option-semigroup-monoid-second
Question
It is possible to define a monoid instance for Option<A> that behaves like that:
| x | y | concat(x, y) |
|---|---|---|
| none | none | none |
| some(a1) | none | some(a1) |
| none | some(a2) | some(a2) |
| some(a1) | some(a2) | some(S.concat(a1, a2)) |
// the implementation is left as an exercise for the reader
declare const getMonoid: <A>(S: Semigroup<A>) => Monoid<Option<A>>
What is the empty member for the monoid?
Answer
none is the empty member for our monoid because with it, all the Monoid laws are true. Let's check the monoid laws on our new monoid:
associative
concat(none, concat(none, concat(none))) === concat(concat(none, none), none)
concat(none, concat(none, concat(some(z)))) === concat(concat(none, none), some(z))
concat(none, concat(some(y), concat(none))) === concat(concat(none, some(y)), none)
concat(none, concat(some(y), concat(some(z)))) === concat(concat(none, some(y)), some(z))
concat(some(x), concat(none, concat(none))) === concat(concat(some(x), none), none)
...
concat(some(x), concat(some(y), concat(some(z)))) === concat(concat(some(x), some(y)), some(z))
right identity
concat(some(x), none) === some(x)
left identity
concat(none, some(x)) === some(x)