What is monoid homomorphism exactly?

Question

I\'ve read about monoid homomorphism from Monoid Morphisms, Products, and Coproducts and could not understand 100%.

The author says (emphasis original):

Answer 1

trait Monoid[T] {
def op(a: T, b: T): T
def zero: T
}

val strMonoid = new Monoid[String] {
  def op(a: String, b: String): String = a ++ b
  def zero: String = ""
}

val lcMonoid = new Monoid[List[Char]] {
  def op(a: List[Char], b: List[Char]): List[Char] = a ::: b
  def zero = List.empty[Char]
}

homomorphism via function f

f{M.op(x,y)} = N.op(f(x),g(y))

for example, using toList available on String

//in REPL
scala> strMonoid.op("abc","def").toList == lcMonoid.op("abc".toList,"def".toList)
res4: Boolean = true

isomorphism via functions f and g

given bi-directional homomorphism between monoids M and N,

f{M.op(x,y)} = N.op(f(x),f(y))
g{N.op(x,y)} = M.op(g(x),g(y))

And if both (f andThen g) and (g andThen f) are identify functions, then monoids M and N are isomorphic via f and g

g{f{M.op(x,y)}} = g{N.op(f(x),f(y))} = M.op(g(f(x)),g(f(y))) = M.op(x,y)

for example, using toList available on String and toString available on List[Char] (where toList andThen toString and toString andThen toList are identity functions)

scala> ( strMonoid.op("abc","def").toList ).toString == ( lcMonoid.op("abc".toList,"def".toList) ).toString
res7: Boolean = true