In Artin’s book he proves the associativity of a $n$-element product. It says as follows: i) the product of one element is the element itself. ii) the product $a_1a_2$ is given by the law of composition. iii) for any natural n, $[a_1a_2…a_n]=[a_1…a_i][a_{i+1}…a_n]$ Proof: The product is defined by (i) and (ii) for $n≤2$ and satisfies […]

Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I want to do (in particular, I seem to get down to the level of relations in objects..). What […]

When we can define a binary operation $\cdot:M\times M\rightarrow M$ on an algebraic structure $(M,*)$ such that $$a*(b*c)=(a\cdot b)*c$$ If $*$ is associative then $\cdot=*$ even if I’m not sure about the uniqueness (But In right-invertible associative structures this is provable) If $*$ is right-invetible then $a\cdot b=(a*(b*c))\setminus c$ only if $a\cdot b$ doesn’t depends […]

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that holds this property? And can you give a specific example?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is “no” — a counterexample would be to let $x_1*x_2*\cdots*x_n$ to be the product of the $x_i$s up to and […]

What’s the correct way to do division using a backslash? If I write $a/bc + d$, will that be equal to $(a/(bc))+d$? Basically, if I place a slash, will I divide by what’s directly behind the slash $(b)$, the term that’s behind the slash $(bc)$, or everything after the slash $(bc+d)$?

I’m looking for some example of closed curve such that $f*(g*h)=(f*g)*h,$ in some topological space $X$. I tried to use $X$ like the Sierpinski space, but I can’t find such closed curve.

Does commutativity imply associativity? I’m asking this because I was trying to think of structures that are commutative but non-associative but couldn’t come up with any. Are there any such examples? NOTE: I wasn’t sure how to tag this so feel free to retag it.

Most properties of a single binary operation can be easily read of from the operation’s table. For example, given $$\begin{array}{c|ccccc} \cdot & a & b & c & d & e\\\hline a & e & d & b & a & c\\ b & d & c & e & b & a\\ c & […]

This question already has an answer here: Does commutativity imply Associativity? 7 answers

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