On Generalization of Unbounded Order Convergence in Riesz Spaces

Fahreezan Sheraz Diyaldin; Made Tantrawan

doi:10.22342/jims.v32i2.1611

Fahreezan Sheraz Diyaldin ⁽¹⁾ , Made Tantrawan ⁽²⁾

(1) Department of Mathematics, Universitas Gadjah Mada, Indonesia,

(2) Department of Mathematics, Universitas Gadjah Mada, Indonesia

https://doi.org/10.22342/jims.v32i2.1611

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
December 27, 2023

Accepted
November 15, 2025

Published
April 23, 2026

Keywords:

Riesz spaces, order convergence, unbounded order convergence

XML PDF

Abstract

Order convergence is a crucial concept in the theory of Riesz spaces. A generalization of order convergence, known as unbounded order convergence (or uo convergence), has been introduced. In this paper, we present a further generalization of uo convergence by using an arbitrary nonempty subset of the positive cone of the space. We then investigate the properties of this generalization, including uniqueness of limits, algebraic properties, and the relationships among these types of convergence. In particular, we show that the generalizations of uo convergence generated by two subsets are equivalent if and only if the bands generated by those subsets are equal.

Full text article

Generated from XML file

References

H. Nakano, “Ergodic theorems in semi-ordered linear spaces,” Annals of Mathematics, vol. 49, no. 3, pp. 538–556, 1948. https://doi.org/10.2307/1969044.

R. DeMarr, “Partially ordered linear spaces and locally convex linear topological spaces,” Illinois Journal of Mathematics, vol. 8, no. 4, pp. 601–606, 1964. https://doi.org/10.1215/ijm/1256059459.

S. Kaplan, “On unbounded order convergence,” Real Analysis Exchange, vol. 23, no. 1, pp. 175–184, 1997. https://sl1nk.com/UvsTy.

N. Gao and F. Xanthos, “Unbounded order convergence and application to martingales without probability,” Journal of Mathematical Analysis and Applications, vol. 415, pp. 931–943, 2014. http://dx.doi.org/10.1016/j.jmaa.2014.01.078.

N. Gao, “Unbounded order convergence in dual spaces,” Journal of Mathematical Analysis and Applications, vol. 419, pp. 347–354, 2014. http://dx.doi.org/10.1016/j.jmaa.2014.04.067.

A. C. Zaanen, Introduction to operator theory in Riesz spaces. Springer, 1997. https://doi.org/10.1007/978-3-642-60637-3.

C. D. Aliprantis and O. Burkinshaw, Positive operators. Springer, 2006. https://doi.org/10.1007/978-1-4020-5008-4.

W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. North-Holland Publishing Company, 1971. https://doi.org/10.1016/s0924-6509(08)70481-1.

Y. Abramovich and G. Sirotkin, “On order convergence of nets,” Positivity, vol. 9, pp. 287–292, 2005. https://doi.org/10.1007/s11117-004-7543-x.

Authors

Fahreezan Sheraz Diyaldin

Department of Mathematics, Universitas Gadjah Mada

fahreezansheraz@mail.ugm.ac.id (Primary Contact)

Made Tantrawan

Department of Mathematics, Universitas Gadjah Mada

Diyaldin, F. S., & Tantrawan, M. (2026). On Generalization of Unbounded Order Convergence in Riesz Spaces. Journal of the Indonesian Mathematical Society, 32(2), 1611. https://doi.org/10.22342/jims.v32i2.1611

Download Citation

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Downloads

Download data is not yet available.

1. INTRODUCTION

Functional analysis is one branch of mathematics that focuses on the study of vector spaces equipped with specific structures. One of the theories within it is the theory of Riesz spaces, first investigated independently by Frigyes Riesz, Leinid Kantorovich, and Hans Freudenthal around 1935. Riesz spaces consist of vector spaces equipped with a partially ordered relation (usually denoted by \(“≤”\)) that forms a lattice and is compatible with the structure of the vector space. Furthermore, there is a structure derived from Riesz spaces known as Banach lattices. This structure comprises Riesz spaces equipped with a norm and satisfying certain axioms.

One of the important concepts in the theory of Riesz spaces is the order convergence. This concept is significant because its definition involves only the order relation inherent in the Riesz space and does not depend on a specific topology.

