A Countable Three-Generated Orthomodular Lattice with No σ-Complete Embedding
Abstract
We give an explicit countably infinite orthomodular lattice generated by three specified elements and a direct argument that it admits no injective homomorphism into any σ-complete orthomodular lattice. In particular, it does not embed into any set-sized complete orthomodular lattice. The source is a generated subalgebra of a cofinite reduced product of finite-dimensional real subspace lattices. A nonzero finite term a is orthogonal to the orbit of a terminal line under every finite word in three Sasaki projections, while another finite term X satisfies πaπX(t)=a. If the image of this countable orbit has a supremum in a target orthomodular lattice, Sasaki adjunction forces that supremum to commute with the generator images. This forces the image of a to vanish. The proof imposes no atomicity, modularity, Hilbert representation, cardinality bound on the target, or preservation of source infinitary operations. All target lemmas are derived from the orthomodular axioms.
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BibTeX
@misc{higuchi2026countable,
title = {{A Countable Three-Generated Orthomodular Lattice with No $σ$-Complete Embedding}},
author = {Joaquim Reizi Higuchi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14859},
eprint = {2610.14859},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14859},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Countable Three-Generated Orthomodular Lattice with No σ-Complete Embedding
AU - Joaquim Reizi Higuchi
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14859
UR - https://arxiv.org/abs/2610.14859
AB - We give an explicit countably infinite orthomodular lattice generated by three specified elements and a direct argument that it admits no injective homomorphism into any σ-complete orthomodular lattice. In particular, it does not embed into any set-sized complete orthomodular lattice. The source is a generated subalgebra of a cofinite reduced product of finite-dimensional real subspace lattices. A nonzero finite term a is orthogonal to the orbit of a terminal line under every finite word in three Sasaki projections, while another finite term X satisfies π_aπ_X(t)=a. If the image of this countable orbit has a supremum in a target orthomodular lattice, Sasaki adjunction forces that supremum to commute with the generator images. This forces the image of a to vanish. The proof imposes no atomicity, modularity, Hilbert representation, cardinality bound on the target, or preservation of source infinitary operations. All target lemmas are derived from the orthomodular axioms.
ER -