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 standard 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 two commutative diagrams suggests the following two identities:

$$ \left. \begin{array}{l} hg(f)=h(g(f))\\ f(hg)=g(f)(h)f(g) \end{array} \right\} $$

The first is compatibility of course but now there is a second identity which indicates that compatibility can have a dual, which I'm going to call co-compatibility.

These identities have applications in rewriting theory, however it has been put 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?