[mcphage]

> Although fold-left is commonly used to accumulate results, it is more general than that. We can use fold-left as a driver for a state machine. The second argument to fold-left is the initial state, and the combining function is the state transition function. The list argument provides a single input to the state machine on each state transition.

At that point you've lost associativity: ((state * transition) * transition) is meaningful, but (state * (transition * transition)) isn't well defined. Which means you're no longer talking about monoids.

Another way to look at it—by associativity, fold-left and fold-right should be equal. If they're not, or if one is defined and the other isn't, then you don't have associativity.