A Finite Orthomodular Lattice Which Does Not Admit a Full Set of States

SIAM Review ◽  
1970 ◽  
Vol 12 (2) ◽  
pp. 267-271 ◽  
Author(s):  
Mary Katherine Bennett
Keyword(s):  
Orthomodular Lattice ◽  
Full Set Of States

scholarly journals Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 164
Author(s):  
Songsong Dai
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Distributive Law ◽  
Complete Orthomodular Lattice ◽  
The Relationship

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.

The (virtual) conceptual necessity of quantum probabilities in cognitive psychology

2013 ◽  
Vol 36 (3) ◽  
pp. 280-281
Author(s):  
Reinhard Blutner ◽  
Peter beim Graben
Keyword(s):  
Cognitive Psychology ◽  
Orthomodular Lattice ◽  
The State ◽  
Input State ◽  
Cognitive Tests ◽  
Quantum Probabilities ◽  
Conceptual Necessity ◽  
Gleason's Theorem ◽  
Gleason’S Theorem ◽  
General Conditions

AbstractWe propose a way in which Pothos & Busemeyer (P&B) could strengthen their position. Taking a dynamic stance, we consider cognitive tests as functions that transfer a given input state into the state after testing. Under very general conditions, it can be shown that testable properties in cognition form an orthomodular lattice. Gleason's theorem then yields the conceptual necessity of quantum probabilities (QP).

Semigroups Co-Ordinatizing Orthomodular Geometries

1965 ◽  
Vol 17 ◽  
pp. 40-51 ◽  
Author(s):  
D. J. Foulis
Keyword(s):  
Orthomodular Lattice ◽  
Orthomodular Lattices

In (2, 3, 4, and 5), the author has established a connection between orthomodular lattices and Baer *-semigroups. In brief, the connection is as follows. The lattice of closed projections of any Baer *-semigroup forms an orthomodular lattice. Conversely, if L is any orthomodular lattice, there exists a Baer *-semigroup S which co-ordinatizes L in the sense that L is isomorphic to the lattice of closed projections in S. In this note we shall assume that the reader is familiar with the results and the notation of the quoted papers.

scholarly journals Classical Logic and Quantum Logic with Multiple and Common Lattice Models

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Mladen Pavičić
Keyword(s):  
Quantum Logic ◽  
Classical Logic ◽  
Orthomodular Lattice ◽  
Lattice Models ◽  
Quantum Space ◽  
Technical Terms

We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical logic which turns out to have disjoint distributive and nondistributive ortholattices. In particular, we prove that both classical logic and quantum logic are sound and complete with respect to each of these lattices. We also show that there is one common nonorthomodular lattice that is a model of both quantum and classical logic. In technical terms, that enables us to run the same classical logic on both a digital (standard, two-subset, 0-1-bit) computer and a nondigital (say, a six-subset) computer (with appropriate chips and circuits). With quantum logic, the same six-element common lattice can serve us as a benchmark for an efficient evaluation of equations of bigger lattice models or theorems of the logic.

Block-Finite Orthomodular Lattices

1979 ◽  
Vol 31 (5) ◽  
pp. 961-985 ◽  
Author(s):  
Günter Bruns
Keyword(s):  
Orthomodular Lattice ◽  
Boolean Algebras ◽  
Orthomodular Lattices ◽  
Special Situation ◽  
The Given ◽  
Insight Into ◽  
Considerable Insight

Introduction. Every orthomodular lattice (abbreviated : OML) is the union of its maximal Boolean subalgebras (blocks). The question thus arises how conversely Boolean algebras can be amalgamated in order to obtain an OML of which the given Boolean algebras are the blocks. This question we deal with in the present paper.The problem was first investigated by Greechie [6, 7, 8, 9]. His technique of pasting [6] will also play an important role in this paper. A case solved completely by Greechie [9] is the case that any two blocks intersect either in the bounds only or have the bounds, an atom and its complement in common. This is, of course, a very special situation. The more surprising it is that Greechie's methods, if skillfully applied, yield considerable insight into the structure of OMLs and provide a seemingly unexhaustible source for counter-examples.

scholarly journals Left residuated lattices induced by lattices with a unary operation

2019 ◽  
Vol 24 (2) ◽  
pp. 723-729
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Residuated Lattice ◽  
Unary Operation ◽  
Residuated Lattices ◽  
Boolean Algebras ◽  
Natural Question ◽  
Similar Technique ◽  
Orthomodular Lattices

Abstract 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.

scholarly journals Mathematical foundations of quantum mechanics: An advanced short course

2016 ◽  
Vol 13 (Supp. 1) ◽  
pp. 1630011
Author(s):  
Valter Moretti
Keyword(s):  
Quantum Mechanics ◽  
Orthomodular Lattice ◽  
Foundations Of Quantum Mechanics ◽  
Quantum Theories ◽  
Short Course ◽  
Mathematical Foundations ◽  
Algebraic Formulation

This paper collects and extends the lectures I gave at the “XXIV International Fall Workshop on Geometry and Physics” held in Zaragoza (Spain) during September 2015. Within these lectures I review the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced viewpoint, essentially based on the orthomodular lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and theorems also related to the representation of physical symmetries. The final step consists of an elementary introduction the so-called ([Formula: see text]-) algebraic formulation of quantum theories.

Commutative rings whose ideal lattices are complemented

2019 ◽  
Vol 12 (03) ◽  
pp. 1950039
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Ideal Lattice ◽  
Ideal Lattices ◽  
The Ideal

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].

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