ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Proof : The kernel is the inverse image of a substructure, since the neutral
element (as a singleton) is a submonoid (T73). A general result (T49) states
that the inverse image of a structure is substructure.
You can meet more monoids in X33 E34 E35 E36
8. 5: Inverse
78. Definition: Invertible element. Inverse.
Suppose that (A, ♢ , e) is a monoid, and let a ∈ A. We shall say that a is
invertible if there exists b ∈ A such that a ♢ b = e and b ♢ a = e. Then b is
said to be an inverse element to a wrt ♢ .
79. Theorem: Inverse is unique.
An invertible element has only one inverse element.
Proof : Let b1 and b2 be inverses of a. Then by associativity b1 = b1 ♢ e =
b1 ♢ (a ♢ b2) = (b1 ♢ a) ♢ b2 = e ♢ b2 = b2.
80. Definition: The inverse element.
The unique inverse is said to be the inverse element for a.
If you insist on having a general way of denoting the inverse you may write
a ♢ −1 . But usually the context can tell you what operation is involved and then
we simply write a −1 .
8. 6: Powers
81. Definition: Powers with negative exponents in monoids
Let a be an invertible element. For n ∈ N we define a −n to be (a −1 ) n , and a 0
to be e (the operation being known from context)
82. Theorem: Power rules
For any invertible element a and for all m, n ∈ Z we have that at a m ♢ a n =
a m+n and (a m ) n = a mn .
52