ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Let f : B → A1 × . . . × An be a mapping into the product, then we call the
mapping pi ◦f the i ′ th component of f or the component of f of index i . Then
f is completely determined by the collection of its components. So to define f
it suffices to define all the pi ◦ f.
So given a map fi : B → Ai for each i ∈ I there exists a unique map f with
pi ◦ f = fi for all i ∈ I.
If all the factors are the same we shall use power notation : A n = A × . . . × A.
For a map f : A → B we define the associated multi map f : A n → B n by
f(a1, . . . , an) = (f(a1), . . . , f(an)).
Occasionally we shall just write f for f.
These definitions also are meaningful when n = 1. It even shows to be appropriate
to make a convention which extends to the case n = 0, that is the case
where there are no factors at all. We do this by defining the empty tuple, the
ordered set with 0 elements and denote it by (). Then we let the product set
of 0 factors be the set with the only element ().
You should remember that this is just a convention, which is a useful way to
avoid a lot of special cases in some formulations. With this convention we also
have that
A 0 = {()}.
2. 3: Definition
1. Definition: Operation
Let A be a set and let n ≥ 0 be an integer. An operation ♢ in A with n
operands is a mapping
♢ : A n → A.
An operation with n operands is also called an operation with the arity n or an
n-ary operation. For n = 0, 1, 2, 3, 4 one uses the special terms 0-ary, unary,
binary, ternary and quaternary.
2. Remark: Constant operations
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