From Mathematics to Generic Programming

8. More Algebraic Structures

For Emmy Noether, relationships among numbers, functions, and operations became transparent, amenable to generalization, and productive only after they have been dissociated from any particular objects and have been reduced to general conceptual relationships.B. L. van der Waerden

When we first introduced Euclid’s algorithm in Chapter 4, it was for computing the greatest common measure of line segments. Then we showed how to extend it to work for integers. Does it work for other kinds of mathematical entities? This is the question we’ll be investigating in this chapter. As we’ll see, attempts to answer it led to important developments in abstract algebra. We’ll also show how some of these new algebraic structures ...