  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.
