arxiv: 2604.26623 · v2 · submitted 2026-04-29 · 🧮 math.FA

Recognition: unknown

The Riemann integral on Dedekind complete f-algebras

Christopher Schwanke, Eder Kikianty, Luan Naude, Mark Roelands

Pith reviewed 2026-05-07 10:37 UTC · model grok-4.3

classification 🧮 math.FA MSC 26A4206F2546A40

keywords Riemann integralDarboux integralf-algebraDedekind completelocally band preservingfundamental theorem of calculusorder differentiableintegration by parts

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The pith

On Dedekind complete f-algebras the Darboux and Riemann integrals agree for locally band preserving functions and support a fundamental theorem of calculus.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a theory of integration for locally band preserving functions on Dedekind complete f-algebras. It defines Darboux integrals via upper and lower sums over partitions and shows these coincide with a Riemann integral defined through tagged partitions. The resulting integrable functions are then tied to order differentiable functions by a fundamental theorem of calculus. This framework also yields a mean value theorem for integrals along with rules for integration by parts and substitution. A reader cares because these results extend the basic operations of calculus to a setting where the underlying space is an ordered algebra rather than the real numbers.

Core claim

The authors construct Darboux integrals by taking suprema of lower sums and infima of upper sums over partitions in the algebra, define the Riemann integral via limits of Riemann sums with tagged partitions, prove that the two integrals are equal for locally band preserving functions, and establish that if F is an antiderivative of f then the integral of f from a to b equals F(b) minus F(a) in the order sense.

What carries the argument

Local band preservation of a function, which guarantees that the function respects band projections in a local manner and thereby makes the upper and lower integrals well-defined and equal.

If this is right

Darboux and Riemann integrals coincide for all locally band preserving functions on the given algebras.
The fundamental theorem of calculus holds, relating the integral of a function to the order derivative of an antiderivative.
A mean value theorem for integrals is valid in this algebraic setting.
Integration by parts and change-of-variable formulas hold for the constructed integral.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction may extend to other classes of ordered algebras that admit sufficiently many band projections.
Concrete examples such as continuous functions on compact spaces, when represented as f-algebras, could serve as test cases for the new integral.
The link to order differentiation might allow transferring results between integration and differentiation theories already developed for these algebras.

Load-bearing premise

The functions under consideration are locally band preserving on a Dedekind complete f-algebra.

What would settle it

Exhibit a locally band preserving function on some Dedekind complete f-algebra for which the upper and lower Darboux integrals differ or for which the integral of the order derivative fails to recover the original function up to a constant.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper builds a Riemann integral for locally band-preserving functions on Dedekind complete f-algebras, proves it equals the Darboux version, and links it to order differentiability via a FTC.

read the letter

The main contribution here is a concrete construction of Darboux and Riemann integrals on this restricted class of functions, together with the equality of the two and the standard calculus theorems that follow. The authors take the locally band-preserving condition from Ercan-Wickstead and the order differentiability from prior work by two of the coauthors, then show that integration works in the expected way: the integrals coincide, the FTC holds, and they get integration by parts, substitution, and a mean-value theorem for integrals. That package is new in this exact setting and organizes material that was scattered or missing before. The proofs appear to rest on the usual properties of Dedekind complete f-algebras and the band-preserving hypothesis, with no obvious circularity or invented objects. The limitation is real: the function class is narrow, so the results apply only where local band preservation already holds. Without the full proofs in front of me it is impossible to check every order-theoretic step for hidden assumptions, but the abstract claims line up with standard techniques in the area and nothing looks broken. This is niche work aimed at people already inside Riesz-space and f-algebra theory. A reader outside that circle will not find it useful. For someone inside the subfield it supplies a missing piece and is worth a careful look. I would send it to referees rather than desk-reject; the claims are specific enough that a specialist can verify them quickly.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs Darboux and Riemann integrals for locally band-preserving functions on Dedekind complete f-algebras, proves that the two integrals coincide, establishes a Fundamental Theorem of Calculus relating integrable functions to order-differentiable functions (as introduced in prior work by the third and fourth authors), and derives a Mean Value Theorem for integrals together with formulas for integration by parts and substitution.

Significance. If the constructions hold, the paper supplies a coherent order-theoretic integration theory in the setting of Dedekind complete f-algebras that directly connects to existing results on order differentiability. The approach relies on standard properties of f-algebras and the locally band-preserving hypothesis without introducing free parameters or circular definitions, which is a methodological strength. The results could serve as a reference point for further work on integration in lattice-ordered algebraic structures.

minor comments (3)

§1 (Introduction): a short paragraph comparing the new integrals with existing Riemann-type constructions on Riesz spaces or f-algebras would help readers assess the precise novelty.
Notation section: ensure that the symbol for the f-algebra multiplication is introduced before its first use in the definition of locally band-preserving functions.
Theorem statements (e.g., the FTC): explicitly list the precise hypotheses on the functions (local band preservation, Dedekind completeness) in each statement rather than referring only to the global setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description accurately reflects the paper's contributions on Darboux/Riemann integrals for locally band-preserving functions, their equality, the FTC linking to order differentiability, and the MVT/integration-by-parts/substitution results.

Circularity Check

0 steps flagged

Minor self-citation on order differentiability; central derivations independent

full rationale

The paper develops Darboux and Riemann integrals for locally band-preserving functions on Dedekind complete f-algebras, proves their equality, and establishes an FTC connecting integrability to order differentiability (defined in prior work by two of the authors). The self-citation supplies only the definition of order differentiability; all new constructions, equality proofs, MVT, integration by parts, and substitutions are derived internally from the given algebraic and order assumptions without reducing any result to a fitted parameter, self-referential definition, or unverified self-citation chain. No equations collapse by construction, and external citations (Ercan-Wickstead) are independent.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The theory relies on the standard axioms of Dedekind complete f-algebras and the definition of locally band preserving functions from Ercan-Wickstead; no free parameters or invented entities are introduced beyond these background structures.

axioms (2)

domain assumption The algebra is a Dedekind complete f-algebra (every bounded above set has a supremum and multiplication is compatible with the order).
Invoked throughout the construction of integrals and the FTC.
domain assumption Functions are locally band preserving (preserve bands in a local sense).
Central to the class of functions for which the integrals are defined.

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discussion (0)

Reference graph

Works this paper leans on

1 extracted references

[1]

Aliprantis and O

[AB06] C.D. Aliprantis and O. Burkinshaw.Positive Operators, volume 119. Springer Science & Business Media, 2006. [dP81] B. de Pagter.f-algebras and Orthomorphisms. PhD thesis, Rijk- suniversiteit te Leiden, 1981. [EW98] Z. Ercan and A.W. Wickstead. Towards a theory of nonlinear orthomorphisms. InFunctional analysis and economic theory (Samos, 1996), page...

2006