ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Let ♢ be a binary operation on A and let a ∈ A. The mapping of A into it self
defined by x ↦→ a ♢ x ([A ∋ x ↦→ a ♢ x ∈ A]) is then called the left translation
by a. Right translation is defined analogously
17. Remark:
This is inspired by addition of vectors (fx in space).
2. 7: Induced operation on function spaces.
For any sets M and A we let F(M, A) denote the set of mappings from M to
A. We are going to use an operation on A to define an operation on F(M, A).
This is
18. Definition: Induced operation on function spaces
If ♢ is an operation on A with n operands and M is a set, then we can define
an operation ♢M in F(M, A) by the formula
♢ (f1, . . . fn)(x) = ♢ (f1(x), . . . , fn(x))
and we call it the the operation on F(M, X) induced by ♢ . Usually we shall
just use the same symbol for the induced operation, which hopefully will cause
no serious ambiguity. If necessary we denote it ♢ M . For binary operations
the definition may be written
(f1 ♢ f2)(x) = f1(x) ♢ f2(x)
For each x ∈ M we let δx denote the mapping from F(M, X) to A defined by
δx(f) = f(x). This mapping is called evaluation at x. Then we can summarize
the definition of ♢ M in the diagram
F(M, A) n δx ✲ A n
♢M
♢
❄ ❄
δx F(M, A)
✲ A
from which we see that ♢ M is exactly that operation which makes δx a homomorphism
for all x.
16