodularity in the ortholattice of closed subspaces of a Hilbert space. Several known methods of producing orthomodular posets are shown to be special cases of this result. These include the construction of an orthomodular poset from a modular lattice and the construction of an orthomodular poset from the idempotents of a ring. Particular attention is paid to decompositions of groups and modules. We develop the notion of a norm on a group with operators and of a projection on such a normed group. We show that the projections of a normed group with operators form an orthomodular poset with a full set of states. If the group is abelian and complete under the metric induced by the norm, the projections form a σ-complete orthomodular poset with a full set of countably additive states. We also describe some properties special to those orthomodular posets constructed from relation algebras. These properties are used to give an example of an orthomodular poset which cannot be embedded into such a relational orthomodular poset, or into an orthomodular lattice. It had previously been an open question whether every orthomodular poset could be embedded into an orthomodular lattice.

Orthomodular posets of idempotents in finite rings of matrices

The idempotents, resp. Hermitian idempotents, of a unital ring, resp. involutive unital ring, form an orthomodular poset. We study these Orthomodular posets for rings of matrices over the integers modulom or over Galois fields. In analogy to the Hilbert space situation we look for idempotent matrices (projections) corresponding to splitting subspaces of finite-dimensional vector spaces.

Gleason-type theorems for signed measures on orthomodular posets of projections on linear spaces

We consider some orthomodular posets which are not lattices and the Gleason-type theorems for signed measures on them.

Tensor product of difference posets and effect algebras

A tensor product of difference posets and/or, equivalently, of effect algebras, which generalize orthoalgebras and orthomodular posets, is defined, and an equivalent condition is presented. The proof uses the notion of D-test spaces generalizing test spaces of Randall and Foulis. In particular, we show that a tensor product for difference posets with a nonempty system of probability measures exists.

Partial Abelian semigroups

Orthomodular lattices and posets, orthoalgebras, and D-posets are all examples of partial Abelian semigroups. So, too, are the event structures of test spaces. The passage from an algebraic test space to its logic (an orthoalgebra) is an instance of a general construction involving a partial Abelian semigroupL and a distinguished subsetM\( \subseteq \)L such that perspectivity with respect toM is a congruence onL. The quotient ofL by such a congruence is always a cancellative, unital PAS, and every such PAS arises canonically as such a quotient.

The logics of orthoalgebras

The notion of unsharp orthoalgebra is introduced and it is proved that the category of unsharp orthoalgebras is isomorphic to the category of D-posets. A completeness theorem for some partial logics based on unsharp orthoalgebras, orthoalgebras and orthomodular posets is proved.