Recall that a net in a Riesz space \(X\) is a function from a directed set to \(X\). A net \(( x _ { \alpha } ) _ { \alpha \in \Lambda }\) in the Riesz space \(X\) is said to converge in order to \(x \in X\), denoted by \(x_{\alpha}\overset{o}{\longrightarrow}x\), if there exists a net \(( z _ { \beta } ) _ { \beta \in \Pi }\) in \(X\) with \(z _ { \beta } \downarrow 0\), and for each \(\beta \in \Pi\), there exists \(\alpha _ { 0 } \in \Lambda\) such that \(| x _ { \alpha } - x | \le z _ { \beta }\) for every \(\alpha \in \Lambda\) with \(\alpha \geq \alpha _ { 0 }\). Throughout this paper, unless ambiguity arises, we write \(\left( x _ { \alpha } \right)\) in place of \(( x _ { \alpha } ) _ { \alpha \in \Lambda }\). The concept of order convergence has many applications not only in functional analysis but also in other fields such as financial mathematics.

Some researchers then attempted to generalize the concept of order convergence. In 1948, Hidegoro Nakano[1]introduced a concept of convergence that he called individual convergence. Subsequently, in 1964, Ralph DeMarr[2]defined the unbounded order convergence, abbreviated as uo convergence, which is essentially equivalent to individual convergence. The term uo convergence later became more popular and widely used to this day. A net \(\left( x _ { \alpha } \right)\) in \(X\) is said to converge uo to \(x \in X\), denoted as \(x _ { \alpha } \ { \stackrel { \mathrm { u o } } { \longrightarrow } } \ x , \operatorname { i f } \ | x _ { \alpha } - x | \wedge u \ { \stackrel { \mathrm { o } } { \to } } \ 0\) for every \(u \in X ^ { + }\). Over time, many researchers have examined the properties and applications of uo convergence, including Kaplan[3]investigating the relationship between uo convergence and weak units, Gao and Xanthos[4]discussing applications of uo convergence in martingales without probability, and Niushan Gao[5]studying uo convergence in dual spaces.

In this paper, we introduce a novel form of convergence that generalizes the concept of unbounded order convergence. This new convergence reduces the condition from "For all \(u\in X^+\)" to “for all \(u\in A\)", where \(A \subseteq X ^ { + }\) is specified, and is referred to as A-unbounded order convergence, abbreviated as A-uo convergence. We will investigate some properties of this new concept, including the limit uniqueness and its relation to other convergences. Before that, we establish a new criterion for \(A \subseteq X ^ { + }\), denoted order density on positive cones, inspired by the notion of order dense Riesz subspaces. It will become apparent later in the paper that this criterion is intricately linked to the uniqueness of the limit of A-uo convergent nets.

2. ORDER DENSE SETS ON POSITIVE CONES

Throughout this paper, \(X\) denotes a Riesz space. The set of all positive elements of \(X\) is denoted by \(X ^ { + }\). Analogously, for all \(A \subseteq X , A ^ { + }\) is the set of all positive elements of \(A\). Moreover, we use the symbols \(I _ { A }\) and \(⟨A⟩\) to represent the ideal and band generated by \(A\), respectively. Definitions and fundamental concepts of Riesz spaces are referenced from[6].

We start this section by recalling the definition of order dense Riesz subspaces given by Aliprantis and Burkinshaw[7]. A Riesz subspace \(Y\) of \(X\) is said to be order dense in \(X\) if for any \(u \in X ^ { + }\), there exist \(y \in Y\) such that \(0 \leq y \leq x\) and \(y \ne 0\). For the rest of the paper, we write "\(a<b\)" to denote \(a \leq b\) with \(a \neq b\). Inspired by this definition, we propose the definition of order density on a positive cone as follows.

Definition 2.1.A set\(A \subseteq X ^ { + }\)is said to be order dense in\(X ^ { + }\)if for any u\(\in X ^ { + }\), there exist\(y \in I _ { A }\)such that\(0 < y \le u\).

Clearly, this definition is related to the concept of order dense Riesz subspace as \(A\) is order dense in \(X ^ { + }\) if

.....

Read Counter : 364 Download : 756

Share Now

On Generalization of Unbounded Order Convergence in Riesz Spaces

Abstract

Full text article

References

Authors

Address

Contact Info

Article Sidebar

Abstract

Full text article

References

Authors

Article Details

Address

Contact Info