Determination of steady states in generalized MAX and MIN multivalued networks

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.