Additive Relation and Algebraic System of Equations

Abstract

Additive relations are defined over additive monoids and additive operation is introduced over these new relations then we build algebraic system of equations. We can generate profuse equations by additive relations of two variables. To give an equation with several known parameters is to give an additive relation taking these known parameters as its variables or value and the solution of the equation is just the reverse of this relation which always exists. We show a core result in this paper that any additive relation of many variables and their inverse can be expressed in the form of the superposition of additive relations of one variable in an algebraic system of equations if the system satisfies some conditions. This result means that there is always a formula solution expressed in the superposition of additive relations of one variable for any equation in this system. We get algebraic equations if elements of the additive monoid are numbers and get operator equations if they are functions.