A multivalued network on a complement-closed set (MCN) is a dynamical network model whose entities can take multiple state values in a totally ordered set equipped with a complement operator. Multivalued networks extend the framework of standard (binary) Boolean networks and several types of multivalued network models. In this work, we study steady states in MCN where an undirected graph, in which the entities may or may not have self-loops, encodes the dependencies in the network, and all the local functions are maximum (respectively minimum)- type. Specifically, we give a necessary and sufficient condition for the existence of steady states and provide a complete structural description of them. Additionally, we completely characterize when the steady states of MCN cannot be directly obtained from those of its binary counterpart. In doing so, we show that a Fixed Point Theorem does not hold for these systems, when they are non-binary. Moreover, this result extends to broader classes of MCN in which more general independent local functions are considered.

In this work, we provide a procedure that allows us to transform certain kinds of deterministic Boolean networks on minterm or maxterm functions into symmetric ones, so inferring that such symmetrizable networks can present only periodic points of periods 1 or 2. In particular, we deal with generalized parallel (or synchronous) dynamical systems (GPDS) over undirected graphs, i.e., discrete parallel dynamical systems over undirected graphs where some of the self-loops may not appear. We also study the class of anti-symmetric GPDS (which are non-symmetrizable), proving that their periodic orbits have period 4. In addition, we introduce a class of non-symmetrizable systems which admit periodic orbits with arbitrary large periods.