ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

This makes the following definition meaningful:
24. Definition: The quotient operation
The formula
[a1] ♢ ∼
. . . ♢ ∼
[an] = [a1 ♢ , . . . , ♢ an]
defines am operation on the set of equivalence classes. We call it the quotient
operation induced by ♢ modulo ∼, and we denote it ♢ ∼.
The definition is summarized by the diagram
♢
A n k ✲ A∼ n
∼
♢
❄ ❄
A
k ✲ A∼
which shows that the quotient operation is the one making the canonical projection
a homomorphism.
25. Theorem: The canonical projection is a homomorphism.
Let ♢ be an operation on A and let ∼ be an equivalence relation compatible
with ♢ and let k be canonical projection of A on A∼ associated with ∼. Then
k is a homomorphism wrt to ♢ and ♢ ∼.
Proof : Since k is the canonical projection, we have that k(a) = [a], the
equivalence class containing a. Then the diagram above proves the theorem.
Lets give a less formalistic proof in the case of a binary operation
k(a1) ♢ k(a2) = [a1] ♢ [a2] = [a1 ♢ a2] = k(a1 ♢ a2)
which is the diagrammatic formulation of the condition for homomorphism
(D6).
The previous theorem show that an equivalence relation which is compatible
with an operation gives rise to a homomorphism. The next theorem shows the
opposite, that any homomorphism generates a compatible equivalence relation.
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