Quantum categories for quantum logic

The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors $\mathsf{[E,Sets]}$ where $\mathsf{E}$ is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called $quantos$ – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative $C^*$-algebras, which in turn are studied in their quantum habitat of noncommutative $C^*$-algebras – more fundamental structures than commutative $C^*$-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors $\mathsf{[E,QSets]}$ is constructed where $\mathsf{QSets}$ is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.

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Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽

10.3390/axioms10030164 ◽

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.

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Semigroups Co-Ordinatizing Orthomodular Geometries

Canadian Journal of Mathematics ◽

10.4153/cjm-1965-005-4 ◽

1965 ◽

Vol 17 ◽

pp. 40-51 ◽

Cited By ~ 9

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.

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Generalized Bounded Linear Logic and its Categorical Semantics

Lecture Notes in Computer Science - Foundations of Software Science and Computation Structures ◽

10.1007/978-3-030-71995-1_12 ◽

2021 ◽

pp. 226-246

Author(s):

Yōji Fukihara ◽

Shin-ya Katsumata

Keyword(s):

Linear Logic ◽

Cut Elimination ◽

Grading System ◽

Bounded Linear ◽

The Core ◽

Categorical Structure ◽

Categorical Semantics

AbstractWe introduce a generalization of Girard et al.’s called (and its affine variant ). It is designed to capture the core mechanism of dependency in , while it is also able to separate complexity aspects of . The main feature of is to adopt a multi-object pseudo-semiring as a grading system of the !-modality. We analyze the complexity of cut-elimination in , and give a translation from with constraints to with positivity axiom. We then introduce indexed linear exponential comonads (ILEC for short) as a categorical structure for interpreting the $${!}$$ ! -modality of . We give an elementary example of ILEC using folding product, and a technique to modify ILECs with symmetric monoidal comonads. We then consider a semantics of using the folding product on the category of assemblies of a BCI-algebra, and relate the semantics with the realizability category studied by Hofmann, Scott and Dal Lago.

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Classical Logic and Quantum Logic with Multiple and Common Lattice Models

Advances in Mathematical Physics ◽

10.1155/2016/6830685 ◽

2016 ◽

Vol 2016 ◽

pp. 1-12 ◽

Cited By ~ 3

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.

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Block-Finite Orthomodular Lattices

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-090-6 ◽

1979 ◽

Vol 31 (5) ◽

pp. 961-985 ◽

Cited By ~ 11

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.

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Left residuated lattices induced by lattices with a unary operation

Soft Computing ◽

10.1007/s00500-019-04461-x ◽

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.

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The computing power of Turing machine based on quantum logic

10.29007/k8cb ◽

2018 ◽

Author(s):

Yun Shang ◽

Xian Lu ◽

Ruqian Lu

Keyword(s):

Quantum Logic ◽

Turing Machine ◽

Turing Machines ◽

Computational Power ◽

Orthomodular Lattices ◽

Locally Finite ◽

Computing Power ◽

Truth Value ◽

Quantum Language

Turing machines based on quantum logic can solve undecidableproblems. In this paper we will give recursion-theoreticalcharacterization of the computational power of this kind of quantumTuring machines. In detail, for the unsharp case, it is proved that&#93101&#8746&#92801&#8838LTd(&#949,&#931)(LTw(&#949,&#931))&#8838&#92802when the truth value lattice is locally finite and the operation &#8743is computable, whereLTd(&#949,&#931)(LTw(&#949,&#931))denotes theclass of quantum language accepted by these Turing machine indepth-first model (respectively, width-first model);for the sharp case, we can obtain similar results for usual orthomodular lattices.

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Varieties of Orthomodular Lattices. II

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-027-4 ◽

1972 ◽

Vol 24 (2) ◽

pp. 328-337 ◽

Cited By ~ 11

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.

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Residuation in orthomodular lattices

Topological Algebra and its Applications ◽

10.1515/taa-2017-0001 ◽

2017 ◽

Vol 5 (1) ◽

pp. 1-5 ◽

Cited By ~ 2

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.

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THE GELFAND SPECTRUM OF A NONCOMMUTATIVE C*-ALGEBRA: A TOPOS-THEORETIC APPROACH

Journal of the Australian Mathematical Society ◽

10.1017/s1446788711001157 ◽

2011 ◽

Vol 90 (1) ◽

pp. 39-52 ◽

Cited By ~ 15

Author(s):

CHRIS HEUNEN ◽

NICOLAAS P. LANDSMAN ◽

BAS SPITTERS ◽

SANDER WOLTERS

Keyword(s):

Quantum Mechanics ◽

Usual Sense ◽

Theoretic Approach ◽

Stone Duality ◽

Complete Lattices ◽

C Algebra ◽

C Algebras ◽

Gelfand Transform ◽

Noncommutative Spaces ◽

Gelfand Duality

AbstractWe compare two influential ways of defining a generalized notion of space. The first, inspired by Gelfand duality, states that the category of ‘noncommutative spaces’ is the opposite of the category of C*-algebras. The second, loosely generalizing Stone duality, maintains that the category of ‘point-free spaces’ is the opposite of the category of frames (that is, complete lattices in which the meet distributes over arbitrary joins). Earlier work by the first three authors shows how a noncommutative C*-algebra gives rise to a commutative one internal to a certain sheaf topos. The latter, then, has a constructive Gelfand spectrum, also internal to the topos in question. After a brief review of this work, we compute the so-called external description of this internal spectrum, which in principle is a fibred point-free space in the familiar topos of sets and functions. However, we obtain the external spectrum as a fibred topological space in the usual sense. This leads to an explicit Gelfand transform, as well as to a topological reinterpretation of the Kochen–Specker theorem of quantum mechanics.

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