In his initial paper on braids E. Artin gave
a presentation of an arbitrary braid group with two generators,
say $\sigma_1$ and $\sigma$, and the following relations:
\begin{cases}
\sigma_1 \sigma^i \sigma_1 \sigma^{-i} &=
\sigma^i \sigma_1 \sigma^{-i} \sigma_1 \ \ \text{for} \ \
2 \leq i\leq {n / 2}, \\
\sigma^n &= (\sigma \sigma_1)^{n-1}.
\end{cases}
The connection with the canonical generators is given by the formulae:
\sigma = \sigma_1 \sigma_{2} \dots \sigma_{n-1},
\sigma_{i+1} =\sigma^i \sigma_1 \sigma^{-i}, \quad i =1, \dots
{n-2}.
We observe analogues of this Artin's presentation for various
generalizations of braids. For example, the singular brad monoid
$SB_n$ has a presentation with
generators $\sigma_1$, $\sigma_1^{-1}$, $\sigma$, $\sigma^{-1}$ and
$x_1$
and relations
\begin{cases}
\sigma_1 \sigma^i \sigma_1 \sigma^{-i} =
\sigma^i \sigma_1 \sigma^{-i} \sigma_1 \ \ \text{for} \ \
2 \leq i\leq {n / 2}, \\
\sigma^n = (\sigma \sigma_1)^{n-1},\\
x_1\sigma^i\sigma_1\sigma^{-i}= \sigma^i\sigma_1\sigma^{-i} x_1
\ \ \text{for} \ \ i=0, 2 \dots {n - 2}, \\
x_1 \sigma^i x_1 \sigma^{-i} =
\sigma^i x_1 \sigma^{-i} x_1 \ \ \text{for} \ \
2 \leq i\leq {n / 2}, \\
\sigma^n x_1 = x_1\sigma^n,\\
x_1\sigma\sigma_1\sigma^{-1}\sigma_1 = \sigma\sigma_1\sigma^{-1}\sigma_1
\sigma x_1\sigma^{-1}, \\
\sigma_1\sigma_1^{-1}=\sigma_1^{-1}\sigma_1 =1,\\
\sigma\sigma^{-1}=\sigma^{-1}\sigma =1.
\end{cases}
The other presentations, as that of J. S. Birman, K. H. Ko and S. J. Lee
or Sergiescu graph presentations are also considered.