# Reference: Cancellability of the Dyck congruence

I consider the Dyck congruence $\equiv$ on a parenthesis alphabet $\Sigma = \{a, \bar a, b, \bar b\}$, i.e. the least congruence on $\Sigma^*$ such that $a \bar a \equiv \varepsilon$ and $b \bar b \equiv \varepsilon$.

It appears that $\equiv$ can be cancelled, i.e. for every $v$, $w_1$, $w_2 \in \Sigma^*$, if $vw_1 \equiv vw_2$, then $w_1 \equiv w_2$ (and analogously for cancellation from the right).

Does someone know a literature reference for this property? I wasn't able to find anything along those lines.

• I would check Lyndon and Schupp's Combinatorial Group Theory, but it's really not hard to prove by induction. Aug 16 '17 at 14:13
• You're right. I just wanted to omit the proof. Thank you for the suggestion! Aug 16 '17 at 14:24
• Ah, crap. I thought you meant the usual semigroup conguence. Aug 27 '17 at 6:38

Your assertion is wrong, the congruence $\equiv$ is not cancellable from the right: for instance $\bar a a \bar a \equiv \bar a$, but $\bar a a \not\equiv 1$.
By the way the quotient $\Sigma^*/{\equiv}$ is not a group, but a monoid, called the polycyclic monoid, first introduced in .