On soft topological ordered spaces

Definition 2.1

Molodtsov (1999) A pair $(G,E)$ is said to be a soft set over X provided that G is a mapping of a set of parameters E into $2^X$ .

Remark 2.2

1. For short, we use the notation $G_E$ instead of $(G,E)$ .

2. A soft set $G_E$ can be defined as a set of ordered pairs $G_E=\{(e,G(e)):e\in E$ and $G(e)\in 2^X\}$ .

Definition 2.3

Molodtsov (1999) For a soft set $G_E$ over X and $x\in X$ , we say that:

1. $x\in G_E$ if $x\in G(e)$ , for each $e\in E$ .

2. $x\notin G_E$ if $x\notin G(e)$ , for some $e\in E$ .

Definition 2.4

Maji et al. (2003) A soft set $G_E$ over X is called:

1. A null soft set, denoting by $\stackrel{\sim }{\varnothing }$ , if $G(e)=\varnothing$ , for each $e\in E$ .

2. An absolute soft set, denoting by $\stackrel{\sim }{X}$ , if $G(e)=X$ , for each $e\in E$ .

Definition 2.5

Maji et al. (2003) The union of soft sets $G_A$ and $F_B$ over X is the soft set $V_D$ , where $D=A\bigcup B$ and a map $V:D\to 2^X$ is defined as follows $$V(d)=\left\{\begin{array}{ccc}\hfill G(d)\hfill & \hfill :\hfill & \hfill d\in A-B\hfill \\ \hfill F(d)\hfill & \hfill :\hfill & \hfill d\in B-A\hfill \\ \hfill G(d)\bigcup F(d)\hfill & \hfill :\hfill & \hfill d\in A\bigcap B\hfill \end{array}\right.$$ It is written briefly, $G_A\bigcup ^{\sim }{F}_B=V_D$ .

Definition 2.6

Pei and Miao (2005) The intersection of soft sets $G_A$ and $F_B$ over X is the soft set $V_D$ , where $D=A\bigcap B$ , and a map $V:D\to 2^X$ is defined by $V(d)=G(d)\bigcap F(d)$ , for all $d\in D$ . It is written briefly, $G_A\bigcap ^{\sim }{F}_B=V_D$ .

Definition 2.7

Pei and Miao (2005) A soft set $G_A$ is a soft subset of a soft set $F_B$ if

1. $A\subseteq B$ .

2. For all $a\in A,G(a)\subseteq F(a)$ .

The soft sets $G_A$ and $F_B$ are soft equal if each of them is a soft subset of the other. The set of all soft sets, over X under a parameter set A, is denoted by $S(X_A)$ .

Definition 2.8

Ali et al. (2009) The relative complement of a soft set $G_E$ , denoted by $G_E^c$ , where $G^c:E\to 2^X$ is the mapping defined by $G^c(e)=X⧹G(e)$ , for each $e\in E$ .

Definition 2.9

Shabir and Naz (2011) A collection τ of soft sets over X under a fixed parameters set E is said to be a soft topology on X if it satisfies the following three axioms:

1. $\stackrel{\sim }{X}$ and $\stackrel{\sim }{\varnothing }$ belong to τ.

2. The intersection of a finite family of soft sets in τ belongs to τ.

3. The union of an arbitrary family of soft sets in τ belongs to τ.

The triple $(X,\tau ,E)$ is called a soft topological space (briefly, STS). Every member of τ is called soft open and its relative complement is called soft closed.

Definition 2.10

Shabir and Naz (2011) A soft set $x_E$ over X is defined by $x(e)=\{x\}$ , for each $e\in E$ .

Proposition 2.11

Shabir and Naz (2011) If $(X,\tau ,E)$ is an STS, then for each $e\in E$ , a family ${\tau }_e=\{G(e):G_E\in \tau \}$ forms a topology on X.

Definition 2.12

Shabir and Naz (2011) Let Y be a non-empty subset of an STS $(X,\tau ,E)$ . Then ${\tau }_Y=\{\stackrel{\sim }{Y}\bigcap ^{\sim }{G}_E:G_E\in \tau \}$ is said to be a soft relative topology on Y and the triple $(Y,{\tau }_Y,E)$ is said to be a soft subspace of $(X,\tau ,E)$ .

