Integration in semirings

TL;DR

This paper explores the concept of integration as an inverse to derivation within semirings, generalizing previous work on rings and lattices, and investigates properties, examples, and conditions under which integrals form a semiring.

Contribution

It introduces the notion of integrals in semirings, analyzes their properties, provides examples, and identifies when integrals form a semiring, extending the theory of integration in algebraic structures.

Findings

Integrals in semirings have specific algebraic properties.

Examples of semirings with derivation and integrals are provided.

Conditions are identified under which integrals form a semiring.

Abstract

The concept of integral as an inverse to that of derivation was already introduced for rings and recently also for lattices. Since semirings generalize both rings and bounded distributive lattices, it is natural to investigate integration in semirings. This is our aim in the present paper. We show properties of such integrals from the point of view of semiring operations. Examples of semirings with derivation where integrals are introduced are presented in the paper. These illuminate rather specific properties of such integrals. We show when the set of all integrals on a given semiring forms a semiring again.