# The Unbounded Fuzzy Order Convergence in Fuzzy Riesz Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (i)
- $\forall \phantom{\rule{4pt}{0ex}}k\in K$$\mu (k,k)=1$ (reflexivity);
- (ii)
- for $k,g\in K$ $\mu (k,g)+\mu (g,k)>1$ implies $k=g$ (antisymmetric);
- (iii)
- for $k,h\in K$ $\mu (k,h)\ge {\bigvee}_{g\in K}[\mu (k,g)\wedge \mu (g,h)]$ (transitivity).

**Definition**

**2.**

**Definition**

**3.**

- (i)
- for $k,g\in K$ if $\mu (k,g)>1/2$ then $\mu (k,g)\le \mu (k+h,g+h)$ for all $h\in K$;
- (ii)
- for $k,g\in K$ if $\mu (k,g)>1/2$ then $\mu (k,g)\le \mu (\lambda k,\lambda g)$ for all $0\le \lambda \in \mathbb{R}$.

**Definition**

**4.**

**Proposition**

**1.**

- (i)
- $k={k}^{+}-{k}^{-}$;
- (ii)
- ${k}^{+}\wedge {k}^{-}=0$;
- (iii)
- $|k|={k}^{+}+{k}^{-}$;
- (iv)
- $|k|=0\iff k=0$;
- (v)
- $\mu (||k|-|g||,|k-g|)>1/2$.

**Definition**

**5.**

**Definition**

**6.**

- (i)
- fuzzy order complete if each non-empty subset of K has a supremum and infimum in K;
- (ii)
- fuzzy $\sigma -$ order complete if each nonempty countable subset of K has a supremum and infimum in K;
- (iii)
- fuzzy Dedekind complete if each non-empty subset of K which is bounded from above has a supremum in K;
- (iv)
- fuzzy $\sigma -$ Dedekind complete if each nonempty countable subset of K that is bounded from above has a supremum in K.

## 3. Unbounded Fuzzy Order Convergence

**Definition**

**7.**

**Example**

**1.**

**Definition**

**8.**

**Definition**

**9.**

**Proposition**

**2.**

- (i)
- ${k}_{\lambda}\stackrel{ufo}{\to}k$ iff $({k}_{\lambda}-k)\stackrel{ufo}{\to}0$;
- (ii)
- if ${k}_{\lambda}\stackrel{ufo}{\to}k$ and ${g}_{\gamma}\stackrel{ufo}{\to}g$, then $a{k}_{\lambda}+b{g}_{\gamma}\stackrel{ufo}{\to}ak+bg$ for each $a,b\in \mathbb{R}$;
- (iii)
- if ${k}_{\lambda}\stackrel{ufo}{\to}k$ and ${k}_{\lambda}\stackrel{ufo}{\to}g$, then $k=g$;
- (iv)
- if ${k}_{\lambda}\stackrel{ufo}{\to}k$, then
- (a)
- ${\left({k}_{\lambda}\right)}^{+}\stackrel{ufo}{\to}{k}^{+}$;
- (b)
- ${\left({k}_{\lambda}\right)}^{-}\stackrel{ufo}{\to}{k}^{-}$.

Furthermore, (a) and (b) imply that$$|{k}_{\lambda}|\stackrel{ufo}{\to}|k|.$$ - (v)
- If a positive net ${k}_{\lambda}\stackrel{ufo}{\to}k$ and $\mu ({k}_{\lambda},{g}_{\gamma})>1/2$, ${g}_{\gamma}\stackrel{ufo}{\to}g$, then $\mu (k,g)>1/2$.

**Proof.**

- (i)
- Suppose ${k}_{\lambda}\stackrel{ufo}{\to}k$. Then $|({k}_{\lambda}-k)-0|\wedge g=|{k}_{\lambda}-k|\wedge g\stackrel{fo}{\to}0$ for each $g\in {K}^{+}$, hence $({k}_{\lambda}-k)\stackrel{ufo}{\to}0$. The converse can be proved analogously.
- (ii)
- Suppose ${k}_{\lambda}\stackrel{ufo}{\to}k$ and ${g}_{\gamma}\stackrel{ufo}{\to}g$. Then we have$$\mu (|({k}_{\lambda}+{g}_{\gamma})-(k+g)|\wedge h,(|{k}_{\lambda}-k|+|{g}_{\gamma}-g|)\wedge h)>1/2$$$$\mu \left(\right(|{k}_{\lambda}-k|+|{g}_{\gamma}-g|)\wedge h,|{k}_{\lambda}-k|\wedge h+|{g}_{\gamma}-g|\wedge h)>1/2$$

