  For special universal $C^*$-algebras associated to $k$-semigraphs we present
the universal representations of these algebras, prove a Cuntz--Krieger
uniqueness theorem, and compute the $K$-theory. These $C^*$-algebras seem to be
the most universal Cuntz--Krieger like algebras naturally associated to
$k$-semigraphs. For instance, the Toeplitz Cuntz algebra is a proper quotient
of such an algebra.
