Regularity in Quantum Logic

In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let ∗ stand for the Łukasiewicz t-norm on [0, 1]2. Let us consider the operation ⋄ on [0, 1] by setting x ⋄ y = (x ⊥ ∗y ⊥)⊥ ∗ (x∗y)⊥, where x ⊥ = 1−x. Let us call the triple L = ([0, 1], ⋄, 1) the Łukasiewicz groupoid. Let B be a Boolean algebra. Denote by L(B) the set of all L-valued measures (L-valued states). We show as a main result of this paper that the family L(B) consists precisely of the union of classical real states and Z 2-valued states. With the help of this result we characterize the L-valued states on orthomodular posets. Since the orthomodular posets are often understood as “quantum logics” in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L-valued state.

Fuzzy set representations of some quantum structures

In the orthodox quantum logic approach to foundations of quantum mechanics originated by Birkhoff and von Neumann [1] the object called quantum logic is not a logic in the very sense of this word but an algebraic structure: orthomodular lattice or poset. These orthomodular structures, abstracted from structures of Hilbert spaces used to describe quantum phenomena are weakenings of Boolean algebras. They can be thought of as representing Lindenbaum algebras of systems of propositions about quantum objects, the propositions belonging to the quantum logic proper. Since Lindenbaum algebras of propositional systems which obey the rules of classical logic are Boolean algebras it is inferred that the quantum logic proper, whatever it means, is non-classical.

We consider the amalgamation of bounded involution posets over a strictly directed graph as applied to orthomodular lattices, orthomodular posets or orthoalgebras. In the finite setting, we show that the order dimension of the amalgamation does not exceed that of the amalgamated structures by more than one. We also present conditions under which equality obtains.

Remark on a paper of M [formula omitted]czyński

09-P-12-Layered germanates with 9-membered rings

Formal Concept Analysis (FCA) can be used to analyze data given in the form of a formal context. In particular, FCA provides efficient algorithms for computing a minimal basis of the implications holding in the context. In this paper, we extend classical FCA by considering data that are represented by relational structures rather than formal contexts, and by replacing atomic attributes by complex formulae defined in some logic. After generalizing some of the FCA theory to this more general form of contexts, we instantiate the general framework with attributes defined in the Description Logic (DL) EL, and with relational structures over a signature of unary and binary predicates, i.e., models for EL. In this setting, an implication corresponds to a so-called general concept inclusion axiom (GCI) in EL. The main technical result of this report is that, in EL, for any finite model there is a finite set of implications (GCIs) holding in this model from which all implications (GCIs) holding in the model follow.

General overlap functions are generalized on the basis of overlap functions, which have better application effects in classification problems, and the (weak) inflationary BL-algebras as the related algebraic structure were also studied. However, general overlap functions are a class of aggregation operators, and their commutativity puts certain restrictions on them. In this article, we first propose the notion of pseudo general overlap functions as a non-commutative generalization of general overlap functions, so as to extend their application range, then illustrate their relationship with several other commonly used aggregation functions, and characterize some construction methods. Secondly, the residuated implications induced by inflationary pseudo general overlap functions are discussed, and some examples are given. Then, on this basis, we show the definitions of inflationary pseudo general residuated lattices (IPGRLs) and weak inflationary pseudo BL-algebras, and explain that the weak inflationary pseudo BL-algebras can be gained by the inflationary pseudo general overlap functions. Moreover, they are more extensive algebraic structures, thus enriching the content of existing non-classical logical algebra. Finally, their related properties and their relations with some algebraic structures such as non-commutative residuated lattice-ordered groupoids are investigated. The legend reveals IPGRLs include all non-commutative algebraic structures involved in the article.

This paper deals with the topics of modeling joint distributions on a generalized probability space. An algebraic structure known as quantum logic is taken as the basic model. There is a brief summary of some earlier published findings concerning a function s-map, which is a mathematical tool suitable for constructing virtual joint probabilities of even non-compatible propositions. The paper completes conclusions published in 2020 and extends the results for three or more random variables if the marginal distributions are known. The existence of a (n+1)-variate joint distribution is shown in special cases when the quantum logic consists of at most n blocks of Boolean algebras.

Besides the physicomathematical controversy concerning the ‘phenomenological justification’ and the specific formal structure of the calculus known as quantum logic (QL), there is a philosophical controversy concerning whether this ‘calculus of experimental propositions’ is properly speaking a logic rather than simply an algebraic structure only analogous to logic properly so called. Directly associated with this controversy is the issue of whether logic is an empirical science on the par with, e.g., physical geometry. This article is not concerned, however, with the general question of the empirical character of logic, either classical or quantal; it is rather concerned only with one seemingly decisive objection against regarding QL as logic, namely, the objection advanced by Jauch and Piron (1970, p. 176), who argue that since QL lacks an essential feature of logic — viz., a deduction scheme — it is `very questionable whether we may properly call the lattice of general quantum mechanics a logic.’ Their argument is based on what they regard to be the general failure of the lattice of QL propositions to admit a material implication connective or conditional operation by means of which the modus ponens deduction scheme can be incorporated into QL. The view of Jauch and Piron is also supported by Greechie and Gudder (1971, 1973), who examine a number of results which suggest that no reasonable material conditional exists in QL.

