Logo Questions Linux Laravel Mysql Ubuntu Git Menu
 

Monoid homomorphism and isomorphism

I am reading the book 'Programming in Scala' (The red book).

In the chapter about Monoids, I understand what a Monoid homomorphism is, for example: The String Monoid M with concatenation and length function f preserves the monoid structure, and hence are homomorphic.

M.op(f(x), f(y)) == M.op(f(x) + f(y))
// "Lorem".length + "ipsum".length == ("Lorem" + "ipsum").length

Quoting the book (From memory, so correct me if I am wrong:

When this happens in both directions, it is named Monoid isomorphisim, that means that for monoids M, N, and functions f, g, f andThen g and g andThen f are the identity function. For example the String Monoid and List[Char] Monoid with concatenation are isomorphic.

But I can't see an actual example for seeing this, I can only think of f as the length function, but what happens with g?

Note: I have seen this question: What are isomorphism and homomorphisms.

like image 741
Alejandro Alcalde Avatar asked Sep 11 '19 11:09

Alejandro Alcalde


People also ask

What is a monoid homomorphism?

A monoid homomorphism is a map between monoids that preserves the monoid operation and maps the identity element of the first monoid to that of the second monoid (the identity element is a 0-ary operation). A group homomorphism is a map between groups that preserves the group operation.

What is monoid homomorphism explain with example?

Here are some examples of a monoid: set of integers with operation = addition and identity = zero. set of integers with operation = multiplication and identity = one. set of lists with operation = appending and identity = empty list. set of strings with operation = concatenation and identity = empty string.

What's the difference between homomorphism and isomorphism?

A function κ:F→G κ : F → G is called a homomorphism if it satisfies equalities (#) and (##). A homomorphism κ:F→G κ : F → G is called an isomorphism if it is one-to-one and onto. Two rings are called isomorphic if there exists an isomorphism between them.

Are monoids isomorphic?

A bijective monoid homomorphism is called a monoid isomorphism. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.


2 Answers

To see the isomorphism between String and List[Char] we have toList: String -> List[Char] and mkString: List[Char] -> String.

length is a homomorphism from the String monoid to the monoid of natural numbers with addition.

A couple of examples of endo-homomorphism of the String monoid are toUpperCase and toLowerCase.

For lists, we have a lot of homomorphisms, many of which are just versions of fold.

like image 160
craigpastro Avatar answered Oct 19 '22 02:10

craigpastro


Here is siyopao's answer expressed as ScalaCheck program

object IsomorphismSpecification extends Properties("f and g") {
  val f: String => List[Char] = _.toList
  val g: List[Char] => String = _.mkString

  property("isomorphism") = forAll { (a: String, b: List[Char]) =>
    (f andThen g)(a) == a && (g andThen f)(b) == b
  }
}

which outputs

+ f and g.isomorphism: OK, passed 100 tests.
like image 2
Mario Galic Avatar answered Oct 19 '22 03:10

Mario Galic