Universal monoid actions: The power of freedom

Abstract:

Motivated by the recent work of Balcerzak and Kania [Proc. Amer. Math. Soc. 151 (2023) pp. 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 [Proc. Amer. Math. Soc. 145 (2017), pp, 251–265] concerning a surjectively universal operator on the classical Banach space $\ell _1$ of summable sequences.