Definition 2.13

Shabir and Naz (2011) For a soft subset $H_E$ of an STS $(X,\tau ,E),\mathit{Int}(H_E)$ is the largest soft open set contained in $H_E$ and $\mathit{Cl}(H_E)$ is the smallest soft closed set containing $H_E$ .

Definition 2.14

Zorlutuna et al. (2012) A soft subset $W_E$ of an STS $(X,\tau ,E)$ is called soft neighborhood of $x\in X$ , if there exists a soft open set $G_E$ such that $x\in G_E\stackrel{\sim }{\subseteq }{W}_E$ .

Definition 2.15

Zorlutuna et al. (2012) A soft mapping between $S(X_A)$ and $S(Y_B)$ is a pair $(f,\varphi )$ , denoted also by $f_{\varphi }$ , of mappings such that $f:X\to Y,\varphi :A\to B$ . Let $G_K$ and $H_L$ be soft subsets of $S(X_A)$ and $S(Y_B)$ , respectively. Then the image of $G_K$ and pre-image of $H_L$ are defined by:

1. $f_{\varphi }(G_K)={(f_{\varphi }(G))}_B$ is a soft subset of $S(Y_B)$ such that $$f_{\varphi }(G)(b)=\left\{\begin{array}{ccc}\hfill {\bigcup ^{\sim }}_{a\in {\varphi }^{-1}(b)\bigcap K}f(G(a))\hfill & \hfill :\hfill & \hfill {\varphi }^{-1}(b)\bigcap K\ne \varnothing \hfill \\ \hfill \varnothing \hfill & \hfill :\hfill & \hfill {\varphi }^{-1}(b)\bigcap K=\varnothing \hfill \end{array}\right.$$ for each $b\in B$ .

2. $f_{\varphi }^{-1}(H_L)={(f_{\varphi }^{-1}(H))}_A$ is a soft subset of $S(X_A)$ such that $$f_{\varphi }^{-1}(H)(a)=\left\{\begin{array}{ccc}\hfill f^{-1}(H(\varphi (a)))\hfill & \hfill :\hfill & \hfill \varphi (a)\in L\hfill \\ \hfill \varnothing \hfill & \hfill :\hfill & \hfill \varphi (a)\notin L\hfill \end{array}\right.$$ for each $a\in A$ .

Definition 2.16

Zorlutuna et al. (2012) A soft map $f_{\varphi }:S(X_A)\to S(Y_B)$ is said to be:

1. Injective if f and ϕ are injective.

2. Surjective if f and ϕ are surjective.

3. Bijective if f and ϕ are bijective.

Proposition 2.17

Nazmul and Samanta (2013) Let $f_{\varphi }:S(X_A)\to S(Y_B)$ be a soft map. Then for each soft subsets $G_A$ and $H_B$ of $S(X_A)$ and $S(Y_B)$ , respectively, we have the following results:

1. $G_A\stackrel{\sim }{\subseteq }{f}_{\varphi }^{-1}{f}_{\varphi }(G_A)$ and $G_A=f_{\varphi }^{-1}{f}_{\varphi }(G_A)$ if $f_{\varphi }$ is injective.

2. $f_{\varphi }{f}_{\varphi }^{-1}(H_B)\stackrel{\sim }{\subseteq }{H}_B$ and $f_{\varphi }{f}_{\varphi }^{-1}(H_B)=H_B$ if $f_{\varphi }$ is surjective.

Definition 2.18

(Nazmul and Samanta, 2013; Zorlutuna et al., 2012) A soft map $f_{\varphi }:(X,\tau ,A)\to (Y,\theta ,B)$ is said to be:

1. Soft continuous if the inverse image of each soft open subset of $(Y,\theta ,B)$ is a soft open subset of $(X,\tau ,A)$ .

2. Soft open (resp. soft closed) if the image of each soft open (resp. soft closed) subset of $(X,\tau ,A)$ is a soft open (resp. soft closed) subset of $(Y,\theta ,B)$ .

3. Soft homeomorphism if it is bijective, soft continuous and soft open.

Definition 2.19

Aygünoǧlu and Aygün (2012)

1. A collection $\{G_{i_E}:i\in I\}$ of soft open sets is called soft open cover of an STS $(X,\tau ,E)$ if $\stackrel{\sim }{X}=\stackrel{\sim }{\bigcup _{i\in I}}{G}_{i_E}$ .

