Free bounded archimedean $\ell$-algebras

TL;DR

This paper investigates the existence of free objects in the category of bounded archimedean $ ext{l}$-algebras, showing they do not exist for sets but do for weighted sets, with implications discussed.

Contribution

It introduces weighted sets and proves the existence of free objects in $bal$ for these, contrasting with the non-existence for plain sets.

Findings

Free objects do not exist in $bal$ for sets.

Free objects do exist in $bal$ for weighted sets.

Discusses consequences of these existence results.

Abstract

We show that free objects on sets do not exist in the category $bal$ of bounded archimedean $\ell$-algebras. On the other hand, we introduce the category of weighted sets and prove that free objects on weighted sets do exist in $bal$. We conclude by discussing several consequences of this result.