On the free Banach lattice generated by a lattice

TL;DR

This paper investigates the structural properties of free Banach lattices generated by a distributive lattice, including conditions for strong units, density character, and lattice homomorphisms.

Contribution

It provides new characterizations and analyses of the structural and geometric properties of free Banach lattices generated by lattices.

Findings

Characterized when $FBL\langle L\rangle$ has a strong unit.

Computed the density character of $FBL\langle L\rangle$.

Studied when $FVL\langle L\rangle$ is order dense in $FBL\langle L\rangle$.

Abstract

We study structural properties of the free Banach lattice $FBL\langle L\rangle$ generated by a distributive lattice $L$. We characterize when $FBL\langle L\rangle$ has a strong unit, compute its density character, analyze the density character of order intervals and study when is $FVL\langle L\rangle$ order dense in $FBL\langle L\rangle$. We also study projection bands, quasi-interior points, and Banach lattice homomorphisms induced by lattice homomorphisms. Finally, we show that $FBL\langle L\rangle$ is lattice isometric to $FBL\langle L^{\mathrm{op}}\rangle$, where $L^{\mathrm{op}}$ denotes the opposite lattice.