Statistical convergence of nets in Riesz spaces

TL;DR

This paper extends the concept of statistical convergence from sequences to nets within Riesz spaces, utilizing finite additive measures on directed sets, and explores its relation to lattice properties.

Contribution

It introduces a new definition of statistical convergence for nets in Riesz spaces and examines its connections with order convergence and lattice operators.

Findings

Established a new framework for statistical convergence of nets in Riesz spaces.

Identified relationships between statistical convergence and lattice properties.

Provided theoretical insights into the structure of Riesz spaces with respect to convergence modes.

Abstract

The statistical convergence is defined for sequences with the asymptotic density on the natural numbers, in general. In this paper, we introduce the statistical convergence for nets in Riesz spaces by using the finite additive measures on directed sets. Moreover, we give some relations among the statistical convergence and the lattice properties such as the order convergence and lattice operators.