Characterization of the universal property of the Lorentz completion

Characterize the normed lattice $F$, the monotone complete Banach lattice $E$, and the order-continuous homomorphism $J:F\to E$ satisfying the stated universal factorization property through all order-continuous homomorphisms from $F$ into monotone complete Banach lattices.

Background

For a demicontinuous normed lattice, the paper identifies the Lorentz completion as a reflection into the category of monotone complete Banach lattices with order-continuous homomorphisms. It then asks for a converse or intrinsic characterization of any triple (F,E,J)(F,E,J) possessing the corresponding universal factorization property, without assuming in advance that EE is the Lorentz completion of FF.

References

Suppose $F$ is a normed lattice, $E$ is a monotone complete Banach lattice, and $J:F\to E$ is an order-continuous homomorphism such that whenever $T:F\to G$ is an order-continuous homomorphism into a monotone complete Banach lattice $G$, there is a unique order-continuous homomorphism $\widehat{T}:E\to G$ such that $\widehat{T}J=T$. What can be said about $F$, $E$, and $J$?

— Variants of order semicontinuity in Banach lattices  (2609.03070 - Bilokopytov, 2 Sep 2026) in Question following the categorical discussion in Section 6, “The Lorentz completion”