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 $π_aπ_X(t)=a$. 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.