We investigate a generalization of Hopf algebra slq (2) by weakening the invertibility of the generator K, i.e. exchanging its invertibility KK−1=1 to the regularity KKK=K. This leads to a weak Hopf algebra wslq (2) and a J -weak Hopf algebra vslq (2) which are studied in detail. It is shown that the monoids of group-like elements of wslq (2) and vslq (2) are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from wslq (2) a quasi-braided weak Hopf algebra U w q is constructed and it is shown that the corresponding quasi-R-matrix is regular RwRˆwRw =Rw.