# Smallest order closed sublattices and option spanning

## TL;DR

This paper investigates the smallest order closed sublattice containing a given sublattice in a vector lattice, revealing conditions under which order closures behave predictably and linking these to properties of Banach lattices and financial option spanning.

## Contribution

It establishes that the smallest order closed sublattice can be obtained via second order closure and characterizes order continuity of the norm in Banach lattices through order closure properties.

## Key findings

- Smallest order closed sublattice equals the second order closure in many cases.
- Order closure of sublattices characterizes order continuity of the norm.
- Provides a framework connecting lattice theory with financial economics.

## Abstract

Let $Y$ be a sublattice of a vector lattice $X$. We consider the problem of identifying the smallest order closed sublattice of $X$ containing $Y$. It is known that the analogy with topological closure fails. Let $\overline{Y}^o$ be the order closure of $Y$ consisting of all order limits of nets of elements from $Y$. Then $\overline{Y}^o$ need not be order closed. We show that in many cases the smallest order closed sublattice containing $Y$ is in fact the second order closure $\overline{\overline{Y}^o}^o$. Moreover, if $X$ is a $\sigma$-order complete Banach lattice, then the condition that $\overline{Y}^o$ is order closed for every sublattice $Y$ characterizes order continuity of the norm of $X$. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset.