Nearly all of my experience with math is in the "applied math" realm, so I haven't had any formal study of rings, or other fundamental algebraic concepts that help to prove all the relevant applied math.
I saw answers like this one talking about matrix multiplication being commutative due to its function composition.
But this didn't quite help me get to the fundamental concept I'm looking for. I guess I have 2 questions:
what other (relatively) common types of "multiplication" are not commutative, other than matrices?
- E.g., we could stretch the definition and say that a convolution is "multiplication-like". However, convolutions are commutative.
is there a relatively accessible way to understand the underlying traits that lead to a function being commutative, other than the obvious "changing the order"?
- E.g., in graph theory, non-planar graphs are extensions or supergraphs of either the utility or $K_5$ graph. Is there some underlying cohesive concept bringing together commutative or non-commutative "multiplication".
To answer your first question, a very accessible example of a non-commutative multiplication is concatenation of words. Say you are given an alphabet $A$ (that is, a set of letters). For the sake of making explicit examples, let's pick $$A = \{a,b,\ldots,x,y,z,\_\},$$ where we will use $\_$ to indicate a space. We define the free group over $A$, denoted by $\langle A\rangle$, as the set of all words (of finite length) we can form with the letters in our alphabet together with the multiplication given by concatenating two words and neutral element given by the empty word $\emptyset$. Examples of such words are $$abcd,\qquad hi\_how\_are\_you,\qquad imokayregnak,$$ and an example of multiplication is $$(do)\cdot g = dog.$$ Obviously, this multiplication is not commutative.
Another example that is very important in (at least) mathematics, physics and robotics is Lie algebras. A Lie algebra is a vector space $\mathfrak{g}$ together with a multiplication $$[\ ,\ ]:\mathfrak{g}\times\mathfrak{g}\longrightarrow\mathfrak{g},$$ called the Lie bracket, which is antisymmetric $$[x,y] = -[y,x]$$ and satisfies the Jacobi identity $$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0.$$ An example of this is $\mathbb{R}^3$ with the cross product.