The Riemann integral on Dedekind complete $f$-algebras

TL;DR

This paper develops a Riemann integral theory for locally band preserving functions on Dedekind complete f-algebras, establishing fundamental calculus results and integration techniques.

Contribution

It introduces a unified Riemann integral framework for these functions and connects it to order differentiability, including fundamental theorems and integration methods.

Findings

Darboux and Riemann integrals are shown to be equivalent.

A Fundamental Theorem of Calculus for these functions is proved.

The paper establishes integration by parts and substitution formulas.

Abstract

In this paper we develop a theory of integration for locally band preserving functions, introduced by Ercan and Wickstead, on Dedekind complete $f$-algebras. Specifically, we construct Darboux and Riemann integrals and show that they are equal. We then connect the theory of integrable functions to the theory of order differentiable functions, introduced by the third and fourth authors, by proving a Fundamental Theorem of Calculus. Furthermore, we show that a Mean Value Theorem for Integrals holds and that we can integrate by parts and substitutions.