# word problem

Let $(X;R)$ be a presentation    for the group $G=\mathrm{Gp}\left\langle X\mid R\right\rangle$. It is well known that $G$ is a quotient group  of the free monoid with involution on $X$, i.e. $G=\left(X\amalg X^{-1}\right)^{\ast}/\theta$ for some congruence     $\theta\subseteq\left(X\amalg X^{-1}\right)^{\ast}\times\left(X\amalg X^{-1}% \right)^{\ast}$. We recall that $R\subset\left(X\amalg X^{-1}\right)^{\ast}$ is a set of words all representing the identity     $1_{G}$ of the group, i.e. $[r]_{\theta}=1_{G}$ for all $r\in R$. The word problem in the category of groups consists in establish whether or not two given words $v,w\in\left(X\amalg X^{-1}\right)^{\ast}$ represent the same element of $G$, i.e. whether or not $[v]_{\theta}=[w]_{\theta}$.

Let $(X;T)$ be a presentation for the inverse    monoid $M=\mathrm{Inv}^{1}\left\langle X\mid T\right\rangle=\left(X\amalg X^{-1}\right% )^{\ast}/\tau$, where $\tau=(\rho_{X}\cup T)^{\mathrm{c}}$. The concept of presentation for inverse monoid is analogous to the group’s one, but now $T$ is a binary relation  on $\left(X\amalg X^{-1}\right)^{\ast}$, i.e. $T\subseteq\left(X\amalg X^{-1}\right)^{\ast}\times\left(X\amalg X^{-1}\right)^% {\ast}$. The word problem in the category of inverse monoids consists in establish whether or not two given words $v,w\in\left(X\amalg X^{-1}\right)^{\ast}$ represent the same element of $M$, i.e. whether or not $[v]_{\tau}=[w]_{\tau}$.

We can modify the last paragraph to introduce the word problem in the category of inverse semigroups as well.

A classical results in combinatorial group theory says that the word problem in the category of groups is undecidable, so it is undecidable also for the larger categories of inverse semigroups and inverse monoids.

## References

Title word problem WordProblem 2013-05-17 17:00:19 2013-05-17 17:00:19 Mazzu (14365) unlord (1) 11 Mazzu (1) Definition msc 20M18 msc 20M05 word problem