Representation and Superposition of Discrete Functions and Equations with Parameterized Operations

Existence results for Hilbert's problem 13th mean that any equation constructed by continue functions can be given solution represented as a superposition of continue functions of one variable or of continue functions of two variables. Constructive results for discrete functions are given in this paper that any equation constructed by functions called discrete 3 function for which field of definition is a set containing only -1,0,1 can be given solution represented as a superposition of discrete 3 functions of one variable or of two variables. Formula solution for equation with parameterized operations can be given after introducing four special operators being correspondence among known operations and new operations. These results can be extended to discrete operators and operator equations constructed by them.