Is there a name for the relationship between $f()$ and $g()$ when $f(f(0,a),b)$ is guaranteed to be equal to $g(f(0,a),f(0,b))$ ?
I'm using $0$ here to represent an initial state. The real problem is that $f()$ will be called repeated any number of times and $g()$ represents a way of combining resulting states from two separate series of calls to find the state that would have resulted from the same sequence of transitions done sequentially.
I'm just looking for a name for this property so I can search more on what can be done when you have it.
Lookup:
Essentially, your $g$ is defined by $f$ composed with itself recursively.