ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Proof : The factors have per definition same type, lets say with aryties (k1, . . . , kn).
Men each of product operations has the same arity as its factors. Therefore we
have that the aryties for the product structure also be (k1, . . . , kn).
3. 5: Induced structure on quotient
54. Definition: Quotient structure modulo an equivalence relation.
Canonical projection.
Let there be given an algebraic structure (A, ♢ 1
, . . . , ♢ n
), where the underlying
set A is equipped with an equivalence relation ∼ compatible with all the
operations.
We then define an algebraic structure, called the quotient structure of (A, ♢ 1
modulo ∼, to be the algebraic structure (A∼, ♢ 1
55. Theorem: Quotient structure type.
The quotient structure is of same type.
Proof : Obvious .
∼, . . . , ♢ n
56. Theorem: The canonical projection is a homomorphism.
Den canonical projection is a homomorphism.
Proof : Obvious.
57. Theorem: Homomorphism gives equivalence.
∼).
, . . . , ♢ n
Assume that f is a homomorphism of the algebraic structure (A, ♢ 1
, . . . , ♢ n
)
into the algebraic structure (B, ♡ 1
, . . . , ♡ n
), and let the relation ∼ in A be the
equivalence relation induced by f (that is: a ∼ b ⇔ f(a) = f(b)). Then we
have that ∼ is compatible with the structure on A.
31
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