In this paper we show that given a positive implicative BCI-algebra $X$
with condition (S), every branch $V(a)$ of $X$ with respect 
to the BCI-ordering $\le$ on $X$ forms an upper semilattice $(V(a);\,\le)$; 
especially, if $V(a)$ is a finite set,
$(V(a);\,\le)$ forms a lattice; moreover, if $(V(a);\,\le)$ is a lattice, 
it must be distributive. We also obtain some interesting identities on $V(a)$.