2. An STS $(X,\tau ,E)$ is called soft compact (resp. soft Lindelöf) provided that every soft open cover of $\stackrel{\sim }{X}$ has a finite (resp. countable) subcover.

Proposition 2.20

Aygünoǧlu and Aygün (2012) Every soft closed subset $H_E$ of a soft compact (resp. soft Lindelöf) space is soft compact (resp. soft Lindelöf).

Definition 2.21

Aygünoǧlu and Aygün (2012) Let $G_A$ and $H_B$ be soft sets over X and Y, respectively. Then the cartesian product of $G_A$ and $H_B$ is denoted by ${(G\times H)}_{A\times B}$ and is defined as $(G\times H)(a,b)=G(a)\times H(b)$ , for each $(a,b)\in A\times B$ .

Theorem 2.22

Aygünoǧlu and Aygün (2012) Let $(X,\tau ,A)$ and $(Y,\theta ,B)$ be two STSs. Let $\mathrm{\Omega }=\{G_A\times F_B:G_A\in \tau$ and $F_B\in \theta \}$ . Then the family of all arbitrary union of elements of $\mathrm{\Omega }$ is a soft topology on $X\times Y$ .

Definition 2.23

Das and Samanta (2013) A soft set $H_E$ over X is called countable (resp. finite) if $H(e)$ is countable (resp. finite), for each $e\in E$ .

Definition 2.24

(Das and Samanta, 2013; Nazmul and Samanta, 2013) A soft subset $P_E$ of $\stackrel{\sim }{X}$ is called soft point if there exists $e\in E$ and there exists $x\in X$ such that $P(e)=\{x\}$ and $P(\alpha )=\varnothing$ , for each $\alpha \in E⧹\{e\}$ . A soft point will be shortly denoted by $P_e^x$ and we say that $P_e^x\in G_E$ , if $x\in G(e)$ .

Definition 2.25

El-Shafei et al. (2018) For a soft set $G_E$ over X and $x\in X$ , we say that

1. $x⋐G_E$ if $x\in G(e)$ , for some $e\in E$ .

2. if $x\notin G(e)$ , for each $e\in E$ .

Definition 2.26

El-Shafei et al. (2018) A soft set $G_E$ in $S(X_E)$ is said to be stable if there exists a subset S of X such that $G(e)=S$ , for each $e\in E$ .

Definition 2.27

El-Shafei et al. (2018) An STS $(X,\tau ,E)$ is said to be:

1. p-soft $T_0$ -space if for every pair of distinct points $x,y\in X$ , there is a soft open set $G_E$ such that $x\in G_E$ , or $y\in G_E$ , .

2. p-soft $T_1$ -space if for every pair of distinct points $x,y\in X$ , there are soft open sets $G_E$ and $F_E$ such that $x\in G_E$ , and $y\in F_E$ , .

3. p-soft $T_2$ -space if for every pair of distinct points $x,y\in X$ , there are disjoint soft open sets $G_E$ and $F_E$ containing x and y, respectively.

4. p-soft regular if for every soft closed set $H_E$ and $x\in X$ such that , there are disjoint soft open sets $G_E$ and $F_E$ such that $H_E\stackrel{\sim }{\subseteq }{G}_E$ and $x\in F_E$ .

5. (Shabir and Naz, 2011) Soft normal if for every two disjoint soft closed sets $H_{1_E}$ and $H_{2_E}$ , there are two disjoint soft open sets $G_E$ and $F_E$ such that $H_{1_E}\stackrel{\sim }{\subseteq }{G}_E$ and $H_{2_E}\stackrel{\sim }{\subseteq }{F}_E$ .

6. p-soft $T_3$ -space if it is both p-soft regular and p-soft $T_1$ -space.

7. p-soft $T_4$ -space if it is both soft normal and p-soft $T_1$ -space.

Lemma 2.28

El-Shafei et al. (2018) If $H_{E_1\times E_2}$ is a soft closed subset of a soft product space $(X\times Y,{\tau }_1\times {\tau }_2,E_1\times E_2)$ , then $H_{E_1\times E_2}=\left[{(G_{E_1})}^c\times \stackrel{\sim }{Y}\right]\bigcup ^{\sim }\left[\stackrel{\sim }{X}\times {(F_{E_2})}^c\right]$ , for some $G_{E_1}\in {\tau }_1$ and $F_{E_2}\in {\tau }_2$ .

