On identities in orthocomplemented difference lattices

Matoušek, Milan; Pták, Pavel

doi:10.2478/s12175-010-0033-7

Cited by 5 publications

(1 citation statement)

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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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