In this paper, the notion of direct sum of branches in hks is introduced and some related properties are investigated. Applying this notion to lower hyper $BCK$-semi lattice, a necessary condition for a hi to be prime is given. Some properties of hkc are studied. It is proved that if $H$ is a hkc and $[a)$ is finite for any $ain H$, then $mid Aut(H)mid=1$.

Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l = 2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X0(11). In addition we show that there do not exist any non-zero abelian v...