ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Suppose that ♢ is an operation on A, that ♡ 1
, . . . , ♡ r
B1, . . . , Br, all with n operands.
Further let B = B1 × · · · × Br
and ♡ = ♡ 1
× · · · × ♡ r
.
and assume that fi : A → Bi are the components of f : A → B.
are operations on
Then f a homomorphism wrt ♢ and ♡ if and only if fi is a homomorphism
wrt to ♢ and ♡ i
for all i
Proof : Since fi = pi ◦ f the only if part follows from T9: 1 and the if part
follows from T9:2, since pi is surjective.
This is used to prove a very famous theorem, see the example E15
2. 9: Induced operation on quotient.
While building product sets is about how to create new objects with many properties
from objects with simpler properties, then building quotients is about to
forget properties, which are not relevant in the context.
A radical example is to forget all properties of a permutation except its parity.
We are then left with only two elements ”even” and ”odd”. When we transfer
some operation on permutations to these object we must assure our selves that
the operation is insensitive to all other properties of the operation.
This is the case with multiplication of permutations, since the parity of the
product only depends on the parity of the factors.
The general framework for this is notion of a partition K of a set X into classes
according to some criterion. Then K is the set of classes. Each class is non
empty and all elements must be in exactly one class. If K is a class and x ∈ K
we say that x is a representative of K and that K is the class of x, which is
also denoted [x]K.
The mapping of X into K which to x assigns its class [x]K is called the canonical
projection and we denote it by kK.
Any mapping f of X into some set Y which is constant on the classes can in
an obvious way be considered to be defined on XK, lets call it fK. Then we
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