\quad
    \varphi_\alpha: \begin{cases} 0 \to 01 \\ 1 \to 1 \end{cases} \quad
    \varphi_\beta: \begin{cases} 0 \to 10 \\ 1 \to 1 \end{cases}$$ and with the monoid $\mathcal{M}$ generated by them, i.e. $\mathcal{M} = \langle \varphi_a, \varphi_b, \varphi_\alpha, \varphi_\beta \rangle $. The monoid $\mathcal{M}$ is also called *special Sturmian monoid*. For a nonempty word $u = u_0\cdots u_{n-1}$ over the alphabet $\{a,b,\alpha,\beta\}$ we put $$\varphi_u = \varphi_{u_0} \circ \varphi_{u_1} \circ \cdots \circ \varphi_{u_{n-1}}.$$

The monoid $\mathcal{M}$ is not free. It is easy to show that for any $k \in \N$ we have $$\varphi_{\alpha a^k\beta } = \varphi_{\beta b^k\alpha}\quad \text{ and } \quad \varphi_{a\alpha^kb} = \varphi_{b\beta^ka}.$$ We can equivalently say that the following rewriting rules hold on the set of words from $\{a,b,\alpha,\beta\}^*$: $$\label{eq:relations}
  \alpha a^k\beta = \beta b^k\alpha \quad \text{ and } \quad  a\alpha^kb = b\beta^ka\qquad \text{ for any $k \in \N$ }.$$

In [@See91], the author reveals a presentation of the Sturmian monoid which includes the special Sturmian monoid $\mathcal{M}= \langle \varphi_a, \varphi_b,\varphi_\alpha, \varphi_\beta\rangle$. A presentation of the special Sturmian monoid follows from this result. It is also given explicitly in [@ReKa07]:

\[thm:relations\] Let $w,v\in \{a,b,\alpha,\beta\}^*$. The morphism $\varphi_w$ is equal to $\varphi_{v}$ if and only if the word $v$ 