I don't think you could even define an associative binary operator on states in the Game of Life because of its computational irreducibility.

CGOL is is turing complete. If you can make a NOR gate, you can make anything.

I know it is Turing-complete; I was instead commenting on its computational irreducibility. My point is that it is impossible to express the rules in the form of an associative operator over a sequence of board states. You could say the same thing about iterating with a sufficiently complex circuit of NOR gates.