The Riemann integral on Dedekind complete $f$-algebras

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.