What can we learn about a magma by studying these monoids?

Question

Given a magma $(X,*)$, we get three monoids in the following way. First, define a pair of functions $L,R : X \rightarrow (X \rightarrow X).$

$$(Lx)(y) = x*y,\quad (Rx)(y) = y*x$$

Then each of the following forms a subset of the functions $X \rightarrow X.$

1. $\mathrm{im} \,L$
2. $\mathrm{im} \,R$
3. $\mathrm{im} \,L \cup \mathrm{im} \,R$

Thus each generates a monoid of functions $X \rightarrow X.$

General Question. What can we learn about the magma $M=(X,*)$ by studying these three monoids? For example, are there any interesting pairs of identities $I,J$ such that $(X,*)$ satisfies identity $I$ iff the monoid generated by $\mathrm{im} \,L$ satisfies identity $J$?

A link or reference would be nice.