We consider semimultiplicative sets. These are sets with a partially
defined associative multiplication. They naturally generalize
groups, groupoids, semigroups and small categories.
One can associate reduced $C^*$-algebras and crossed products to them.
For instance, Toeplitz-Cuntz-Krieger algebras of higher rank graphs
can be realized as the reduced $C^*$-algebra of the path space of the
We are interested in the $K$-theory
of such $C^*$-algebras
and propose a semimultiplicative set equivariant $KK$-theory,
which generalizes Kasparov's equivariant $KK$-theory for groups,
as a possible tool to tackle such $K$-theory computations.