We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}\left(
2\right) $ by weakening the invertibility of the generator $K$, i.e.
exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$.
This leads to a weak Hopf algebra $w\mathfrak{sl}_{q}\left( 2\right) $ and a
$J$-weak Hopf algebra $v\mathfrak{sl}_{q}\left( 2\right) $ which are studied
in detail. It is shown that the monoids of group-like elements of
$w\mathfrak{sl}_{q}\left( 2\right) $ and $v\mathfrak{sl}_{q}\left(
2\right) $ are regular monoids, which supports the general conjucture on the
connection betweek weak Hopf algebras and regular monoids. Moreover, from
$w\mathfrak{sl}_{q}\left( 2\right) $ a quasi-braided weak Hopf algebra
$\overline{U}_{q}^{w}$ is constructed and it is shown that the corresponding
quasi-$R$-matrix is regular $R^{w}\hat{R}^{w}R^{w}=R^{w}$.