Basic semirings

2019 ◽  
Vol 69 (3) ◽  
pp. 533-540
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Classical Logic ◽  
Boolean Algebras ◽  
Basic Algebra ◽  
The Other ◽  
Natural Question ◽  
Orthomodular Lattices ◽  
Common Generalization ◽  
Boolean Rings ◽  
Mv Algebras

Abstract Basic algebras were introduced by Chajda, Halaš and Kühr as a common generalization of MV-algebras and orthomodular lattices, i.e. algebras used for formalization of non-classical logics, in particular the logic of quantum mechanics. These algebras were represented by means of lattices with section involutions. On the other hand, classical logic was formalized by means of Boolean algebras which can be converted into Boolean rings. A natural question arises if a similar representation exists also for basic algebras. Several attempts were already realized by the authors, see the references. Now we show that if a basic algebra is commutative then there exists a representation via certain semirings with involution similarly as it was done for MV-algebras by Belluce, Di Nola and Ferraioli. These so-called basic semirings, their ideals and congruences are studied in the paper.

Basic algebras, logics, trends and applications

2015 ◽  
Vol 08 (03) ◽  
pp. 1550040 ◽  
Author(s):  
Ivan Chajda
Keyword(s):  
Quantum Mechanics ◽  
20Th Century ◽  
19Th Century ◽  
Classical Logic ◽  
Boolean Algebras ◽  
Heyting Algebras ◽  
Fuzzy Logics ◽  
Orthomodular Lattices ◽  
Mv Algebras ◽  
Intuitionistic Logics

The classical logic was axiomatized algebraically by means of Boolean algebras in 19th century by George Boole. Similar attempts went on 20th century for algebraic axiomatization of non-classical logics, e.g. intuitionistic logics (Brouwer and Heyting algebras), many-valued logics (Łukasiewicz, Chang’s MV-algebras, Post algebras), the logic of quantum mechanics (orthomodular lattices and posets) and fuzzy logics (residuated lattices). In this paper, we are focused in a common generalization of MV-algebras and orthomodular lattices. The resulting algebras, called basic algebras, have surprisingly strong and interesting properties and they can be investigated in their own. The aim of the paper is to get an overview of results reached during the last decade.

Effect-like algebras induced by means of basic algebras

2010 ◽  
Vol 60 (1) ◽  
Author(s):  
Ivan Chajda
Keyword(s):  
Distributive Lattice ◽  
Effect Algebra ◽  
Binary Operation ◽  
Basic Algebra ◽  
Natural Question ◽  
Lattice Effect Algebra ◽  
Mv Algebra ◽  
Lattice Effect ◽  
Mv Algebras

AbstractHaving an MV-algebra, we can restrict its binary operation addition only to the pairs of orthogonal elements. The resulting structure is known as an effect algebra, precisely distributive lattice effect algebra. Basic algebras were introduced as a generalization of MV-algebras. Hence, there is a natural question what an effect-like algebra can be reached by the above mentioned construction if an MV-algebra is replaced by a basic algebra. This is answered in the paper and properties of these effect-like algebras are studied.

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 An algebraic analysis of categorical syllogisms by using Carroll’s diagrams

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 367-383 ◽  
Author(s):  
Ibrahim Senturk ◽  
Tahsin Oner
Keyword(s):  
Boolean Algebras ◽  
Algebraic Methods ◽  
Quantitative Relation ◽  
Algebraic Structures ◽  
Elimination Method ◽  
Logical Calculus ◽  
Algebraic Analysis ◽  
Boolean Rings ◽  
Algebraic Properties ◽  
Mv Algebras

In this paper, we analyze the algebraic properties of categorical syllogisms by constructing a logical calculus system called Syllogistic Logic with Carroll Diagrams (SLCD).We prove that any categorical syllogism is valid if and only if it is provable in this system. For this purpose, we explain firstly the quantitative relation between two terms by means of bilateral diagrams and we clarify premises via bilateral diagrams. Afterwards, we input the data taken from bilateral diagrams, on the trilateral diagram. With the help of the elimination method, we obtain a conclusion that is transformed from trilateral diagram to bilateral diagram. Subsequently, we study a syllogistic conclusion mapping which gives us a conclusion obtained from premises. Finally, we allege valid forms of syllogisms using algebraic methods, and we examine their algebraic properties, and also by using syllogisms, we construct algebraic structures, such as lattices, Boolean algebras, Boolean rings, and many-valued algebras (MV-algebras).

scholarly journals On ring-like structures of lattice-Ordered numerical events

2021 ◽  
pp. 2150186
Author(s):  
Dietmar Dorninger ◽  
Helmut Länger
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Boolean Algebra ◽  
Special Interest ◽  
Physical System ◽  
Boolean Algebras ◽  
Boolean Rings ◽  
Term Functions ◽  
Ordered Algebras