- Let $\mu (|k-g|,|k-{k}_{\lambda}|+|g-{k}_{\lambda}|)>1/2$ for each $\lambda $. Let $h=|k-g|$. Observe that $|k-g|=|k-g|\wedge h$. Also$$\mu (|k-g|\wedge h,|k-{k}_{\lambda}|\wedge h+|g-{k}_{\lambda}|\wedge h)>1/2.$$
- Suppose $|{k}_{\lambda}-k|\stackrel{ufo}{\to}0$. As $\mu (|{\left({k}_{\lambda}\right)}^{+}-{k}^{+}|,|{k}_{\lambda}-k|)>1/2$ for each $\lambda $. So $|{\left({k}_{\lambda}\right)}^{+}-{k}^{+}|\stackrel{ufo}{\to}0$. Hence, ${\left({k}_{\lambda}\right)}^{+}\stackrel{ufo}{\to}{k}^{+}$. Thus $-{k}_{\lambda}\stackrel{ufo}{\to}-k$ this gives that ${\left({k}_{\lambda}\right)}^{-}\stackrel{ufo}{\to}{k}^{-}$. The final statement follows from $\mu (||{k}_{\lambda}|-|k||,|{k}_{\lambda}-k|)>1/2$.
- By Proposition 2, ${k}_{\lambda}=|{k}_{\lambda}|\stackrel{ufo}{\to}|k|$. $k=|k|$ by uniqueness of fuzzy order limit. As $\mu (0,{g}_{\gamma}-{k}_{\lambda})>1/2$, then ${g}_{\gamma}-{k}_{\lambda}\stackrel{ufo}{\to}g-k$, and we have $\mu (k,g)>1/2$.

**Remark**

**1.**

**Proposition**

**3.**

- (i)
- Let ${\left({k}_{n}\right)}_{n\in N}$ be a disjoint sequence in a σ-Dedekind complete FRS $(K,\mu )$. Then ${k}_{n}\stackrel{ufo}{\to}0$ in K.
- (ii)
- Let ${\left({k}_{n}\right)}_{n\in N}$ be a sequence in an FRS $(K,\mu )$. If ${k}_{n}\stackrel{ufo}{\to}0$, then ${inf}_{m}\left({k}_{{n}_{m}}\right)=0$ for each increasing sequence $\left({n}_{m}\right)$ of natural numbers.

**Proof.**

- (i)
- Fix $k\in {K}^{+}$. We will show that ${lim\; sup}_{n}(|{k}_{n}|\wedge k)=0$. Indeed, let $g\in {K}^{+}$ such that $\mu (g,{sup}_{n}(|{k}_{n}|\wedge k))>1/2$. Therefore,$$\mu (g\wedge |{k}_{n}|,(\underset{n+1}{sup}(|{k}_{n+1}|\wedge k)\wedge |{k}_{n}|)>1/2\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\underset{n+1}{sup}(|{k}_{n+1}|\wedge |{k}_{n}|\wedge k)=0.$$$$g=g\wedge \underset{n\ge 1}{sup}(|{k}_{n}|\wedge k)=\underset{n\ge 1}{sup}(g\wedge |{k}_{n}|\wedge k)=0.$$
- (ii)
- Suppose ${k}_{n}\stackrel{ufo}{\to}0$. Take $\left({n}_{m}\right)$ as an increasing sequence of natural numbers. Clearly, ${k}_{{n}_{m}}\stackrel{ufo}{\to}0$. Let $\mu (k,{k}_{{n}_{m}})>1/2$ for each $m\in \mathbb{N}$, and $k\in {K}^{+}$. Therefore, $k={k}_{{n}_{m}}\wedge k\stackrel{fo}{\to}0$ implies that $k=0$. Hence, ${inf}_{m}\left({k}_{{n}_{m}}\right)=0$.

## 4. Fuzzy Weak Order Unit

**Definition**

**10.**

- (i)
- A subset C of K is said to be fuzzy order closed (fo-closed for short) if for any net $\left({k}_{\lambda}\right)\subset C$ and $k\in K$ with ${k}_{\lambda}\stackrel{fo}{\to}k$ in K implies $k\in C$.
- (ii)
- A subset S of K is called fuzzy solid if $\mu (|k|,|g|)>1/2$ and $g\in S$ implies $k\in S$.
- (iii)
- A fuzzy solid vector subspace I of K is called a fuzzy ideal of K.
- (iv)
- A fuzzy order closed ideal in K is said to be a fuzzy band.

