Just happened upon these illuminative insights for "Developing an intuition for Monoid" — here are the essentials, distilled and/or paraphrased-for-clarity.
Just a fancy moniker for any sets of values with both:
x <> y
(or mappend) that is associative ie. (x <> (y <> z)) == ((x <> y) <> z)
,id
(or mempty, or call it nil / default / unit if you will.. you get the idea) such that (x <> id) == (id <> x) == x
(n + 0) == n
id = 0
with (<>) = (+)
is a monoid for numbers(n × 1) == n
id = 1
with (<>) = (×)
is a monoid for numbers(b || False) == b
id = False
with (<>) = (||)
is a monoid for booleans(b && True) == b
id = True
with (<>) = (&&)
is a monoid for booleans(s ++ "") == s
; for all lists, (l ++ []) == l
id = []
/ id = ""
with (<>) = (++)
is a monoid for lists/strings
So intuitively, if there's an obvious (associative) combining function <>
from both [X]-to-X and []-to-Xdefault/empty (where also X == (X <> Xdefault/empty)
, always) — then that <> with that Xdefault/empty make a monoid for all X.
()
, why booleans/numbers can be seen as monoids but aren't Monoid
instances by default, etc..
Given a Data.Foldable
instance for some container (list, map, etc..), you can write a generic fold
function that only takes one argument (the container).
f a
(with f
an instance of Foldable
and a
an instance of Monoid
), then instead of foldr someFunc startVal container
you can just write: fold container
.Foldable
part is what's important for the container. Monoid
has no concept of a "container" whatsoever, it simply provides a "zero" value and a way to combine two values together.
So at the end of the day, this all just means that monoids serve as the language used by folds to tear down bigger structures. What exactly that language expresses is up to the Monoid
, and what it means to be torn down is up to the instance of Foldable
.
Grasping all of the above, we can now parse Diehl: «A monoid is a combination of a unit and a single associative operation over a set of values.» So far, I haven't encountered (neither before nor upon having absorbed the above findings) any pressing need of defining and wiring up custom Monoid
instances for explicit use with any Foldable
s (or any other purpose), but it'll probably come up some obscure way when least expected.