Is this monoid recognizable as free group?

Question

Based on a set $X=\left\{ a,b,\ldots\right\} $ of symbols define $X'=\left\{ a,a^{-1},b,b^{-1},\ldots\right\} $ as a set of symbols. Let $R$ be the set of words of the form $xx^{-1}$ or $x^{-1}x$. Let $M$ be the monoid having $X'$ as generators and $R$ as relations. Then $M$ is a group since word $x_{n}^{-\varepsilon_{n}}\cdots x_{1}^{-\varepsilon_{1}}$ serves as inverse of word $x_{1}^{\varepsilon_{1}}\cdots x_{n}^{\varepsilon_{n}}$ (with $\varepsilon_{i}\in\left\{ 1,-1\right\} $ and $x^{1}:=x$). Can it be shown that this group is free over set $X$?

Answer 1

The definition of your relations is slightly ambiguous. First, your relations should be written as $xx^{-1} = 1$ and $x^{-1}x = 1$. Further, I suppose you assume that $x$ is an element of $X$ in your definition. Then your answer is yes. See

http://en.wikipedia.org/wiki/Free_group