An algebra $(G,\cdot,0)$ of type $(2,0)$ is called a weak BCC-algebra iff $xy=yx=0$ implies $x=y$ and if the conditions $(yz)((xy)(xz))=0$, $xx=0$, $0x=x$ hold for all $x,y,z\in G$. In the paper it is proved that a weak BCC-algebra is a BCI-algebra, i.e. satisfies $((xy)xz))(zy)=0$ and $x0=x$ iff it is a Boolean group. It is proved also that every left (right) alternative weak BCC-algebra is a Boolean group. In the end the so-called para-associative weak BCC-algebras are considered, i.e. weak BCC-algebras with the condition: $(x_1x_2)x_3= x_i(x_jx_k)$, where $(i,j,k)$ is a fixed permutation of the set $\{1,2,3\}$. It is proved that these week BCC-algebras are Boolean groups.