Switching function canonical forms based on commutative and associative binary operations

It is often convenient to consider the arguments of a switching function to be the components 0 or 1 of a vector. In order to investigate systematically the properties of switching functions, it is important to have standard algebraic forms for their representation and manipulation. Ease of manipulation is attained by selecting binary operations that are commutative and associative, and such that the secondary connective is distributive over the primary connective. Four distributive laws hold among the four commutative and associative operations. The operations in each distributive law are used as the connectives of a standard, or canonical, form of an arbitrary switching function F(x) of n arguments. The main result relating the partial ordering of logical vectors to the parity of binomial coefficients is established. The partial difference operation is used to expand an arbitrary switching function about its arguments. Transformations among the canonical forms are given.