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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.

goblin GONE
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    Something could be said: if I'm not wrong associativity of $X$ is equivalent to requiring that every element of $\text{im} L$ commute with every element of $\text{im} R$. I'm not aware if there are other properties of this kind. – Giorgio Mossa Jan 07 '14 at 19:39

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