Definition 2.29

Kelley (1975) a binary relation $⪯$ on a non-empty set X is called a partial order relation if it is reflexive, anti-symmetric and transitive.

$\left\{(x,x)\text{: for each}x\in X\right\}$ is the equality relation on X and it is indicated by ▵.

Definition 2.30

Kelley (1975) Let $(X,⪯)$ be a partially ordered set. An element $a\in X$ is called:

1. A smallest element of X provided that $a⪯x$ , for all $x\in X$ .

2. A largest element of X provided that $x⪯a$ , for all $x\in X$ .

Definition 2.31

Nachbin (1965) A triple $(X,\tau ,⪯)$ is said to be a topological ordered space, where $(X,⪯)$ is a partially ordered set and $(X,\tau )$ is a topological space.

Definition 2.32

McCartan (1968) A topological ordered space $(X,\tau ,⪯)$ is called:

1. Lower (Upper) $T_1$ -ordered if for each $x⋠y$ in X, there is an increasing (resp. a decreasing) neighborhood W of $a($ resp. $b)$ such that $b($ resp. $a)$ belongs to $W^c$ .

2. $T_0$ -ordered if it is lower $T_1$ -ordered or upper $T_1$ -ordered.

3. $T_1$ -ordered if it is lower $T_1$ -ordered and upper $T_1$ -ordered.

4. $T_2$ -ordered if for each $x⋠y$ in X, there are disjoint neighborhoods $W_1$ and $W_2$ of x and y, respectively, such that $W_1$ is increasing and $W_2$ is decreasing.

Definition 3.1

Let $⪯$ be a partial order relation on a non-empty set X and let E be a set of parameters. A triple $(X,E,⪯)$ is said to be a partially ordered soft set.

Definition 3.2

Let $(X,E,⪯)$ be a partially ordered soft set. We define an increasing soft operator $i:(S(X_E),⪯)\to (S(X_E),⪯)$ and a decreasing soft operator $d:(S(X_E),⪯)\to (S(X_E),⪯)$ as follows, for each soft subset $G_E$ of $S(X_E)$

1. $i(G_E)={(\mathit{iG})}_E$ , where iG is a mapping of E into X given by $\mathit{iG}(e)=i(G(e))=\{x\in X:y⪯x$ , for some $y\in G(e)\}$ .

2. $d(G_E)={(\mathit{dG})}_E$ , where dG is a mapping of E into X given by $\mathit{dG}(e)=d(G(e))=\{x\in X:x⪯y$ , for some $y\in G(e)\}$ .

Definition 3.3

A soft subset $G_E$ of a partially ordered soft set $(X,E,⪯)$ is said to be:

1. Increasing if $G_E=i(G_E)$ .

2. Decreasing if $G_E=d(G_E)$ .

Proposition 3.4

We have the following results for a soft subset $G_E$ of a partially ordered soft set $(X,E,⪯)$ .

1. $G_E$ is increasing if and only if for each $P_e^x\in i(G_E)$ , then $P_e^x\in G_E$ .

2. $G_E$ is decreasing if and only if for each $P_e^x\in d(G_E)$ , then $P_e^x\in G_E$ .

3. If $G_E$ is increasing, then for each $x\in i(G_E)$ , we have $x\in G_E$ .

4. If $G_E$ is decreasing, then for each $x\in d(G_E)$ , we have $x\in G_E$ .

Proof

We only prove case (i), and the other follow similar lines.

Necessity: It comes immediately from Definition (3.3).

Sufficiency: By hypothesis, $P_e^x\in i(G_E)$ implies that $P_e^x\in G_E$ . Then $x\in G(e)$ . Since $⪯$ is reflexive, then $x\in i(G_E)$ . So $P_e^x\in i(G_E)$ . This means that $i(G_E)\stackrel{\sim }{\subseteq }{G}_E$ . Thus $G_E=i(G_E)$ . Hence a soft set $G_E$ is increasing.   □

Proposition 3.5

Let $\{G_{j_E}:j\in J\}$ be a collection of increasing (resp. decreasing) soft subsets of a partially ordered soft set $(X,E,⪯)$ . Then:

1. $\stackrel{\sim }{\bigcup _{j\in J}}{G}_{j_E}$ is increasing (resp. decreasing).

2. $\stackrel{\sim }{\bigcap _{j\in J}}{G}_{j_E}$ is increasing (resp. decreasing).

