Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices

2013/06/08 by Carmen Constantin, Constantin, Carmen, Andreas Doering +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #03G12 #06C15 #81P10 #Advanced Algebra and Logic #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Logic (math.LO) #Quantum Physics (quant-ph) #math.LO #msc:03G12 #msc:06C15 #msc:81P10 #quant-ph

paper · pdf · doi:10.48550/arxiv.1306.1950

12 pages, no figures; v2: minor corrections, improved presentation

openalex publication_date 2013/06/08 · arxiv created 2013/12/05 · arxiv updated 2013/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B(L) of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=P(H), the projection lattice of the algebra B(H) of bounded operators on Hilbert space.

Cited by

• Sheaf-Theoretic Methods in Quantum Mechanics and Quantum Information Theory

Related

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• Sublattices and Δ-blocks of orthomodular posets
• Logical and algebraic properties of generalized orthomodular posets
• Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets
• Varieties corresponding to classes of complemented posets