The standard notion for comparing structures is that of morphism.
Definition
Let (M,*,e) and (M',*',e') be two monoids. A morphism (of monoids) is a map f : M -> M' with the following properties.
- f(e) = e';
- for all a, b
M:
f(a * b) = f(a) *' f(b).
If f is bijective, then we call f
an isomorphism.
If two monoids are isomorphic (that is, there is an isomorphism from one to the other), then they are of the `same shape' (morph=shape).
Theorem
If f : M -> M' is an isomorphism of monoids, then
- the cardinalities of M and M' are equal;
- the inverse map f-1 : M' -> M is also an isomorphism of monoids.