# Congruence Representations via Soft Ideals in Soft Topological Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1**

**Definition 2**

**Definition 3**

**Definition 4**

**Definition 5**

**Definition 6**

**Definition 7**

- 1.
- The soft union of $({E}_{i},\beta )$ is a soft set $(E,\beta )={\tilde{\cup}}_{i\in I}({E}_{i},\beta )$ such that $E\left(\mu \right)={\cup}_{i\in I}{E}_{i}\left(\mu \right)$ for all $\mu \in \beta $.
- 2.
- The soft intersection of $({E}_{i},\beta )$ is a soft set $(E,\beta )={\tilde{\cap}}_{i\in I}({E}_{i},\beta )$ such that $E\left(\mu \right)={\cap}_{i\in I}{E}_{i}\left(\mu \right)$ for each $\mu \in \beta $.

**Definition 8**

- 1.
- The soft set difference $(E,\beta )$ and $(G,\beta )$ is defined to be the soft set $(H,\beta )=(E,\beta )-(G,\beta )$, where $H\left(\mu \right)=E\left(\mu \right)-G\left(\mu \right)$ for all $\mu \in \beta $.
- 2.
- The soft symmetric difference of $(E,\beta )$ and $(G,\beta )$ is defined by $(E,\beta )\tilde{\Delta}(G,\beta )=[(E,\beta )-(G,\beta )]\tilde{\cup}[(G,\beta )-(E,\beta )]$.

**Definition 9**

- 1.
- $\Phi ,\widehat{X}\in \tau $,
- 2.
- $(E,\beta ),(F,\beta )\in \tau $ implies $(E,\beta )\tilde{\cap}(F,\beta )\in \tau $, and
- 3.
- $\{({E}_{i},\beta ):i\in I\}\tilde{\subseteq}\tau $ implies ${\tilde{\cup}}_{i\in I}({E}_{i},\beta )\in \tau $.

**Definition 10**

**Lemma 1**

**Definition 11**

**Definition 12**

**Definition 13**

**Lemma 2**

**Lemma 3**

**Definition 14**

**Lemma 4**

**Definition 15**

**Definition 16**

- 1.
- If $(E,\beta ),(G,\beta )\in \tilde{I}$, then $(E,\beta )\tilde{\cup}(G,\beta )\in \tilde{I}$, and
- 2.
- If $(G,\beta )\in \tilde{I}$ and $(E,\beta )\tilde{\subseteq}(G,\beta )$, then $(E,\beta )\in \tilde{I}$.

**Definition 17**

- 1.
- $\Phi \in \mathcal{A}$,
- 2.
- If $(E,\beta )\in \mathcal{A}$, then ${(E,\beta )}^{c}\in \mathcal{A}$, and
- 3.
- If $({E}_{m},\beta )\in \mathcal{A}$, for all $m=1,2,\cdots ,n$, then ${\tilde{\cup}}_{m=1}^{n}({E}_{m},\beta )\in \mathcal{A}$.

**Definition 18**

**Definition 19.**

- 1.
- Soft clopen [35] if $(E,\beta )$ is both soft open and soft closed.
- 2.
- Soft regular open [36] if $int\left(cl\right(E,\beta \left)\right)=(E,\beta )$.
- 3.
- 4.
- Soft codense [22] if $int(E,\beta )=\Phi $.
- 5.
- Soft ${G}_{\delta}$[22] if $(E,\beta )={\tilde{\cap}}_{n=1}^{\infty}({G}_{n},\beta )$, where $({G}_{n},\beta )\in \tau $.
- 6.
- Soft ${F}_{\sigma}$[22] if $(E,\beta )={\tilde{\cup}}_{n=1}^{\infty}({E}_{n},\beta )$, where $({E}_{n},\beta )\in {\tau}^{c}$.
- 7.
- Soft nowhere dense [21] if $int\left(cl\right(E,\beta \left)\right)=\Phi $.
- 8.

**Remark 1**

**Definition 20**

**Lemma 5**

**Proposition 1**

**Definition 21**

**Lemma 6**

**Definition 22**

**Lemma 7**

**Lemma 8**

**Lemma 9**

**Definition 23**

**Remark 2.**

**Lemma 10**

**Theorem 1**

**Lemma 11**

- 1.
- If $(E,\beta )\tilde{\subseteq}(G,\beta )$, then ${C}_{2}(E,\beta )\tilde{\subseteq}{C}_{2}(G,\beta )$.
- 2.
- ${C}_{2}\left((E,\beta )\tilde{\cap}(G,\beta )\right)\tilde{\subseteq}{C}_{2}(E,\beta )\tilde{\cap}{C}_{2}(G,\beta )$.
- 3.
- ${C}_{2}\left((E,\beta )\tilde{\cup}(G,\beta )\right)={C}_{2}(E,\beta )\tilde{\cup}{C}_{2}(G,\beta )$.
- 4.
- ${C}_{2}(E,\beta )-{C}_{2}(G,\beta )\tilde{\subseteq}{C}_{2}\left((E,\beta )-(G,\beta )\right)$.
- 5.
- ${C}_{2}(E,\beta )\tilde{\subseteq}cl(E,\beta )$.
- 6.
- ${C}_{2}(E,\beta )\in {\tau}^{c}$.
- 7.
- ${C}_{2}\left[{C}_{2}(E,\beta )\right]={C}_{2}(E,\beta )$.

