In this note first we define the notion
of {\it dual positive implicative hyper $K$-ideal of type 1},
where for simplicity is written by $DPIHKI-T1$. Then we obtain
some basic related results. After that we determine
hyper $K$-algebras of order 3,which have
$D_{1}=\{1\}$, $D_{2}=\{1,2\}$ and $D_{3}=\{0,1\}$
as a $DPIHKI-T1$. Finally we give some connections
between the notions of dual positive implicative hyper
$K$-ideals of types 1, 2, 3 and 4.