Rough Approximation Operators on a Complete Orthomodular Lattice

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

Left residuated lattices induced by lattices with a unary operation

In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

Commutative rings whose ideal lattices are complemented

We characterize those commutative rings [Formula: see text] whose ideal lattice [Formula: see text] endowed with the annihilation operation is an ortholattice. Moreover, we provide an analogous characterization for the annihilator lattice [Formula: see text] endowed with the annihilation operation. Since the ideal lattice of [Formula: see text] is modular, [Formula: see text] is already an orthomodular lattice provided it is an ortholattice. However, there exist also commutative rings whose ideal lattices are complemented but the complementation differs from annihilation. We present an example of such a ring and develop a procedure producing infinitely many rings with this property. Finally, we provide a sufficient condition for double annihilation to be a homomorphism from [Formula: see text] onto [Formula: see text].

Quantum categories for quantum logic

The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors $\mathsf{[E,Sets]}$ where $\mathsf{E}$ is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called $quantos$ – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative $C^*$-algebras, which in turn are studied in their quantum habitat of noncommutative $C^*$-algebras – more fundamental structures than commutative $C^*$-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors $\mathsf{[E,QSets]}$ is constructed where $\mathsf{QSets}$ is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.

Wigner's theorem for an infinite set

It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space ${\mathcal H}$ correspond to direct product decompositions $\mathcal{H}\simeq A\times A^\perp$ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.

Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups

It is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular latticeX, admits a natural embedding into a group{G(X)}with a lattice ordering so that{G(X)}determinesXup to isomorphism. The embedding{X\hookrightarrow G(X)}appears to be a universal (non-commutative) group-valued measure onX, while states ofXturn into real-valued group homomorphisms on{G(X)}. The existence of completions is characterized by a generalized archimedean property which simultaneously applies toXand{G(X)}. By an extension of Foulis’ coordinatization theorem, the negative cone of{G(X)}is shown to be the initial object among generalized Baer{{}^{\ast}}-semigroups. For finiteX, the correspondence betweenXand{G(X)}provides a new class of Garside groups.

Kadison’s antilattice theorem for a synaptic algebra

We prove that if A is a synaptic algebra and the orthomodular lattice P of projections in A is complete, then A is a factor if and only if A is an antilattice.We also generalize several other results of R. Kadison pertaining to infima and suprema in operator algebras.

Residuation in orthomodular lattices

We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.