ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

26. Theorem: Criterion for homomorphy on the quotient
Let ∼ be an equivalence relation on A compatible with the operation ♢ . Let k
denote the canonical projection of A on A∼. Suppose that ♡ is an operation
on B and f is a mapping of A∼ into B. Then f is a homomorphism wrt
( ♢ ∼, ♡ ) if and only if f ◦ k is a homomorphism wrt ( ♢ , ♡ )
Proof : Consider the diagram
A
✒ k
A∼
❅ f
❅❅❘
f◦k ✲ B
The if part follows from T9:2 (since k is surjective) and the only if part follows
from T9:1
27. Theorem: Homomorphism and compatibility.
Suppose that f : A → B is a homomorphism wrt ♢ and ♡ . Let ∼ be the
equivalence relation induced by f, (that means a ∼ b ⇔ f(a) = f(b)). Then ∼
will be compatible with ♢ .
Proof : Suppose that a1 ∼ a ′ 1, . . . , an ∼ a ′ n. Then
and so
f( ♢ (a1, . . . , an)) = ♡ (f(a1), . . . , f(an))
= ♡ (f(a ′ 1), . . . , f(a ′ n)) = f( ♢ (a ′ 1, . . . , a ′ n))
♢ (a1, . . . , an) ∼ ♢ (a ′ 1, . . . , an) ′
2. 10: Axioms for operations.
28. Definition: Associative operation
A binary operation ♢ on A is said to be associative if
a ♢ (b ♢ c) = (a ♢ b) ♢ c
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