ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

classes in K have the form {x: k(x) = K} for some K ∈ K. But this is clear
from the definitions.
In our examples we see that
x ↦→ [x]r is the canonical indicator for the partition into remainder classes
(x, y) ↦→ the vertical line(x, y) is the canonical indicator for the partition into
vertical lines
Many indicators are made by selecting a special well suited representative for
each class to represent all the members of the class, so to speak some sort of
principal representative .
Some examples:
1) The principal remainder modulo some integer
2) The projection on the x-axis (that is (x, 0))
3) Rn in the class of linear spaces of dimension n in the partioning of
the set of finite dimensional linear spaces with the dimension as the
indicator.
4) the principal argument of a complex number. To be more explicit:
In the real numbers you have an equivalence relation which makes to
real numbers equivalent if theire difference is a multiple of 2π. As
5)
the indicator of a class you often choose the (unique) member in the
interval [0, 2π[.
The principal value of arcsin by the partition which has sin as its
indicator function. (the pricipal value of arcsin y is that solution x to
the equation sin x = y which is in the interval [ − π
]
π
2 , 2 . This is the
one which computer programmes chooses if not otherwise instructed.
For mathematicians arcsin is the equivalence class itself, since this
leads to the simplest rules for calculation.
We promised to motivate, explain, excuse the terminology quotient. We do it
by an example: Let Z = X × Y , and let pX and pY denote the projections on
the factors. If we use pX as the indicator function we get a partition into classes
of the form K x = {(x, y): y ∈ Y } and the mapping X ∋ x ↦→ K x ∈ X/pA is a
bijection. We use this to identify X/f with X and so Z = K × Y . This could
be expressed that K is Z divided by Y .