The inverse of an element from a monoid need not exist, but if it does,
it is unique:
Corollary
Every element of a monoid
has at most one inverse.
The inverse of an invertible element g is denoted by
g-1.
Theorem
Suppose that (M,*,e) is a monoid.
Then
- e is invertible;
-
if g and h are invertible, then also g * h
is invertible with inverse h-1 * g-1;
-
if g is invertible, then g-1
is also invertible with inverse g;
-
the subset of invertible elements of M
is a submonoid in which every element is invertible.
The theorem implies that if g is invertible than gn is invertible for positive n. The inverse of gn is (g-1)n and is denoted by g-n.