Normal Forms and Decompositions of Monotone Ternary Functions

Among the ternary functions, the subset of monotone functions undoubtedly has the most applications. For this reason, monotone ternary functions have been intensively studied in previous research, although sometimes under different names. For example, Mukaidono called them regular ternary functions and proved that these are exactly the ternary functions that can be represented by propositional logic formulas.In this paper, we present a decomposition theorem for monotone ternary functions to split a given function into four functions by eliminating one of the argument variables. The decomposition theorem presented in this paper is the basis for recursive algorithms on monotone ternary functions that eliminate one of the used variables in each recursion step. As an example, we present such a recursive algorithm that translates a given monotone function into a propositional logic formula, and we easily prove this way Mukaidono's theorems on disjunctive normal forms of regular ternary, i.e., monotone, and B-ternary functions.