The statistically unbounded $\tau$-convergence on locally solid Riesz spaces

TL;DR

This paper introduces the concept of statistically unbounded $ au$-convergence in locally solid Riesz spaces, exploring its properties, related notions, and connections with order convergence.

Contribution

It defines and studies the properties of $st$-$u_ au$-closed sets, $st$-$u_ au$-Cauchy sequences, and $st$-$u_ au$-continuity in locally solid Riesz spaces, extending the theory of convergence.

Findings

Defined $st$-$u_ au$-convergence and related notions.

Established relationships between order and $st$-$u_ au$-convergence.

Provided conditions for $st$-$u_ au$-completeness.

Abstract

A sequence $(x_n)$ in a locally solid Riesz space $(E,\tau)$ is said to be statistically unbounded $\tau$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}\big\lvert\{k\leq n:\lvert x_k-x\rvert\wedge u\notin U\}\big\rvert\to 0$ as $n\to\infty$. In this paper, we introduce this concept and give the notions $st$-$u_\tau$-closed subset, $st$-$u_\tau$-Cauchy sequence, $st$-$u_\tau$-continuous and $st$-$u_\tau$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_\tau$-convergence.