scholarly journals Quantum logic as an implication algebra

1970 ◽  
Vol 2 (1) ◽  
pp. 101-106 ◽  
Author(s):  
P.D. Finch
Keyword(s):  
Boolean Algebra ◽  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Binary Operation ◽  
Unary Operation ◽  
Material Implication ◽  
Logical Implication ◽  
Partially Ordered ◽  
Implication Algebra

For the purpose of this paper a logic is defined to be a non-empty set of propositions which is partially ordered by a relation of logical implication, denoted by “≤”, and which, as a poset, is orthocomplemented by a unary operation of negation. The negation of the proposition x is denoted by NX and the least element in the logic is denoted by 0, we write NO = 1.A binary operation “→” is introduced into a logic, the operation is interpreted as material implication so that “x → y” is a proposition of the logic and is read as “x materially implies y”. If material implication has the properties11. (x → 0) = NX, 12. if x ≤ y then (z → x) ≤ (z → y), 13. if x ≤ y then x ≤ (y ≤ z)= x → z, 14. x ≤ {y → N(y → Nx)}, then the logic is an orthomodular lattice. The lattice operations of join and meet are given by x ∨ y = Nx → N(Nx → Ny) x ∧ y = N(X → N(x → y)) and, in terms of the lattice operations, the material implication is given by (x → y) = (y ∧ x) ∨ NX.Moreover the logic is a Boolean algebra if, and only if, in addition to the properties above, material implication satifies 15. (x → y) = (Ny → Nx).

An axiomatisation of quantum logic

1973 ◽  
Vol 38 (3) ◽  
pp. 389-392 ◽  
Author(s):  
Ian D. Clark
Keyword(s):  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Binary Operation ◽  
Ordered Set ◽  
Unary Operation ◽  
Axiom System ◽  
Partial Ordering ◽  
Orthomodular Poset ◽  
Implication Algebra ◽  
Image Position

The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset.In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa.Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following:(1) the greatest lower bound of a and Na exists and is 0,(2) a ≤ b implies Nb ≤ Na,(3) NNa = a.Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfyingwhere a ≤ b means a ∧ b = a, as usual.Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following:An orthomodular lattice gives a Finch implication algebra by defining → byA Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ byThe orthocomplementation is unchanged in both cases.

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.

Information Retrieval Systems, an Algebraic Approach I1

1981 ◽  
Vol 4 (3) ◽  
pp. 551-603
Author(s):  
Zbigniew Raś
Keyword(s):  
Information Retrieval ◽  
Boolean Algebra ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Algebraic Approach ◽  
Partially Ordered ◽  
Retrieval Systems ◽  
Information Retrieval Systems ◽  
Present Part ◽  
L Systems

This paper is the first of the three parts of work on the information retrieval systems proposed by Salton (see [24]). The system is defined by the notions of a partially ordered set of requests (A, ⩽), the set of objects X and a monotonic retrieval function U : A → 2X. Different conditions imposed on the set A and a function U make it possible to obtain various classes of information retrieval systems. We will investigate systems in which (A, ⩽) is a partially ordered set, a lattice, a pseudo-Boolean algebra and Boolean algebra. In my paper these systems are called partially ordered information retrieval systems (po-systems) lattice information retrieval systems (l-systems); pseudo-Boolean information retrieval systems (pB-systems) and Boolean information retrieval systems (B-systems). The first part concerns po-systems and 1-systems. The second part deals with pB-systems and B-systems. In the third part, systems with a partial access are investigated. The present part discusses the method for construction of a set of attributes. Problems connected with the selectivity and minimalization of a set of attributes are investigated. The characterization and the properties of a set of attributes are given.

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.

