On Set-Representable Orthocomplemented Difference Lattices

Su, Jianning

doi:10.1007/s11083-020-09522-7

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…In a similar vein, we can find Abbott algebras without any state or with a precisely one state (see [ 7 ] and [ 12 ]). Also, we find that the free Abbott algebra over 2 generators contains precisely 128 elements and the free algebra over 3 generators is infinite (see [ 10 , 11 ], etc.). A specific line of algebraic nature is the notion of modularity in the Abbott algebras.…”

Section: Resultsmentioning

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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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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Section: Resultsmentioning

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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Quantum Logics that are Symmetric-difference-closed

Burešová

Pták

2021

Int J Theor Phys

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On Set-Representable Orthocomplemented Difference Lattices

Cited by 4 publications

References 7 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

Quantum Logics that are Symmetric-difference-closed

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