John and Keith
Thanks for the correction. I just realized that the associativity diagram I drew is just an extension of commutivity as shown below:
So it isn't the real associativity diagram since you can have associativity without commutivity. So the simplest way to draw any of these are just wires = wires with labels but that is the same as writing them out which is fine but I think it loses a lot of information of whats really going on.
Below is a better diagram of associativity that is not dependent on commutivity:
>but a nice feature of wiring diagrams it that we can leave associativity implicit!
It might be because of poetic justice, but I think this may not be true? When you draw, there is an implicit order to the placement of arrows, circle or whatever it is you are drawing. So the following two pictures are actually not equal until you give it that rule!
Once you define commutivity, then you can say that order doesn't matter :
So when you first start off with a wiring diagram before you define any of the rules, there is an implicit order to the placement of the wires and then rules are added to allow permutations. I guess it really depends on what rules you want to start off with but I just thought by doing this you can show more with the diagrams within the context of monodical preorders. But then again, I only know part of the picture so please fill me in.