You know what, just for shits and giggles I actually DID open an introductory textbook on abstract algebra, just in case my undergraduate degree in maths failed

You know what, just for shits and giggles I actually DID open an introductory textbook on abstract algebra, just in case my undergraduate degree in maths failed me. Here's what the first chapter says about addition ('+') in Z: Here are 4 elementary properties that + satisfies: • (Associativity): a + (b + c) = (a + b) + c ∀a, b, c ∈ Z • (Existence of additive identity) a + 0 = 0 + a = a ∀a ∈ Z. • (Existence of additive inverses) a + (−a) = (−a) + a = 0 ∀a ∈ Z • (Commutativity) a + b = b + a ∀a, b ∈ Z. Also: Key Observation: There are naturally occuring sets (other than Z and Q) which come equipped with a concept of + and ×, whose most basic properties are the same as those of the usual addition and multiplication on Z or Q. I could go on with rings, fields, and vector spaces which also rely on the concept of addition as defined in Z, but I'm really curious to learn about addition being commonly used as a non-commutative operation, especially in the presence of an accompanying multiplication operation. edit: I forgot to provide proper references, so here's another example including a reference: When dealing with Abelian groups, it is customary to use additive notation. That is, the group operation will be called "addition," and we will "add" two elements rather than "multiply" them. We write g + h instead of g • h or gh, and ng replaces g^n. [Walker, Elbert A., "Introduction to Abstract Algebra", 1987; Ch. 2, Pg. 70]