Monoids
Monoids are the most basic way to combine any values. A monoid is algebra that is closed under an associative binary operation and has an identity element.
We can think of a monoid as a design pattern that allows us to quickly reduce (or fold) on a collection of a single type in a parallel way.
Monoid rules
A monoid is anything that satisfies the following rules:
- Closure rule
- Associativity rule
- Identity rule
Let's discuss these rules in brief.
Closure rule
“If you combine two values of same type, you get another value of the same type.”
Given two inputs of the same type, a monoid returns one value of the same type as the input.
Closure rule examples
1 + 2 = 3, and 3 is an integer.
1 + 2 + 3 also equals an integer.
1 + 2 + 3 + 4 also equals an integer.
Our binary operation has been extended into an operation that works on lists!
Closure axiom
If a, b ∈ S, then a + b ∈ S.
That says, if a and b are any two values in the set S of integers and if we apply the binary operation + to any two values, then...