**Definition**

**11.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Definition**

**12.**

**Remark**

**2.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### Fuzzy Ideals and Completeness with Respect to Ufo-Convergence

**Remark**

**3.**

**Proposition**

**6.**

**Proof.**

**Definition**

**13.**

**Proposition**

**7.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Zadeh, L.A. Similarity relations and fuzzy ordering. Inf. Sci.
**1971**, 8, 177–200. [Google Scholar] [CrossRef] - Venugopalan, P. Fuzzy Ordered Sets. Fuzzy Sets Syst.
**1992**, 46, 221–226. [Google Scholar] [CrossRef] - Ajmal, N.; Thomas, K.V. Fuzzy lattices. Inf. Sci.
**1994**, 79, 271–291. [Google Scholar] [CrossRef] - Bodenhofer, U. Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets Syst.
**2003**, 137, 113–137. [Google Scholar] [CrossRef] - Chon, I. Partial order relations and fuzzy lattices. Korean J. Math.
**2009**, 17, 361–374. [Google Scholar] - Revanasiddappa, M.B.; Harish, B.S. A New Feature Selection Method based on Intuitionistic Fuzzy Entropy to Categorize Text Documents. Int. J. Interact. Multimed. Artif. Intell.
**2018**, 5, 103–117. [Google Scholar] [CrossRef] - Sanchez, H. Resolution of composite fuzzy relation equation. Inf. Control
**1976**, 30, 38–48. [Google Scholar] [CrossRef] - Yuan, B.; Wu, W. Fuzzy ideals on a distributive lattice. Fuzzy Sets Syst.
**1990**, 35, 231–240. [Google Scholar] - Beg, I.; Islam, M. Fuzzy ordered linear spaces. J. Fuzzy Math.
**1995**, 3, 659–670. [Google Scholar] - Beg, I.; Islam, M. Fuzzy Riesz Spaces. J. Fuzzy Math.
**1994**, 2, 211–241. [Google Scholar] - Beg, I.; Islam, M. Fuzzy Archimedean spaces. J. Fuzzy Math.
**1997**, 5, 413–423. [Google Scholar] - Beg, I. Extenssion of fuzzy positive linear operator. J. Fuzzy Math.
**1998**, 6, 849–855. [Google Scholar] - Beg, I. On fuzzy order relations. J. Nonlinear Sci. Appl.
**2012**, 5, 357–378. [Google Scholar] [CrossRef] - Hong, L. Fuzzy Riesz subspaces, fuzzy ideals, fuzzy bands and fuzzy band projections. Ann. West Univ. Timis. Math. Comput. Sci.
**2015**, 53, 77–108. [Google Scholar] [CrossRef] [Green Version] - Park, C.; Movahednia, E.; Mosadegh, S.M.; Mursaleen, M. Riesz fuzzy normed spaces and stability of a lattice preserving functional equation. J. Comput. Anal. Appl.
**2018**, 24, 569–579. [Google Scholar] - Amorós, C.; Argyros, I.K.; González, R.; Magreñán, A.A.; Orcos, L.; Sarría, I. Study of a High Order Family: Local Convergence and Dynamics. Mathematics
**2019**, 7, 225. [Google Scholar] [CrossRef] - Argyros, I.K.; Gonzalez, D. Local convergence for an improved Jarratt-type method in Banach space. Int. J. Interact. Multimed. Artif. Intell.
**2015**, 3, 20–25. [Google Scholar] [CrossRef] - Abramovich, Y.; Sirotkin, G. On order convergence of nets. Positivity
**2005**, 9, 287–292. [Google Scholar] [CrossRef] - Aliprintis, C.D.; Burkinshaw, O. Positive Operator; Spriger: Dordrecht, The Netherlands, 2006. [Google Scholar]
- Gao, N.; Troitsky, V.G.; Xanthos, F. Uo-convergence and its applications to Cesàro means in Banach lattices. Isr. J. Math.
**2017**, 220, 649–689. [Google Scholar] [CrossRef] - Gao, N.; Xanthos, F. Unbounded order convergence and application to martingales without probability. J. Math. Anal. Appl.
**2014**, 415, 931–947. [Google Scholar] [CrossRef] - Kaplan, S. On unbounded order convergence. Real Anal. Exch.
**1997**, 23, 175–184. [Google Scholar]

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**MDPI and ACS Style**

Iqbal, M.; Malik, M.G.A.; Bashir, Y.; Bashir, Z.
The Unbounded Fuzzy Order Convergence in Fuzzy Riesz Spaces. *Symmetry* **2019**, *11*, 971.
https://doi.org/10.3390/sym11080971

**AMA Style**

Iqbal M, Malik MGA, Bashir Y, Bashir Z.
The Unbounded Fuzzy Order Convergence in Fuzzy Riesz Spaces. *Symmetry*. 2019; 11(8):971.
https://doi.org/10.3390/sym11080971

**Chicago/Turabian Style**

Iqbal, Mobashir, M. G. Abbas Malik, Yasir Bashir, and Zia Bashir.
2019. "The Unbounded Fuzzy Order Convergence in Fuzzy Riesz Spaces" *Symmetry* 11, no. 8: 971.
https://doi.org/10.3390/sym11080971