Free and Projective Banach Lattices

TL;DR

This paper introduces the concept of free and projective Banach lattices, establishing their existence, fundamental properties, and structural characterizations based on the number of generators, while exploring their relationships and examples.

Contribution

It defines free Banach lattices in the category of Banach lattices, proves their existence, and characterizes their structure for finite and countable generators, also linking projectivity with free lattices.

Findings

Existence of free Banach lattices proven.

Structural characterizations for finite and countable generators.

Connection established between projective and free Banach lattices.

Abstract

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms and establish some of their fundamental properties. We give much more detailed results about their structure in the case that there are only a finite number of generators and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice $P$ to be projective if whenever $X$ is a Banach lattice, $J$ a closed ideal in $X$, $Q:X\to X/J$ the quotient map, $T:P\to X/J$ a linear lattice homomorphism and $\epsilon>0$ there is a linear lattice homomorphism $\hat{T}:P\to X$ such that (i) $T=Q\circ \hat{T}$ and (ii) $\|\hat{T}\|\le (1+\epsilon)\|T\|$. We establish the connection between projective Banach lattices and free Banach lattices and describe several families of Banach lattices that are projective as well as proving that some are not.