Proof

$(\mathbf{i})$ : We prove this case when a collection consists of increasing soft sets. Let $P_e^x\in \stackrel{\sim }{\bigcup _{j\in J}}{G}_{j_E}$ . Then there exists $j_0\in J$ such that $P_e^x\in G_{{j_0}_E}$ . Therefore $i(P_e^x)\stackrel{\sim }{\subseteq }i(G_{{j_0}_E})=G_{{j_0}_E}\stackrel{\sim }{\subseteq }\stackrel{\sim }{\bigcup _{j\in J}}{G}_{j_E}$ . Thus a soft set $\stackrel{\sim }{\bigcup _{j\in J}}{G}_{j_E}$ is increasing.

A similar proof is given for the case between parentheses.

By analogy with $(\mathbf{i})$ , one can prove $(\mathbf{ii})$ .   □

Corollary 3.6

A collection of all increasing (resp. decreasing) soft subsets of a partially ordered soft set $(X,E,⪯)$ forms a soft topology on X.

Proposition 3.7

A soft subset $G_E$ of a partially ordered soft set $(X,E,⪯)$ is increasing (resp. decreasing) if and only if $G_E^c$ is decreasing (resp. increasing).

Proof

Let $G_E$ be an increasing soft set. Suppose, to the contrary, that $G_E^c$ is not decreasing. Then there exists $P_e^x\in d(G_E^c)$ and $P_e^x\notin G_E^c$ . So $x\in d(G^c(e))$ and $x\notin G^c(e)$ . This means that there exists $y\in G^c(e)$ such that $x⪯y$ . Since $x\in G(e)$ and the soft set $G_E$ is increasing, then $y\in G(e)$ . But this contradicts that $G(e)\bigcap G^c(e)=\varnothing$ . Hence $G_E^c$ is decreasing. Similarly, one can prove the proposition in case of $G_E$ is decreasing.   □

Theorem 3.8

The finite product of increasing (resp. decreasing) soft sets is increasing (resp. decreasing).

Proof

We only prove the theorem for two soft sets in case of increasing soft sets and one can prove it similarly for finite soft sets.

Let $G_A$ and $F_B$ be two increasing soft subsets of $(X,A,{⪯}_1)$ and $(Y,B,{⪯}_2)$ , respectively. Setting $H_{A\times B}=G_A\times F_B$ such that $H(a,b)=G(a)\times F(b)$ , for each $(a,b)\in A\times B$ . Suppose, to the contrary, that $H_{A\times B}$ is not increasing. Then there exists a soft point $P_{(\alpha ,\beta )}^{(x,y)}$ such that $P_{(\alpha ,\beta )}^{(x,y)}\in i(H_{A\times B})$ and $P_{(\alpha ,\beta )}^{(x,y)}\notin H_{A\times B}$ . This means that $(x,y)\in i(H(\alpha ,\beta ))$ and $(x,y)\notin H(\alpha ,\beta )$ . So $(x,y)\in i(G(\alpha )\times F(\beta ))$ implies that

$$x\in i(G(\alpha ))=G(\alpha )\mathit{and}y\in i(F(\beta ))=F(\beta )$$ and $(x,y)\notin G(\alpha )\times F(\beta )$ implies that

(2)

$$x\notin G(\alpha )\mathit{or}y\notin F(\beta )$$ From (1) and (2), we obtain a contradiction. Since the contradiction arises by assuming that the soft set $H_{A\times B}$ is not increasing, then $H_{A\times B}$ is increasing.

A similar proof is given for the case between parentheses.   □

Proposition 3.9

Let $G_E$ and $F_E$ be two soft subsets of $(X,E,⪯)$ and let $i:(S(X_E),⪯)\to (S(X_E),⪯)$ be an increasing soft operator. Then:

1. $i(\stackrel{\sim }{\varnothing })=\stackrel{\sim }{\varnothing }$ .

2. $G_E\stackrel{\sim }{\subseteq }i(G_E)$ .

3. $i(i(G_E))=i(G_E)$ .

4. $i\left[G_E\bigcup ^{\sim }{F}_E\right]=i(G_E)\bigcup ^{\sim }i(F_E)$ .

Proof

The proof of items $(\mathbf{i})$ and $(\mathbf{ii})$ are obvious.

