PreprintCC BY 4.0arXiv
Aliprantis's questions on locally solid topologies
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.
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BibTeX
@misc{munozlahoz2026aliprantiss,
title = {{Aliprantis's questions on locally solid topologies}},
author = {David Muñoz-Lahoz and Mitchell A. Taylor and Pedro Tradacete},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29498},
eprint = {2609.29498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29498},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Aliprantis's questions on locally solid topologies
AU - David Muñoz-Lahoz
AU - Mitchell A. Taylor
AU - Pedro Tradacete
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29498
UR - https://arxiv.org/abs/2609.29498
AB - 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.
ER -