Domination and Multistate Systems

Domination theory has been studied extensively in the context of binary monotone systems, where the structure function is a sum of products of the component state variables, and with coefficients given by the signed domination function. Using e.g., matroid theory, many useful properties of the signed domination function has been derived. In this paper we show how some of these results can be extended to multistate systems. In particular, we show how the signed domination function can be extended to such systems. Using M\"{o}bius inversion we show how the signed domination function can be expressed in terms of a multistate structure function. Moreover, using this expression we show how calculating the signed domination function of a multistate system can be reduced to calculating the signed domination function of an associated binary system. This way many results from binary theory can easily be extended to multistate theory.