$(\mathbf{iii})$ : From $(\mathbf{ii})$ , we get that $i(G_E)\stackrel{\sim }{\subseteq }i(i(G_E))$ . On the other hand, let $x⋐i(i(G_E))$ . Then there exists $y⋐i(G_E)$ such that $y⪯x$ . Also, there exists $z⋐G_E$ such that $z⪯y$ . Since $⪯$ is transitive, then $z⪯x$ . So $x⋐i(G_E)$ . Thus $i(i(G_E))\stackrel{\sim }{\subseteq }i(G_E)$ . This completes the proof of this property.

$(\mathbf{iv})$ : Obviously, $i(G_E)\bigcup ^{\sim }i(F_E)\stackrel{\sim }{\subseteq }i\left[G_E\bigcup ^{\sim }{F}_E\right]$ . On the other hand, $G_E\bigcup ^{\sim }{F}_E\stackrel{\sim }{\subseteq }i(G_E)$ $\bigcup ^{\sim }i(F_E)$ . From $(\mathbf{iii})$ and Definition (3.3), we infer that $i(G_E)$ and $i(F_E)$ are increasing. From Proposition (3.5), we infer that $i(G_E)$ $\bigcup ^{\sim }i(F_E)$ is increasing. So $i\left[G_E\bigcup ^{\sim }{F}_E\right]\stackrel{\sim }{\subseteq }i(G_E)\bigcup ^{\sim }i(F_E)$ . Hence this part of the proposition holds.   □

Proposition 3.10

Let $G_E$ and $F_E$ be two soft subsets of $(X,E,⪯)$ and let $d:(S(X_E),⪯)\to (S(X_E),⪯)$ be a decreasing soft operator. Then:

1. $d(\stackrel{\sim }{\varnothing })=\stackrel{\sim }{\varnothing }$ .

2. $G_E\stackrel{\sim }{\subseteq }d(G_E)$ .

3. $d(d(G_E))=d(G_E)$ .

4. $d\left[G_E\bigcup ^{\sim }{F}_E\right]=d(G_E)\bigcup ^{\sim }d(F_E)$ .

Proof

The proof is similar to that of Proposition (3.9).   □

Proposition 3.11

The following two results hold for a soft map $f_{\varphi }:S(X_A)\to S(Y_B)$ .

1. The image of each soft point is soft point.

2. If $f_{\varphi }$ is bijective, then the inverse image of each soft point is soft point.

Proof

1. Consider $P_{\alpha }^x$ is a soft point in the domain. Then $f_{\varphi }(P_{\alpha }^x)={(f_{\varphi }(P))}_B$ such that $f_{\varphi }(P)(b)=\bigcup _{a\in {\varphi }^{-1}(b)}\varphi (P(a))$ . Since $$P(a)=\left\{\begin{array}{ccc}\hfill \mathit{singleton}\mathit{element}\mathit{of}X\hfill & \hfill :\hfill & \hfill a=\alpha \hfill \\ \hfill \varnothing \hfill & \hfill :\hfill & \hfill a\ne \alpha \hfill \end{array}\right.,$$ then this part of proposition holds.

2. Consider $P_{\beta }^y$ is a soft point in the codomain. Then $f_{\varphi }^{-1}(P_{\beta }^y)={(f_{\varphi }^{-1}(P))}_A$ such that $f_{\varphi }^{-1}(P)(a)=\bigcup f^{-1}(P(\varphi (a)))$ . Since $\varphi (a)$ is a singleton element in B and $P_{\beta }^y$ is a soft point, then $$P(\varphi (a))=\left\{\begin{array}{ccc}\hfill y\hfill & \hfill :\hfill & \hfill \varphi (a)=\beta \hfill \\ \hfill \varnothing \hfill & \hfill :\hfill & \hfill \varphi (a)\ne \beta \hfill \end{array}\right.$$ Since ϕ and f are bijective, then $$f^{-1}(P(\varphi (a)))=\left\{\begin{array}{ccc}\hfill \mathit{singleton}\mathit{element}\mathit{in}X\hfill & \hfill :\hfill & \hfill \varphi (a)=\beta \hfill \\ \hfill \varnothing \hfill & \hfill :\hfill & \hfill \varphi (a)\ne \beta \hfill \end{array}\right.$$ This completes the proof of this part of proposition.   □

Definition 3.12

Let $P_{\alpha }^x$ and $P_{\alpha }^y$ be two soft points in a partially ordered soft set $(X,E,⪯)$ . We say that $P_{\alpha }^x⪯P_{\alpha }^y$ if $x⪯y$ .

