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 Li iaoiAOp(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."

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