> This is because free monoids are ordered lists, the most fundamental data structure in all of programming. Well, I think unsigned integers are even more fund

> This is because free monoids are ordered lists, the most fundamental data structure in all of programming. Well, I think unsigned integers are even more fundamental than ordered lists, and those behave as the ring of integers mod 2^bit_width . So aren't rings even more important in programming than monoids? My answer would be that neither really is. You're confusing a few examples of important things happening to be X, with the concept of X being an important concept. When people use ordered lists, they are not thinking of them as monoids, or using any interesting theorems from monoid theory. If mathematicians had never come up with the definition of monoids, C and C++ would have been designed in exactly the same way as they are now.