Properties of free Banach lattices

Characterize the normed spaces $E$ for which the free Banach lattices over $E$ possess the semicontinuity, demicontinuity, monotone boundedness, completeness, and related order properties studied in the paper.

Background

The paper proves a complete characterization for the free AM-space Cph(BE∗,w∗)C_{ph}(B_{E^{*},\mathrm{w}^{*}}) in terms of the dimension of EE, including regularity, semicontinuity, pointwise demicontinuity, and anti-demicontinuity. It then points out that the corresponding question for free Banach lattices is broader and remains unresolved, with only partial results available.

References

Characterize those normed spaces $E$ such that the free Banach lattices over $E$ have the properties considered in this article.

— Variants of order semicontinuity in Banach lattices  (2609.03070 - Bilokopytov, 2 Sep 2026) in Question following Remark \ref{noidea}, Section 8, “Semicontinuity-like properties in free AM-spaces”