it's not exactly order independence, it's associativity: (a • b) • c == a • (b • c) i.e. "no matter where you put the parentheses, it'll give the same res

it's not exactly order independence, it's associativity: (a • b) • c == a • (b • c) i.e. "no matter where you put the parentheses, it'll give the same result". this DOESN'T mean that a • b == b • a though it may be true for some monoids like addition/multiplication over numbers¹. you'll notice that associativity indeed holds for strings (or arrays of some T), where concatenation is the monoidal operation: ("he" ++ "ll") ++ "o!" == "he ++ ("ll" ++ "o!") == "hello!" though at first blush associativity may not seem very useful, it is! it means folding/"summing" a sequence of monoidal values is trivially parallelizable. i.e. if you've got a sequence as = [a0, a1, a2, ..., an-1, an] and you want to "sum"/fold it using a's monoidal operation "•": r = (a0 • a1 • a2 • ... • an) you can split it into chunks across multiple workers and then "sum" the results of summing each chunk: r0 = (a0•a1•a2) // worker 0 r1 = (a3•a4•a5) // worker 1 ... rm = (an-2,an-1,an) // worker m r = r0•r1•...•rm // sum up what the workers produced the nice thing is, as long as we don't reorder the elements, we'll always get the same result no matter how we chunk up `as`! and this will hold for any monoid `(a, •)`. btw, an example of a less trivial monoid is `sorted lists of T `: - `•`, the monoidal operation, is `merge(a, b)`, which combines two sorted lists into a new sorted list: `merge([1,2,3,5], [2,4]) == [1,2,2,3,4,5]` - `e`, the "neutral element", is just the empty list `[]` this is a (commutative) monoid, and you could say that mergesort relies on its monoidal properties to work! --- ¹ if you can swap values around and still get the same result, you've got a "commutative monoid".