Verifying associativity of monoids using number theoretic argument

Question

I'm studying monoids and trying to test whether a given construction is a monoid. Given a set $M=\{x_0, x_1, ..., x_6\}$ and a multiplication $x_i \cdot x_j = x_{i+j-k}$ where $k$ is the largest multiple of 3 contained in $i+j-4$, we want to test whether M is a monoid. Now, $7 \choose 2$ is $21$ so, I first wanted to list all $21$ products $x_0 \cdot x_1$, $x_0 \cdot x_2$, ..., $x_5 \cdot x_6$ and check all $7\choose 3$ combinations but this is obviously impractical. I suspect that I am to use some number theoretic trick to verify associativity but I don't know what it is. Any hints will be helpful.

Answer 1

Okay... this is the cyclic monoid generated by $x_1$, with the following rule: you go $$x_1\to x_2\to x_3\to x_4\to x_5\to x_6\to x_4 \to x_5\to x_6\to x_4\to\cdots$$

In other words: add the indices. If they exceed $x_6$, you wrap back around to $x_4$. That’s why the rule is, essentialy, “subtract the largest (positive) multiple of $3$ less than or equal to $i+j-4$“.

This should make it easy to verify that multiplication is associative; you can pretend that you are just adding indices, but under the further assumption that $4\sim 4+3k$, $5\sim 5+3k$, and $6\sim 6+3k$ with $k\geq 0$ (in fact, it is a quotient of the free monoid in one generator modulo the congruence that makes $x^i\equiv x^j$ if and only if $i=j$ or $i\equiv j\pmod{3}$ and $i,j\geq 4$).

(Every cyclic monoid corresponds to a pair of nonnegative integers $r,s$, with $x^i=x^j$ if and only if $i=j$, or $i,j\geq r$ and $i\equiv j\pmod{s}$; the infinite cyclic monoid, isomorphic to the nonnegative integers, corresponds to $r=s=0$; the trivial monoid to $r=0$, $s=1$; this monoid to $r=4$ and $s=3$).