Additive Relation and Algebraic System of Equations

TL;DR

This paper introduces an algebraic framework based on additive relations over monoids, demonstrating that complex multi-variable equations can be decomposed into superpositions of single-variable relations, ensuring solvability.

Contribution

It develops a novel algebraic system of equations using additive relations, showing that multi-variable equations can be expressed as superpositions of single-variable relations under certain conditions.

Findings

Any multi-variable additive relation can be expressed as a superposition of single-variable relations.

Solutions to these equations always exist within the algebraic system.

The framework applies to both numerical and functional elements, leading to algebraic and operator 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.