Jauch-Piron orthomodular posets and propositional systems: A comparison

We compare the structures obtained via orthomodular posets and via propositional systems, discussing some examples of the links between them. Despite some analogies, the two structures are fundamentally different.     

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.     

Minimal orthomodular lattices

The lattices calledminimal orthomodular (MOL) arise in a special exclusion problem concerning the class of all orthomodular lattices (OML) and the subclass of all modular orthocomplemented lattices. This problem was given in G. Kalmbach's book,Orthomodular Lattices. We prove that an exclusion system necessarily must contain an infinite lattice. We prove that, except one, all the finite, irreducible MOLs have only blocks with eight elements. We characterize finite MOLs by a covering property related to equational classes generated by the modular ortholattices MOn.     

Conditional probabilities on orthomodular lattices

A definition of generalized probability on an orthomodular lattice which includes as particular cases the classical probability space and non-commutative probability theory on a von Neumann algebra is proposed. In this generalized structure the problem of conditioning with respect to Boolean σ-subalgebras is examined.     

Measures on infinite-dimensional orthomodular spaces

We classify the measures on the lattice of all closed subspaces of infinite-dimensional orthomodular spaces (E, ) over fields of generalized power series with coefficients in . We prove that every -additive measure on can be obtained by lifting measures from the residual spaces of (E, ). The measures being lifted are known, for the residual spaces are Euclidean. From the classification we deduce, among other things, that the set of all measures on is not separating.Research supported by the Swiss National Science Foundation.     

A particular non-atomistic orthomodular poset

Every atomic orthomodular lattice is atomistic. We show that the corresponding statement for orthomodular posets fails. The result is of interest in the study of the Algebraic Structure of Quantum Mechanics, see [5].     

Quantum MV-Algebras and Commutativity

We introduce a notion of commutativity inquantum MV-algebras (QMV-algebras) and we investigatethe corresponding structure of the center. It turns outthat the center of a QMV-algebra is a multivalued algebra (MV-algebra). Finally, we prove thatthe center of the QMV-algebra of all effects on aHilbert space is irreducible.     

Sites and tours in orthoalgebras and orthomodular lattices

A block of an orthoalgebra (or of an orthomodular lattice) is a maximal Boolean subalgebra. A site is the intersection of two distinct blocks. L is block (site)-finite if there are only finitely many blocks (sites). We introduce a certain type of subalgebra of an orthoalgebra which is a subortholattice if the orthoalgebra is an ortholattice (and therefore an orthomodular lattice) and which is block finite if the orthoalgebra is site finite. The construction yields a cover of a site-finite orthoalgebra or orthomodular lattice L by block-finite substructures of the same type and having the same center as L. Every site-finite orthomodular lattice is commutator finite.In memory of Charles H. Randall.     

Contraexamples in difference posets and orthoalgebras

We show that every orthoalgebra (difference orthoposet) uniquely determines a difference orthoalgebraic structure. We give examples of posets on which there exist more than one difference operation. In spite of that, every finite chain is a uniquely determined difference poset. On a difference poset there need not exist any orthoalgebraic operation, but the category of difference orthoposets is isomorphic with the category of orthoalgebras. But a difference poset which is also an orthoposet need not be a difference orthoposet. Moreover, there exist complete lattices on which there does not exist any difference operation. Finally, we show that difference operations and orthoalgebraic operations need not be extendable on a MacNeille completion of the base poset.     

Commutativity of Quantum Family Algebras

Recently, A. A. Kirillov introduced an important notion of classical and quantum family algebras. Here the criterion of commutativity is given. The quantum eigenvalues of are computed.     

Commutativity equations and dressing transformations

We study dressing transformations that generate all solutions to commutativity equations and, after picking up special coordinates, all solutions to WDVV equations. We conjecture that the homological tensor product of solutions to the commutativity equations corresponds to the tensor product of matrices of the dressing transformation and check this in the first nontrivial case.     

Extensions of real-valued difference posets

The difference poset of real-valued functions (fuzzy sets) is investigated. The difference posets on some subsets of the unit real interval [0, 1] are characterized. It is shown that although it is not always possible to represent the difference by some generator, for dense subsets such a representation exists and is unique.     

Algebras of bounded operators on nonclassical orthomodular spaces

A Hermitian space is called orthomodular if the Projection Theorem holds: every orthogonally closed subspace is an orthogonal summand. Besides the familiar real or complex Hilbert spaces there are non-classical infinite dimensional examples constructed over certain non-Archimedeanly valued, complete fields. We study bounded linear operators on such spaces. In particular we construct an operator algebraA of von Neumann type that contains no orthogonal projections at all. For operators inA we establish a representation theorem from which we deduce thatA is commutative. We then focus on a subalgebra which turns out to be an integral domain with unique maximal ideal. Both analytic and topological characterizations of are given.     

Embedding of posets into lattices in quantum logic

It has been proposed that some posets of quantum logic could be embedded into lattices in order to recover the lattice structure avoiding the introduction of ad hoc axioms. We consider here the embedding _s of any posetS into the complete lattice _s of its closed ideals (normal embedding ofS) and show that _s can be characterized (up to a lattice isomorphism) either by means of a density property or by means of a minimality property. Both of these suggest that the normal embedding satisfies some intuitive conditions which make it preferable with respect to other possible embeddings ofS. We consider the poset of all the effects associated to yes-no experiments and briefly comment on the application of the normal embedding in this case. The possibility of giving a physical interpretation to the elements of is also discussed.Research sponsored by CNR and INFN (Italy).     

Symmetric chain partitions of orthocomplemented posets

It is shown that any orthocomplemented posetP of finite width admits a chain partition of cardinality 2[2/3 width(P)] which is symmetrical with respect to the orthocomplement. This cardinality is the best possible.     

Spectral decompositions of operators on non-Archimedean orthomodular spaces

The most central property of an infinite-dimensional Hilbert space is expressed by the projection theorem: Every orthogonally closed linear subspace is an orthogonal summand. Besides the obvious Hilbert spaces, there exist other infinite-dimensional orthomodular spaces. Here we study bounded linear operators on an orthomodular spaceE constructed over a field of generalized power series with real coefficients. Our main result states that every bounded, self-adjoint operator gives rise to a representation ofE as the closure of an infinite orthogonal sum of invariant subspaces each of which is of dimension 1 or 2. The proof combines the technique of reduction modulo the residual spaces with theorems on orthogonal decompositions of finite matrices over fields of power series.     

Automorphisms of orthomodular lattices and symmetries of quantum logics

Given a group G, there is a proper class of pairwise nonembeddable orthomodular lattices with the automorphism group isomorphic to G. While the validity of the above statement depends on the used set theory, the analogous statement for groups of symmetries of quantum logics is valid absolutely.     

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.