ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

We are now going to move any algebraic structure on A to be a structure
on F(M, A) (See the introduction to D 18), which we shall call the induced
operation:
50. Definition: Induced structure on function spaces
Antag at (A, ♢ 1
, . . . , ♢ n
) is a algebraic structure and at M is a set. Then kalder
vi den algebraic structure on F(M, A) der som operations har de inducerede
operations for den inducerede structure. We tænker os altid, uden at behøve at
fremhæve det, at F(M, A) is equipped with denne structure.
51. Theorem: The structure on the function space is of same type
Using the notation from the previous theorem we have that F(M, A) is of the
same type asA
Proof : Oplagt.
3.4.1: Induced structure on product.
Since we can construct the product of a number of operations we can also
construct the product of structures by taking the product of the involved operations.
This is made precise in the following
52. Definition: Direct product of structures
We take the case of two factors first: Suppose that (A, ♢ 1
, . . . , ♢ n
) and (B, ♡ 1
, . . . , ♡ n
are two structures of same type. Then we define their product as the structure
(A × B, ♢ 1
×♡ 1
, . . . , ♢ n
× ♡ n
)
Next lets take m factors each of which have m operations:
Suppose that (Ai , ♢ i 1, . . . , ♢ i n) for i = 1, . . . , m are algebraic structures all of
same type. Then we define the product structure as (A, ♢ 1
, . . . , ♢ n
), where
A = A1 × · · · × An ♢ i
= ♢ i 1 × · · · × ♢ i m.
53. Theorem: The product structure is of same type
The product structure has the same type as the factors.
30
)