Arithmetic and k-maximality of the cyclic free magma

Abstract

We survey free magmas and we explore the structure of their submagmas. By equipping the cyclic free magma with a second distributive operation we obtain a ringoid-like structure with some primitive arithmetical properties. A submagma is k-maximal when there are only k − 1 submagmas between it and the free magma itself. These two tools, arithmetic and maximality, allow us to study the lattice of the submagmas of a free magma.