Classical theorems from analysis for locally band preserving functions on Dedekind complete Φ-algebras

2025/08/23 by Kikianty, Eder, Naude, Luan, Roelands, Mark +1
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2508.16923

Abstract

In this paper we explore the concept of locally band preserving functions, introduced by Ercan and Wickstead, on Dedekind complete Φ-algebras. Specifically, we show that all super order differentiable functions are locally band preserving. Furthermore, some foun- dational results from classical analysis are proved in this setting, such as the Intermediate Value Theorem, the Extreme Value Theorem, and the Mean Value Theorem. Moreover, we show that these generalisations can fail for functions that are not locally band pre- serving. With the goal in mind to further develop the theory of complex differentiation in Dedekind complete complex Φ-algebras, a complex version of the Mean Value Theorem is also provided.

Citations

• Differentiable, Holomorphic, and Analytic Functions on Complex Φ-Algebras
• Series and power series on universally complete complex vector lattices
• Boolean-valued models as a foundation for locally L⁰-convex analysis and Conditional set theory

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