Let [Formula: see text] be a set of states of a physical system. The probabilities [Formula: see text] of the occurrence of an event when the system is in different states [Formula: see text] define a function from [Formula: see text] to [Formula: see text] called a numerical event or, more accurately, an [Formula: see text]-probability. Sets of [Formula: see text]-probabilities ordered by the partial order of functions give rise to so-called algebras of [Formula: see text]-probabilities, in particular, to the ones that are lattice-ordered. Among these, there are the [Formula: see text]-algebras known from probability theory and the Hilbert-space logics which are important in quantum-mechanics. Any algebra of [Formula: see text]-probabilities can serve as a quantum-logic, and it is of special interest when this logic turns out to be a Boolean algebra because then the observed physical system will be classical. Boolean algebras are in one-to-one correspondence to Boolean rings, and the question arises to find an analogue correspondence for lattice-ordered algebras of [Formula: see text]-probabilities generalizing the correspondence between Boolean algebras and Boolean rings. We answer this question by defining ring-like structures of events (RLSEs). First, the structure of RLSEs is revealed and Boolean rings among RLSEs are characterized. Then we establish how RLSEs correspond to lattice-ordered algebras of numerical events. Further, functions for associating lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs are studied. It is shown that there are only two ways to assign lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs if one restricts the corresponding mappings to term functions over the underlying orthomodular lattice. These term functions are the very functions by which also the Boolean algebras can be assigned to Boolean rings.

scholarly journals Residuated Structures and Orthomodular Lattices

Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Algebraic Logic ◽  
Residuated Lattices ◽  
Quantum Structures ◽  
Heyting Algebras ◽  
Quantum Algebras ◽  
Orthomodular Lattices ◽  
Lattice Of Subvarieties ◽  
Residuated Structures ◽  
Common Framework ◽  
Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

scholarly journals Coordinate representation for non-Hermitian position and momentum operators

2017 ◽  
Vol 473 (2205) ◽  
pp. 20170434 ◽  
Author(s):  
F. Bagarello ◽  
F. Gargano ◽  
S. Spagnolo ◽  
S. Triolo
Keyword(s):  
Quantum Mechanics ◽  
The Self ◽  
The Other ◽  
Alternative Strategy ◽  
Coordinate Representation ◽  
Suitable Sense ◽  
Adjoint Operators ◽  
Ordinary Quantum

In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q ^ and p ^ similar, in a suitable sense, to the self-adjoint position and momentum operators q ^ 0 and p ^ 0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions , and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with q ^ and p ^ , based on the so-called quasi *-algebras .

Would You Like Some World Music with your Latte? Starbucks, Putumayo, and Distributed Tourism

2004 ◽  
Vol 1 (2) ◽  
pp. 209-223 ◽  
Author(s):  
ANAHID KASSABIAN
Keyword(s):  
Quantum Mechanics ◽  
Public Spaces ◽  
World Music ◽  
The Other ◽  
Retail Stores ◽  
Theoretical Perspectives ◽  
The One ◽  
Coffee Shops ◽  
Insight Into

Through an examination of the labels Hear Music and Putumayo and their place in coffee shops and retail stores on the one hand, and of world music scholarship on the other, I argue that listening to world music in public spaces demands new theoretical perspectives. The kinds of tourism that take place in listening to world music inattentively suggest a kind of bi-location. Borrowing from quantum mechanics, I suggest that the term ‘entanglement’ might offer some insight into this bi-location and the ‘distributed tourism’ that I argue is taking place.

Radical of a-Ideals in Mv -Modules

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
F. Forouzesh ◽  
E. Eslami ◽  
A. Borumand Saeid
Keyword(s):  
Maximal Ideal ◽  
The Other ◽  
Subject Classification ◽  
Mv Algebra ◽  
Mv Algebras

Abstract In this paper, we introduce the notion of the radical of an ideal in MV - algebras. Several characterizations of this radical is given. We define the notion of a semi-maximal ideal in an MV -algebra and prove some theorems which give relations between this semi-maximal ideal and the other types of ideals in MV -algebras. Also we prove that A/I is a semi-simple MV -algebra if and only if I is a semi-maximal ideal of an MV -algebra A. The above notions are used to define the radical of A-ideals in MV -modules and investigate some properties. Mathematics Subject Classification 2010: 03B50, 03G25, 06D35

scholarly journals WKB FORMALISM AND A LOWER LIMIT FOR THE ENERGY EIGENSTATES OF BOUND STATES FOR SOME POTENTIALS

2007 ◽  
Vol 22 (35) ◽  
pp. 2675-2687 ◽  
Author(s):  
LUIS F. BARRAGÁN-GIL ◽  
ABEL CAMACHO
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Field ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Approximation Method ◽  
Bound States ◽  
The Other ◽  
Homogeneous Gravitational Field ◽  
Definition Of ◽  
Ground Energy

In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.

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