In this review, we highlight recent advancements on pillararene-based assemblies. The driving forces for the formation of the pillararene-based assemblies are discussed first. The host-guest interactions are deemed as not only general strategy for constructing assemblies but also essential components for preventing the assemblies from the dissociation. Solvent effect is also important in the assembling process, since it could influence the host-guest interactions and provide solvophobic effect on pillararenes for the assembly. Then, several pillararene-based assembly architectures are introduced, including pillararene-based interlocked structures, such as (poly)pseudorotaxanes, (poly)rotaxanes, and daisy chains, classified by their topological structures and synthetic strategy. The morphologies of the supramolecular assemblies are divided into several types, for example, nanospheres, nanotubes and supramolecular polymers. Furthermore, the functions and potential applications are summarized accompanied with related assembly structures. The review not only provides fundamental findings, but also foresights future research directions in the research area of pillararene-based assemblies.

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">As a common seismic facies visualization analysis method, SOM projects the waveform or seismic attribute vectors into a two-dimensional topological plane in a nonlinear way, which can effectively and efficiently discover the topological structure of a dataset. SOM does not need to set the number of classes in prior, and has friendly visualization characteristics and excellent generalization, which are conducive to seismic facies interpretation using unlabeled data. However, due to the competitive learning used in SOM and the imbalance of data distribution in real world, the samples from majority classes are expanded on the topological plane and the minority classes are compressed. As a result, the plane cannot accurately describe the global structure of data distribution. To improve the visualization precision by modifying the topological relationship of the prototype vectors of SOM, we utilize UMAP (Uniform Manifold Approximation and Projection), a novel manifold learning technique for dimension reduction, to correct the prototype vectors generated by SOM. By combing the advantages of SOM and UMAP in the representation of data topological structure, the global relationship between seismic data samples can be properly established, and the internal relative spatial structure of majority class samples can be retained as much as possible, resulting in a more reliable classification. Meanwhile, the framework maintains the advantages of SOM in visualization. In the modeling tests and real data experiments, we have demonstrated the effectiveness and rationality of UMAP-SOM on the spatial structure representation of three-dimensional seismic data</i> .

It is shown that an orthomodular lattice can be axiomatized as an ortholattice with a unique operation of identity (bi-implication) instead of the operation of implication, and a corresponding algebraic unified quantum logic is formulated. A statistical YES-NO physical interpretation of the quantum logical propositions is then provided to establish a support for a novel YES-NO representation of quantum logic which prompts a conjecture about a possible completion of quantum logic by means of probabilistic forcing. In this paper we shall anwser two basic questions of quantum' logics, present a novel probabilistic representation of the logic and of quantum measurements, and propose a probabilistic forcing as a possible tool for completing quantum logic. The questions are, first, whether there is a unique object language operation which can take over the role of the unique classical operation of implication (conditional, set-theoretic inclusion), and, second, whether there is a relation which is more appropriate for set-theoretic representation of quantum-theoretic measurements than the usual irreflexive and symmetric orthogonality relation. We answer the first question in the positive by substituting the biimplication for the implication and the equality for the ordering relation, thus at the same time making the ordering within quantum sets irrelevant. This renders the usual techniques of quantum logic as a deductive inferential theory inappropriate and ascribes quantum deductive logic a particular equational meaning. The result is obtained in Section 2.

The notion of observable is a quantum mechanical variant of that of a random variable (see [32], [69], [20], [29], etc.). It is supposed to help in treating noncompatible events. In the logico-algebraic approach an observable is defined as a σ-additive measure with values in a “quantum logic” (= with values in an abstract σ-complete orthomodular poset). Though an individual observable ranges in a Boolean σ-algebra, non-classical phenomena occur when one develops “noncomutative” probability theory. First, the range Boolean σ-algebra does not have to be set-representable (even in the fundamental case of the Hilbert space logic!). Second, when one treats collections of observables, the respective range Boolean σ-algebras vary and often do not allow for an umbrella Boolean σ-algebra. In this article we want to survey a few lines of recent research on observables. We also want to formulate — in the main text and in the notes at the end (referred to as ) — some open questions which seem related to quantum axiomatics and noncommutative probability theory. An extensive bibliography of papers dealing with observables is also provided.

Quantum Logics Defined by Sets of Numerical Events