Demicontinuity versus anti-semicontinuity

Determine whether a demicontinuous normed lattice can also be anti-semicontinuous, where anti-semicontinuity means that the order closure of its unit ball equals the entire lattice.

Background

The paper distinguishes demicontinuity, which requires the order adherence of the unit ball to be norm bounded, from anti-semicontinuity, which requires the order closure of the unit ball to equal the whole lattice. It observes that anti-demicontinuity implies anti-semicontinuity and that anti-semicontinuity is incompatible with topological semicontinuity, but does not resolve whether demicontinuity and anti-semicontinuity can coexist.

References

However, it is unknown whether it is possible for a demicontinuous normed lattice to also be anti-semicontinuous.

— Variants of order semicontinuity in Banach lattices  (2609.03070 - Bilokopytov, 2 Sep 2026) in Remark following the summary of properties in Section 4, “Pointwise demicontinuity and anti-demicontinuity”