| Trying to be helpful here, sorry if it doesn’t map to your field of which I am not familiar. The _algebra_ you learn in school is elementary algebra, using variables. In modern math, _algebra_ or _modern_ algebra or _abstract_algebra, is the study of structures over sets with defined operations on the elements of that set. ADTs are an example of an algebraic structure, specifically called one that converts non-trivial semantic (runtime) properties to trivial ones (T/F). This post is dealing with the structure in another way. If you understand magmas, monoids, etc.. that can be helpful. But the lay description I find useful is the algebra is what _arises_ from that defined set domain and operations. The key point is studying the structure, which is the algebraic structure, or the algebra. It is basically what pops out, not what you start with, although that is flawed. Almost all modern math will use that _modern_ meaning of algebra. |
I actually wish we taught more of this math early on[0]. Children seem to be quick to grasp many of the fundamentals of important structures like groups, fields, and algebras. I find that many of these concepts have fundamentally shifted how I think and can be used on a daily basis, without the need of writing formulas or using formal semantics.
It's odd that it takes getting up upper division undergraduate education in math (or sometimes from neighboring fields) to learn what the field is even fundamentally about. It's akin to teaching people that programming by teaching people how to use a word editor. It's such a narrow aspect and no surprise so many are so fundamentally confused.
[0] I'm unconvinced the "new maths" programs were a complete failure. Just because we didn't get it right on the first attempt doesn't mean we should have thrown the baby out with the bath water.