**Lemma 12**

**Lemma 13**

- 1.
- ${C}_{2}\left({C}_{1}(E,\beta )\right)=\Phi $, i.e., ${C}_{2}((E,\beta )-{C}_{2}(E,\beta ))=\Phi $
- 2.
- ${C}_{2}(E,\beta )=cl\left(int\left({C}_{2}(E,\beta )\right)\right)$.

## 3. Congruence Modulo a Soft Ideal

**Definition 24.**

**Lemma 14.**

**Proof.**

**Lemma 15.**

- 1.
- $(D,\beta )\tilde{\cup}(G,\beta )\approx (E,\beta )\tilde{\cup}(H,\beta )$.
- 2.
- $(D,\beta )\tilde{\cap}(G,\beta )\approx (E,\beta )\tilde{\cap}(H,\beta )$.
- 3.
- $(D,\beta )-(G,\beta )\approx (E,\beta )-(H,\beta )$.

**Proof.**

**Lemma 16.**

**Proof.**

## 4. Soft Open Sets Modulo a Soft Ideal

**Definition 25.**

**Proposition 2.**

**Proof.**

**Theorem 2.**

- 1.
- If $\mathcal{N}\left(\tau \right)\tilde{\subseteq}\tilde{I}$, then ${\mathfrak{B}}_{r}(\tau ,\tilde{I})$ is a soft algebra.
- 2.
- If ${\mathfrak{B}}_{r}(\tau ,\mathcal{C}\left(\tau \right))$ is a soft algebra, then $\mathcal{N}\left(\tau \right)\tilde{\subseteq}\mathcal{C}\left(\tau \right)$.

**Proof.**

## 5. Soft Open Sets Modulo Soft Nowhere Dense Sets

**Definition 26.**

**Lemma 17.**

**Proof.**

**Proposition 3.**

**Proof.**

**Theorem 3.**

**Proof.**

## 6. Soft Sets with the Baire Property

**Definition 27.**

**Remark 3.**

**Proposition 4.**

**Proof.**

**Proposition 5.**

**Proof.**

**Proposition 6.**

**Proof.**

**Theorem 4.**

**Proof.**

**Proposition 7.**

**Proof.**

**Proposition 8.**

**Proof.**

**Proposition 9.**

**Proof.**

**Proposition 10.**

**Proof.**

**Proposition 11.**

**Proof.**

**Proposition 12.**

**Proof.**

**Lemma 18.**

**Proof.**

**Proposition 13.**

**Proof.**

**Proposition 14.**

**Proof.**

**Proposition 15.**

**Proof.**

**Proposition 16.**

**Proof.**

**Theorem 5.**

- 1.
- $(E,\beta )\in {\mathfrak{B}}_{r}(\tau ,\mathcal{M}\left(\tau \right))$.
- 2.
- If $(E,\beta )=(H,\beta )\tilde{\Delta}(P,\beta )$, where $(H,\beta )$ is soft regular open and $(P,\beta )\in \mathcal{M}\left(\tau \right)$.
- 3.
- If $(E,\beta )=(C,\beta )\tilde{\Delta}(Q,\beta )$, where $(C,\beta )\in {\tau}^{c}$ and $(Q,\beta )\in \mathcal{M}\left(\tau \right)$.
- 4.
- If $(E,\beta )=[(D,\beta )-(R,\beta )]\tilde{\cup}(S,\beta )$, where $(D,\beta )\in {\tau}^{c}$ and $(R,\beta ),(S,\beta )\in \mathcal{M}\left(\tau \right)$.
- 5.
- If $(E,\beta )=[(G,\beta )-(M,\beta )]\tilde{\cup}(N,\beta )$, where $(G,\beta )\in \tau $ and $(M,\beta ),(N,\beta )\in \mathcal{M}\left(\tau \right)$.
- 6.
- If $(E,\beta )=(U,\beta )\tilde{\cup}(L,\beta )$, where $(U,\beta )$ is a soft ${G}_{\delta}$ set and $(L,\beta )\in \mathcal{M}\left(\tau \right)$.
- 7.
- If $(E,\beta )=(W,\beta )-(T,\beta )$, where $(W,\beta )$ is a soft ${F}_{\sigma}$ set and $(T,\beta )\in \mathcal{M}\left(\tau \right)$.
- 8.
- If there exists $(V,\beta )\in \mathcal{M}\left(\tau \right)$ such that $(E,\beta )-(V,\beta )$ is soft clopen in ${(V,\beta )}^{c}$.
- 9.
- ${C}_{2}(E,\beta )\tilde{\cap}{C}_{2}\left({(E,\beta )}^{c}\right)\in \mathcal{N}\left(\tau \right)$.
- 10.
- ${C}_{2}(E,\beta )-(E,\beta )\in \mathcal{M}\left(\tau \right)$.

**Proposition 17.**

**Proof.**

**Proposition 18.**

**Proof.**

**Definition 28.**

**Theorem 6.**

**Proof.**

**Example 1.**

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ameen, Z.A.; Alqahtani, M.H.
Congruence Representations via Soft Ideals in Soft Topological Spaces. *Axioms* **2023**, *12*, 1015.
https://doi.org/10.3390/axioms12111015

**AMA Style**

Ameen ZA, Alqahtani MH.
Congruence Representations via Soft Ideals in Soft Topological Spaces. *Axioms*. 2023; 12(11):1015.
https://doi.org/10.3390/axioms12111015

**Chicago/Turabian Style**

Ameen, Zanyar A., and Mesfer H. Alqahtani.
2023. "Congruence Representations via Soft Ideals in Soft Topological Spaces" *Axioms* 12, no. 11: 1015.
https://doi.org/10.3390/axioms12111015