Year: 2026
Type: preprint
Abstract: In 1974, C. D. Aliprantis posed several questions concerning topological completions of locally solid vector lattices. We answer all of them in the negative. We construct Hausdorff locally convex-solid vector lattices showing that, without metrizability, neither the $σ$-Lebesgue property nor property (B, i) need pass to the completion; a positive element of the completion need not be the limit of a decreasing sequence of upper elements; the generalized (A, 0) property need not make the canonical image a regular sublattice; and regularity of this image need not imply order density. All five counterexamples have been formalized in Lean 4 using the Banach lattice Lean library.
Source: arXiv (Cornell University)
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G149162043, G1908868068, G6364649569
OpenAlex ID: W7214331140