States on systems of sets that are closed under symmetric difference

Simone, Anna De; Navara, Mirko; Pták, Pavel

doi:10.1002/mana.201500029

Cited by 11 publications

(6 citation statements)

References 14 publications

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“…Recall ([ 4 ]) that a state on is defined as follows : If . It is easy to see that the state space of A is isomorphic (via the isomorphism of Theorem 1.4 ) with the state space of .…”

Section: Resultsmentioning

confidence: 99%

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Generalized $$\mathbb {XOR}$$ Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

Burešová

2023

Int J Theor Phys

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Considering the inference rules in generalized logics, J.C. Abbott arrives to the notion of orthoimplication algebra (see Abbott (1970) and Abbott (Stud. Logica. 2:173–177, XXXV)). We show that when one enriches the Abbott orthoimplication algebra with a falsity symbol and a natural $$\mathbb {XOR}$$ XOR -type operation, one obtains an orthomodular difference lattice as an enriched quantum logic (see Matoušek (Algebra Univers. 60:185–215, 2009)). Moreover, we find that these two structures endowed with the natural morphisms are categorically equivalent. We also show how one can introduce the notion of a state in the Abbott $$\mathbb {XOR}$$ XOR algebras strenghtening thus the relevance of these algebras to quantum theories.

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“…Recall ([ 4 ]) that a state on is defined as follows : If . It is easy to see that the state space of A is isomorphic (via the isomorphism of Theorem 1.4 ) with the state space of .…”

Section: Resultsmentioning

confidence: 99%

“…Recently there has been an effort to soundly introduce and study the notion of a symmetric difference in orthomodular lattices and posets (see [ 3 , 4 , 8 , 9 , 12 ], etc.). In the relation to quantum axiomatics, the idea has been to enrich the “quantum logics” with a kind of a operation.…”

Section: Introduction and Basic Notionsmentioning

confidence: 99%

Generalized $$\mathbb {XOR}$$ Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

Burešová

2023

Int J Theor Phys

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“…The investigation of symmetric-difference-closed orthomodular posets and lattices has recently brought a substantial contribution to the theory of orthocomplemented structures (see [1][2][3][4][5][6][7][8][9]). In this note, we ask on the presence of the "Greechie phenomenon" [10] in ODLs: Is there a stateless ODL?…”

Section: Notionsmentioning

confidence: 99%

A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless

Voráček

Pták

2022

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This paper carries on the investigation of the orthomodular lattices that are endowed with a symmetric difference. Let us call them ODLs. Note that the ODLs may have a certain bearing on "quantum logics" -the ODLs are close to Boolean algebras though they capture the phenomenon of non-compatibility. The initial question in studying the state space of the ODLs is whether the state space can be poor. This question is of a purely combinatorial nature. In this note, we exhibit a finite ODL whose state space is empty (respectively, whose state space is a singleton).

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“…Conversely, all lines intersecting at the same point in the parameter space correspond to collinear points in the image space. Considering the high resolution of symmetrical difference SAR images and the vast image scene, by using the excellent direction selectivity of the Hough transform and the idea of integration, the edge direction is obtained according to the local linear feature orientation, and the symmetrical difference SAR is completed by judging the accumulative degree of intersection points [31].…”

Section: Calculation Of Edge Direction By Hough Transformmentioning

confidence: 99%

Edge Detection Algorithm of a Symmetric Difference Kernel SAR Image Based on the GAN Network Model

Zhang

Liu

et al. 2019

Symmetry

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The symmetrical difference kernel SAR image edge detection algorithm based on the Canny operator can usually achieve effective edge detection of a single view image. When detecting a multi-view SAR image edge, it has the disadvantage of a low detection accuracy. An edge detection algorithm for a symmetric difference nuclear SAR image based on the GAN network model is proposed. Multi-view data of a symmetric difference nuclear SAR image are generated by the GAN network model. According to the results of multi-view data generation, an edge detection model for an arbitrary direction symmetric difference nuclear SAR image is constructed. A non-edge is eliminated by edge post-processing. The Hough transform is used to calculate the edge direction to realize the accurate detection of the edge of the SAR image. The experimental results show that the average classification accuracy of the proposed algorithm is 93.8%, 96.85% of the detection edges coincide with the correct edges, and 97.08% of the detection edges fall into the buffer of three pixel widths, whichshows that the proposed algorithm has a high accuracy of edge detection for kernel SAR images.

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States on systems of sets that are closed under symmetric difference

Cited by 11 publications

References 14 publications

Generalized $$\mathbb {XOR}$$ Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

Generalized $$\mathbb {XOR}$$ Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless

Edge Detection Algorithm of a Symmetric Difference Kernel SAR Image Based on the GAN Network Model

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