[Update: I've now asked the same question on mathoverflow.]
For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes:
$$ \forall f,g,h\in G:hg(f)=h(g(f)) $$
Now suppose there is additional axiom, or constraint if you prefer, called consistency:
$$ \forall f,g\in G: f(g)f=g(f)g $$
This can be represented by a commutative diagram:
If I chain two of these diagrams together I get the following:
The consistency of $f$ and $hg$ can be represented by the following:
Comparing these last two commutative diagrams suggests the following two identities:
$$ \left. \begin{array}{l} hg(f)=h(g(f))\\ (hg)=g(f)(h)f(g) \end{array} \right\} $$
The first is compatibility of course but now there is a second identity which suggests that compatibility has a dual if we require consistency.
These identities have applications in rewriting theory, however it has been suggested to me that a semigroup or monoid with a consistent left action on itself may have interesting mathematical properties in its own right. Is this true? Have semigroups or monoids such as this ever been studied?