Universal monoid actions: the power of freedom

TL;DR

This paper demonstrates that every countable monoid can act universally on a free object over a countable set, and provides new insights into universal operators on Banach spaces, broadening understanding of algebraic and functional analytic universality.

Contribution

It establishes a universal action for all countable monoids on free objects and introduces an abstract concept of being generated by a set, simplifying existing proofs.

Findings

Every countable monoid has a universal action on a free object over a countable set.

A new abstract concept of 'being generated by a set' is introduced.

Provides a simpler proof for a universal operator on $\\ell_1$.

Abstract

Motivated by a recent work of Balcerzak and Kania [Proc. Amer. Math. Soc. 151 (2023) 3737--3742], we show that every countable monoid has a universal action on the free object over a countable infinite set. This is a general result concerning concrete categories with a left adjoint (free) functor. On the way, we introduce an abstract concept of ``being generated by a set". At the same time we obtain a simpler proof of the result of Darji and Matheron concerning a surjectively universal operator on the classical Banach space $\ell_1$ of summable sequences.