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Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ (empty word) whenever $\{a,b\} \notin E(G)$ and $c \in I$ is arbitrary.

I have the following questions about the monoid $M(G)$.

  1. Is this monoid $M(G)$ well studied in the literature?

  2. What are some algebraic combinatorics or general combinatorial significance of this monoid?

Thanks for your time and have a good day.

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    $\begingroup$ It is called Right Angled Coxeter Group (RACG). Google it. $\endgroup$
    – user6976
    Commented Mar 8, 2018 at 20:46
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    $\begingroup$ I agree with YCor, you should have edited your old question. $\endgroup$
    – Benjamin Steinberg
    Commented Mar 9, 2018 at 0:58
  • $\begingroup$ Perhaps the Right-Angled Coxeter Monoid (for the complement of $G$)? $\endgroup$
    – Alex Suciu
    Commented Mar 9, 2018 at 4:19
  • $\begingroup$ @AlexSuciu as Mark has mentioned, it's a monoid presentation obviously defining a group (as does any Coxeter presentation). $\endgroup$
    – YCor
    Commented Mar 9, 2018 at 9:16

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