The "isomorphism" notion

A typical mathematical field is usually described by a class of sets that are

endowed with a "structure" concept that characterizes the subject matter of

the field. Schematically, a typical branch of study in mathematics consists

of pairs (X, 6), where X is a set and & is the "structure" imposed on the

set X that is characteristic to the area. The structure & can be expressed

in terms of algebraic or topological properties or a mixture of the two. Here

are a few examples of mathematical areas.

(1) Groups: Here for the typical object (X, S), the structure & repre-

sents the algebraic structure on X given by the operation of mul-

tiplication (x,y) i— • xy and of the inverse function x \-+

x~l.

(2) Topological spaces: Here for the typical object (X, ©), the structure

& represents the topology of the set X.

(3) Vector spaces: Here for the typical object (X, (3), the structure &

represents the algebraic structure imposed on X by means of the

addition (x, y) \-^ x + y and the scalar multiplication (A, x) i— Ax .

(4) Topological vector spaces: Here for the typical object (X, 6) , the

structure 6 represents the mixture of the vector space structure

of X and the topological structure of X that makes the algebraic

operations of X continuous.

(5) Ordered vector spaces: Here for the typical object (X, 6) , the struc-

ture & represents the algebraic structure of X together with the

vector ordering on X.

Once one deals with the objects of a mathematical field, one would

like to have a way of identifying two objects of the field that look "alike."

Xiii