Convergence via filter in locally solid Riesz spaces

TL;DR

This paper introduces a new convergence concept in locally solid Riesz spaces based on filters, exploring its properties and implications for the structure of these spaces.

Contribution

It defines and analyzes a filter-based convergence in locally solid Riesz spaces, providing foundational properties and insights into this convergence mode.

Findings

Established basic properties of filter convergence in locally solid Riesz spaces

Connected filter convergence to neighborhood structures in the space

Provided a framework for further study of convergence modes in vector lattices

Abstract

Let $(E,\tau)$ be a locally solid vector lattice. A filter $\mathcal{F}$ on the set $E$ is said to be converge to a vector $e\in E$ if, each zero neighborhood set $U$ containing $e$, $U$ belongs to $\mathcal{F}$. We study on the concept of this convergence and give some basic properties of it.