States and Boolean Algebra

2008 ◽  
Vol 15 (04) ◽  
pp. 649-652
Author(s):  
Nabila N. Mikhaeel ◽  
Basim Samir Labib
Keyword(s):  
Mathematical Logic ◽  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Quantum Theories

We investigate subadditive measures on orthomodular lattices. We show as the main result that the Boolean algebra, the special metric orthomodular lattice and the orthomodular lattice which is unital with respect to subadditive states are equivalent. This result may find an application in the foundation of quantum theories and mathematical logic.

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.

A Finiteness Criterion for Orthomodular Lattices

1978 ◽  
Vol 30 (02) ◽  
pp. 315-320 ◽  
Author(s):  
Günter Bruns
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Infinite Chain ◽  
The Other ◽  
Finitely Generated ◽  
Orthomodular Lattices ◽  
Other Hand ◽  
Finiteness Criterion ◽  
Element Set

The main result of this paper is the following: THEOREM. Every finitely generated orthomodular lattice L with finitely many maximal Boolean subalgebras (blocks) is finite. If L has one block only, our theorem reduces to the well-known fact that every finitely generated Boolean algebra is finite. On the other hand, it is known that a finitely generated orthomodular lattice without any further restrictions can be infinite. In fact, in [2] we constructed an orthomodular lattice which is generated by a three-element set with two comparable elements, has infinitely many blocks and contains an infinite chain.

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.

Introduction

2017 ◽  
Author(s):  
Paul J. Nahin
Keyword(s):  
Mathematical Logic ◽  
Boolean Algebra ◽  
Electrical Engineering ◽  
Analytical Tool ◽  
Aristotelian Logic ◽  
Design Engineers ◽  
Electronic Circuitry ◽  
Huge Impact ◽  
The Subject

This introductory chapter considers the work of mathematician George Boole (1815–1864), whose book An Investigation of the Laws of Thought (1854) would have a huge impact on humanity. Boole's mathematics, the basis for what is now called Boolean algebra, is the subject of this book. It is also called mathematical logic, and today it is a routine analytical tool of the logic-design engineers who create the electronic circuitry that we now cannot live without, from computers to automobiles to home appliances. Boolean algebra is not traditional or classical Aristotelian logic, a subject generally taught in college by the philosophy department. Boolean algebra, by contrast, is generally in the hands of electrical engineering professors and/or the mathematics faculty.

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.

Varieties of Orthomodular Lattices. II

1972 ◽  
Vol 24 (2) ◽  
pp. 328-337 ◽  
Author(s):  
Günter Bruns ◽  
Gudrun Kalmbach
Keyword(s):  
Orthomodular Lattice ◽  
Oral Communication ◽  
Boolean Algebras ◽  
Vertical Line ◽  
Orthomodular Lattices ◽  
Lattice Of Varieties

In this paper we continue the study of equationally defined classes of orthomodular lattices started in [1].The only atom in the lattice of varieties of orthomodular lattices is the variety of all Boolean algebras. Every nontrivial variety contains it. It follows from B. Jónsson [4, Corollary 3.2] that the variety [MO2] generated by the orthomodular lattice MO2 of Figure 1 covers the variety of all Boolean algebras. I t was first shown by R. J. Greechie (oral communication) and is not difficult to see that every variety not consisting of Boolean algebras only contains [MO2]. Again it follows from the result of Jónsson's mentioned above that the varieties generated by one of the orthomodular lattices of Figures 2 to 5 cover [MO2]. The Figures 4 and 5 are to be understood in such a way that the orthocomplement of every element is on the vertical line through this element.

scholarly journals Residuation in orthomodular lattices

2017 ◽  
Vol 5 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Residuated Lattice ◽  
Orthomodular Lattices ◽  
Double Negation ◽  
Converse Statement ◽  
One To One ◽  
Double Negation Law

Abstract 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.

Sign in / Sign up

Export Citation Format

Share Document