  Consider a $k$-valued network. Two kinds of (control) invariant subspaces,
called state and dual invariant subspaces, are proposed, which are subspaces of
state space and dual space respectively. Algorithms are presented to verify
whether a dual subspace is a dual or dual control invariant subspace. The
bearing space of $k$-valued (control) networks is introduced. Using the
structure of bearing space, the universal invariant subspace is introduced,
which is independent of the dynamics of particular networks. Finally, the
relationship between state invariant subspace and dual invariant subspace of a
network is investigated. A duality property shows that if a dual subspace is
invariant then its perpendicular state subspace is also invariant and vice
versa.
