JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (5): 74-78.doi: 10.6040/j.issn.1671-9352.0.2023.441

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New characterizations of orthogonal modular lattices based on quantum logic

YANG Xiaofei1, XIAO Feihu1, MA Yingcang1, XIN Xiaolong1,2*   

  1. 1. School of Science, Xian Polytechnic University, Xian 710048, Shaanxi, China;
    2. School of Mathematics, Northwest University, Xian 710127, Shaanxi, China
  • Published:2025-05-19

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Abstract: In order to study some new characterizations of orthomodular law in orthogonal modular lattices, we study orthomodular law from an algebraic perspective. By using complement operations, the reciprocal nature of partial addition and subtraction operations,and residuated property of global multiplication and implication, the equivalent characterizations of orthomodular law are given, respectively. These facts reveal the inherent laws of the generation of orthomodular law. By some examples, it is shown that in general the global operations of addition and multiplication on orthogonal modular lattices are non-associative and non-commutative.

Key words: orthogonal modular lattice, orthogonal lattice, Boolean algebra, Hilbert space, quantum logic

CLC Number: 

  • O153.1
[1] KALMBACH G. Orthomodular lattices[M]. London: Academic Press, 1983.
[2] FREYTES H. An equational theory for σ-complete orthomodular lattices[J]. Soft Computing, 2020, 24:10257-10264.
[3] FAZIO D, LEDDA A, PAOLI F. Residuated structures and orthomodular lattices[J]. Studia Logica, 2021, 9:1201-1239.
[4] WU Yali, YANG Yichuan. Orthomodular lattices as L-algebras[J]. Soft Computing, 2020, 24:14391-14400.
[5] IORGULESCU A. On quantum-MV algebras, part Ⅰ: the orthomodular algebras[J]. Scientific Annals of Computer Science, 2021, 31(2):163-222.
[6] MCDONALD J, BIMBÓ K. Topological duality for orthomodular lattices[J]. Mathematical Logic Quarterly, 2023, 69(2):174-191.
[7] BONZIO S, CHAJDA I. A note on orthomodular lattices[J]. International Journal of Theoretical Physics, 2017, 56:3740-3743.
[8] BLYTH T S. Lattices and ordered algebraic structures[M]. London: Springer, 2005.
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