Free Banach lattices and their properties

A free Banach lattice on a Banach space $E$ is the Banach lattice $X$ with the following properties: (i) $X$ contains $E$ as a subspace (isometrically); (ii) $E$ generates $X$ as a Banach lattice; (iii) Any operator $T : E \to Z$ (where $Z$ is a Banach lattice) extends to a lattice homomorphism $\widehat{T} : X \to Z$ of the same norm (this extension is unique). In a similar fashion, we can define a free $p$-convex Banach lattice. We show how to construct free Banach lattices, and examine their properties (such as the existence of non-trivial convexity, properties of sublattices, as well as the structure of operators between free Banach lattices).
This is a joint work with with M.Taylor, P.Tradacete, and V.Troitsky.