ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

2. 8: Induced operation on product spaces.
On the linear space R n we have defined addition by component wise addition:
(x1, . . . , xn) + (y1, . . . , yn) = (x1 + y1, . . . , xn + yy)
Here we have a set R n = R × . . . × R which is a set product of factors on
which an addition + is known and from these component additions we create
an addition on the product.
To create this component wise operation it is however not needed neither that
the factors nor the operations are the same. So we generalize by allowing the
different factors to have its own operation. The definition may look a little
scary at first
19. Definition: Product operation. Direct product
Let there be given r sets Ai, i = 1, . . . r and on each of these an operation ♢ i
with s operands.
We then define the product operation ♢ = ♢ 1
× · · · × ♢ r
to be the operation uniquely defined by the formula
(⋆) pi ◦ ♢ = ♢ i
◦ pi
on A = A1 × · · · × Ar
(A map into a product is determined by its projections (as mentioned in the
background terminology))
which simply states that to find the component with index i of the result of the
operation you just take component with index i in each of the factors and use
the operation with that index on those.
To see a more explicit formula we restrict to the case where the operations are
binary and we get
(a1, . . . , ar) ♢ (b1, . . . , br) =
(
a1 ♢ 1
b1, . . . , ar ♢ r
The definition may also be summarized in the diagram
17
br
)
.