Is Monoid[String] really a Monoid in scala

Question

I am currently learning about category theory in scala and the law of associativity says

(x + y) + z = x + (y + z)

Thats just fine when working with more than two values

("Foo" + "Bar") + "Test" == "Foo" + ("Bar" + "Test") // true

In that case order doesn't matter. But what if there are only two values. In case for numbers its still working ( commutative ) but when doing the same thing with strings it fails.

3+1==1+3 // True
("Foo" + "Bar") == ("Bar" + "Foo") // Not commuative

So is it legal to say that associativity requires commudativity to fullfill the monoid law ? And so is a String Monoid valid anyway ?

Answer 1

So is it legal to say that associativity requires commutativity to fulfill the monoid law?

No. A binary operation does not need to be commutative to be associative. The fact that ("Foo" + "Bar") == ("Bar" + "Foo") is false is irrelevant to the fact that + is associative for String.

And so is a String Monoid valid anyway?

Yes, you can have a Monoid[String] .

By definition:

A monoid is a set that is closed under an associative binary operation + and has an identity element I in S such that for all a in S , I + a = a + I = a .

A monoid must contain at least one element.

Let + be the binary operation for a Monoid[String] . For any two strings, a and b , a + b is also a String , so the binary operation is closed over the type String . Without rigorous proof, we also know that it is associative.

ie for all strings a , b , and c :

(a + b) + c == a + (b + c)

We also have an identity element "" (the empty string), because for any string a , a + "" == a and "" + a == a .

A monoid whose binary operation is also a commutative is called a commutative monoid . And you clearly cannot have a commutative monoid for String using the + operation.