  Results and tools on discrete interaction networks are often concerned with
Boolean variables, whereas considering more than two levels is sometimes
useful. Multivalued networks can be converted to partial Boolean maps, in a way
that preserves the asynchronous dynamics. We investigate the problem of
extending these maps to non-admissible states, i.e. states that do not have a
multivalued counterpart. We observe that attractors are preserved if a stepwise
version of the original function is considered for conversion. Different
extensions of the Boolean conversion affect the structure of the interaction
graphs in different ways. A particular technique for extending the partial
Boolean conversion is identified, that ensures that feedback cycles are
preserved. This property, combined with the conservation of the asymptotic
behaviour, can prove useful for the application of results and analyses defined
in the Boolean setting to multivalued networks, and vice versa. As a first
application, by considering the conversion of a known example for the discrete
multivalued case, we create a Boolean map showing that the existence of a
cyclic attractor and the absence of fixed points are compatible with the
absence of local negative cycles. We then state a multivalued version of a
result connecting mirror states and local feedback cycles.
