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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".
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Mike Williamson
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  • An example I use a lot: The Clifford product isn't commutative. And the exterior and cross products are anticommutative. The Cartesian product is another example of a noncommutative product. The commutator is also anticommutative -- but it's designed to be. –  Nov 21 '15 at 23:07
  • There's a reason matrix multiplication seems like it's the 'only' noncommutativity you see: every algebra over a field is isomorphic to a ring of matrices over a field (the matrices may have infinite sides.) the finite dimensional algebras are the ones isomorphic to rings of square matrices. If you previously felt confined by this, now might be a good time to instead be thankful that a great deal of rings can be reduced to this form. – rschwieb Nov 22 '15 at 17:39
  • "Commutative function"? Do you mean 'commutative binary operation'? Is there some underlying cohesive concept bringing together commutative or non-commutative "multiplication". Among binary operations, there are commutative ones, and the complement of those are the noncommutative operations. It seems the only concept bringing them together into a whole is "binary operation." – rschwieb Nov 22 '15 at 17:45

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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.

Daniel Robert-Nicoud
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