An algebraic approach to the set of intervals

TL;DR

This paper introduces an algebraic framework for interval arithmetic by modeling intervals as a normed vector space and a four-dimensional algebra, enabling new notions of divisibility and differential calculus with applications.

Contribution

It presents a novel algebraic approach to interval arithmetic, including a new product, divisibility concepts, and differential calculus applicable to intervals.

Findings

Intervals modeled as a normed vector space.

Defined a four-dimensional associative algebra for interval multiplication.

Developed notions of divisibility and Euclidean division for intervals.

Abstract

This paper is devoted to a new approach of the arithmetic of intervals. We present the set of intervals as a normed vector space. We define also a four-dimensional associative algebra whose product gives the product of intervals in any cases. This approach allows to give a notion of divisibility and in some cases an euclidian division. We introduce differential calculus and give some applications.