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 Operators related to idempotent generated and monoid completely regular semigroups

1990 ◽  
Vol 49 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Mario Petrich ◽  
Norman R. Reilly
Keyword(s):  
Binary Operation ◽  
Unary Operation ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups

AbstractThe class CR of completely regular semigroups (unions of groups or algebras with the associative binary operation of multiplication and the unary operation of inversion subject to the laws x = xx-1, (x−1)-1 = x and xx-1 = x-1x) is a variety. Among the important subclasses of CR are the classes M of monoids and I of idempotent generated members. For each C ∈ {I, M}, there are associated mappings ν → ν ∩ C and ν → (Ν ∩ C), the variety generated by ν ∩ C. The lattice theoretic properties of these mappings and the interactions between these mappings are studied.

scholarly journals Free right type A semigroups

1991 ◽  
Vol 33 (2) ◽  
pp. 135-148 ◽  
Author(s):  
John Fountain
Keyword(s):  
Binary Operation ◽  
Type A ◽  
Unary Operation ◽  
Adequate Semigroup ◽  
Green’S Relation

The relation ℒ* is defined on a semigroup S by the rule that a ℒ*b if and only if the elements a, b of S are related by Green's relation ℒ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S) of idempotents is a subsemilattice of S. A right adequate semigroup is an E-semigroup in which every ℒ*-class contains an idempotent. It is easy to see that, in fact, each ℒ*-class of a right adequate semigroup contains a unique idempotent [8]. We denote the idempotent in the ℒ*-class of a by a*. Then we may regard a right adequate semigroup as an algebra with a binary operation of multiplication and a unary operation *. We will refer to such algebras as *-semigroups. In [10], it is observed that viewed in this way the class of right adequate semigroups is a quasi-variety.

Injective Hulls of Semilattices

1970 ◽  
Vol 13 (1) ◽  
pp. 115-118 ◽  
Author(s):  
G. Bruns ◽  
H. Lakser
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Binary Operation ◽  
Ordered Set ◽  
Upper Bounds ◽  
Partial Ordering ◽  
Partially Ordered ◽  
Image Position ◽  
Injective Hulls

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relationdefines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.

Convex structures in a new kind of ordered fuzzy group1

2020 ◽  
Vol 39 (3) ◽  
pp. 4245-4257
Author(s):  
Hongping Liu ◽  
Ruiju Wei ◽  
Qian Ge
Keyword(s):  
Convex Hull ◽  
Binary Operation ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Convex Subgroup ◽  
Partially Ordered ◽  
Convex Structure ◽  
Fuzzy Group ◽  
Fuzzy Setting ◽  
Convex Spaces

By means of a fuzzy binary operation defined on partially ordered sets, a new kind of ordered fuzzy group is proposed in this paper. Some properties of this ordered fuzzy group are studied. Following that, its substructures, such as subgroup and convex subgroup, as well as its homomorphisms, along with their properties are explored. It is shown that each family of these substructures forms a convex structure, where the convex hull of a subset is exactly the (convex) subgroup generated by itself, and the homomorphisms between two ordered fuzzy groups are convexity-preserving mappings between the corresponding convex spaces. In addition, when these substructures are extended to fuzzy setting, several L-convex structures are constructed and investigated.

scholarly journals Study of T-norms and Quantum Logic Functions on BL-algebra and Their Relationships to the Classical Probability Structures

2020 ◽  
pp. 591-599
Author(s):  
Ahmed AL-Adilee ◽  
Habeeb Kareem Abdullah ◽  
Hawraa A. AL-Challabi
Keyword(s):  
Quantum Logic ◽  
Probability Space ◽  
Binary Operation ◽  
The Other ◽  
Symmetric Difference ◽  
Logic Function ◽  
Classical Probability ◽  
Binary Operations ◽  
Basic Probability ◽  
Logic Functions

This paper is concerned with the study of the T-norms and the quantum logic functions on BL-algebra, respectively, along with their association with the classical probability space. The proposed constructions depend on demonstrating each type of the T-norms with respect to the basic probability of binary operation. On the other hand, we showed each quantum logic function with respect to some binary operations in probability space, such as intersection, union, and symmetric difference. Finally, we demonstrated the main results that explain the relationships among the T-norms and quantum logic functions. In order to show those relations and their related properties, different examples were built.

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