In this note first we define the
notions of positive implicative hyper $K$-ideals of types 1,
2,..., 26, 27 . Also we define the notions of commutative hyper
$K$-ideals of types 1, 2, ...,8, 9. Then we give many examples to
show that these notions are different together. Finally we prove
some theorems and  obtain some related results . In particular we
determine the relationships between the implicative hyper
$K$-ideals and commutative hyper $K$-ideals of a hyper
$K$-algebra of order 3, which satisfies the
simple condition.