Arguably then, anything which is an instance of Monad should be a monad.
I would certainly argue strongly for that. If it ain't true, then small, irrelevant changes in a piece of code could break. For example, is it safe to make this change: do f y ==> do x <- f y return x I'd hope it would be. (And the change might be useful if I want to insert a check or a debugging printf in the code.) But if Monads don't have to satisfy this axiom, then I can't make a change like this without having to write additional conditions in the documentation or know all possible instances of the code.
And '+' should be addition. etc. etc.
I think it is well accepted that addition in computers is rarely commutative, associative, etc. across the whole range of values.
Haskell doesn't require any of these to be true, and similarly it doesn't require that every instance of '==' is equality.
Though it is, perhaps, worth noting that all standard instances of Eq use observational equality which is plain, ordinary equality for normal values and, if we look at the common extensions, is 'pointer equality' for mutable objects like IORefs. (For abstract datatypes like balanced binary trees, the appropriate equality would probably be to test whether the trees contain the same set of values even if they are balanced slightly differently.) -- Alastair Reid www.haskell-consulting.com