XIV

The "isomorphism" notion

This is done with the "isomorphism" concept. The idea is very simple:

We say that two objects

( X L , 6 I )

and (^2,62) from a mathematical field

are isomorphic if there exists a one-to-one surjective (i.e., onto) function

f:Xi — X2 (called an isomorphism) such that /(@i) = ©2- The last

identity should be intrepreted in the sense that / "preserves" the structures

61 and &2 of the sets X\ and X2, respectively. For instance, if X\ and X2

are isomorphic vector spaces via the isomorphism / , then (besides / being

one-to-one and onto) it also satisfies f(ax -\- (3y) — otf{x) + (3f(y) for all

x j G l i and all scalars a and (3. Likewise, by saying that two ordered

vector spaces X\ and X2 are isomorphic via / , we mean that / : X\ — • X2

is a one-to-one surjective (linear) operator such that f(x) f(y) holds in

X2 if and only if x y holds in X\.

The word "isomorphism" is the English version of the Greek word

LiiaoiAOp(j)Lcrn6s"

which etymologically is the composition of the Greek

words

"LCTOS"

(which means equal, even, the same) and "fjopfiri" (which

means form, figure, shape, appearance, structure). So, when we say that

two mathematical objects are "isomorphic," we simply express the fact that

they have the same (or similar or identical) shape (or form or appearance)

and the concept of an "isomorphism" simply designates the state of being

"isomorphic."