Rough convergence on Riesz spaces

TL;DR

This paper generalizes the concept of rough convergence to Riesz spaces, introducing a new form of convergence for nets that accounts for a fixed roughness degree, and explores its properties and implications.

Contribution

It develops the theory of rough $ ext{c}$-convergence in Riesz spaces, establishing its axioms, properties, and relationship with order boundedness and limit points.

Findings

Rough $ ext{c}$-convergence satisfies formal convergence axioms

The set of limit points may be non-unique

Order boundedness relates to non-empty limit point sets

Abstract

This paper extends the theory of rough convergence from normed linear spaces to the more abstract setting of Riesz spaces. We introduce and systematically develop the concept of rough $\mathbb{c}$-convergence ($rc$-convergence) for nets. A net $(x_\alpha)_{\alpha\in A}$ in a Riesz space $E$ is said to be rough $\mathbb{c}$-convergent to $x\in E$ if there exists a net $(y_\alpha)_{\alpha\in A}$ in $E$ with $y_\alpha \xrightarrow[]{\mathbb{c}} \theta$ for a given background convergence $\mathbb{c}$, such that $|x_\alpha-x| \leq y_\alpha + \mathbb{r}$ holds for all $\alpha\in A$, where $\mathbb{r}$ is a fixed positive vector in $E$ representing the roughness degree. The study first establishes that this new construction satisfies the axioms of a formal convergence structure. Key properties of $\mathbb{rc}$-convergence are then investigated, including its relationship with linearity and the continuity of lattice operations. Since the limit of an $\mathbb{rc}$-convergent net is not necessarily unique, the paper dedicates significant analysis to the set of rough $\mathbb{c}$-limit points. Furthermore, a crucial connection is established between the order boundedness of a net and the non-emptiness of its set of $\mathbb{rc}$-limit points. This work provides a foundational framework for further exploration of convergence in Riesz spaces.