Homomorphisms are functions that preserve structure in models.
However, it does not preserve elements of the models.
We do have a type of function that does preserve elements, a bijection.
We also want a slightly stronger version of the homomorphism, where things are related in one if and only if they are related in the other.
We call a bijective homomorphism an isomorphism
In a sense, this allows a much more natural way of saying two things have the “same” structure.
Since all statements about a structure are preserved under isomorphism, we can prove statements in one model, and move to another when the other model makes certain statements easier to work with.
Isomorphisms have allowed mathematicians to unify lots of different similarly behaving structures - numbers, symmetry, rotations, reärrangements, etc.
One isomorphism that revolutionized computation is the realization that isomorphism f(x)=e^x connects the group (R,+,0) and the group (R+,*,1).
It also allowed people to prove things about the roots of polynomials, that are made clearer when they become reärrangements.
In fact, over 100 mathematicians across the entire 20th Century worked on classifying all finite simple groups (the bits that combine to make any finite group). This means, if you take any finite simple group, there is an isomorphism to a group on that list.
Noticing an isomorphism between two different configurations of the Petersen graph makes the 3-fold and 5-fold symmetry the same.
In November 2015, László Babai announced an algorithm that finds if there is an isomorphism between two graphs in a much quicker method than the previous algorithms. His breakthrough in the field of graph isomorphisms might have implications for solving many extant problems in theoretical computer science.
Lineär Order isomorphisms:
It can be proved that there is an isomorphism between any two finite lineär orders.
But, extending this to infinite orders shows that this is not the case. However, there is a way to make a sequence of orders, such that any lineär order has an isomorphism to something in that sequence.
This order sequence allows us to nicely tame the many sizes of infinity, in a way.
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