Definition 3.13

A soft map $f_{\varphi }:(S(X_A),{⪯}_1)\to (S(Y_B),{⪯}_2)$ is said to be:

1. Increasing if $P_{\alpha }^x{⪯}_1{P}_{\alpha }^y$ , then $f_{\varphi }(P_{\alpha }^x){⪯}_2{f}_{\varphi }(P_{\alpha }^y)$ .

2. Decreasing if $P_{\alpha }^x{⪯}_1{P}_{\alpha }^y$ , then $f_{\varphi }(P_{\alpha }^y){⪯}_2{f}_{\varphi }(P_{\alpha }^x)$ .

3. Ordered embedding if $P_{\alpha }^x{⪯}_1{P}_{\alpha }^y$ if and only if $f_{\varphi }(P_{\alpha }^x){⪯}_2{f}_{\varphi }(P_{\alpha }^y)$ .

Theorem 3.14

The following two results hold for a soft map $f_{\varphi }:(S(X_A),{⪯}_1)\to (S(Y_B),{⪯}_2)$ .

1. If $f_{\varphi }$ is increasing, then the inverse image of each increasing (resp. decreasing) soft subset of $\stackrel{\sim }{Y}$ is an increasing (resp. a decreasing) soft subset of $\stackrel{\sim }{X}$ .

2. If $f_{\varphi }$ is decreasing, then the inverse image of each increasing (resp. decreasing) soft subset of $\stackrel{\sim }{Y}$ is a decreasing (resp. an increasing) soft subset of $\stackrel{\sim }{X}$ .

Proof

$(\mathbf{i})$ : Let $G_K$ be an increasing soft subset of $\stackrel{\sim }{Y}$ . Suppose that $f_{\varphi }^{-1}(G_K)$ is not increasing. Then there exists $x\in X$ and there exists $\alpha \in A$ such that $P_{\alpha }^x\in i(f_{\varphi }^{-1}(G_K))$ and $P_{\alpha }^x\notin f_{\varphi }^{-1}(G_K)$ . So we infer that there exists $P_{\alpha }^y\in f_{\varphi }^{-1}(G_K)$ such that $P_{\alpha }^y{⪯}_1{P}_{\alpha }^x$ . Since $f_{\varphi }$ is increasing, then $f_{\varphi }(P_{\alpha }^y){⪯}_2{f}_{\varphi }(P_{\alpha }^x)$ and since $f_{\varphi }(P_{\alpha }^y)\in f_{\varphi }(f_{\varphi }^{-1}(G_K))\stackrel{\sim }{\subseteq }{G}_K$ , then $f_{\varphi }(P_{\alpha }^x)\in G_K$ . This implies that $P_{\alpha }^x\in f_{\varphi }^{-1}(G_K)$ . But this contradicts that $P_{\alpha }^x\notin f_{\varphi }^{-1}(G_K)$ . Hence the soft set $f_{\varphi }^{-1}(G_K)$ is increasing.

A similar proof is given for the case between parentheses.

By analogy with $(\mathbf{i})$ , one can prove $(\mathbf{ii})$ .   □

Theorem 3.15

Let $f_{\varphi }:(S(X_A),{⪯}_1)\to (S(Y_B),{⪯}_2)$ be a bijective ordered embedding soft map. Then the image of each increasing (resp. decreasing) soft subset of $\stackrel{\sim }{X}$ is an increasing (resp. a decreasing) soft subset of $\stackrel{\sim }{Y}$ .

Proof

Let $G_L$ be an increasing soft subset of $\stackrel{\sim }{X}$ . Suppose that $f_{\varphi }(G_L)$ is not increasing. Then there exists $y\in Y$ and there exists $\beta \in B$ such that $P_{\beta }^y\in i(f_{\varphi }(G_L))$ and $P_{\beta }^y\notin f_{\varphi }(G_L)$ . So we infer that there exists $P_{\beta }^z\in f_{\varphi }(G_L)$ such that $P_{\beta }^z{⪯}_2{P}_{\beta }^y$ . Since $f_{\varphi }$ is ordered embedding, then $f_{\varphi }^{-1}(P_{\beta }^z){⪯}_1{f}_{\varphi }^{-1}(P_{\beta }^y)$ and since $f_{\varphi }^{-1}(P_{\beta }^z)\in f_{\varphi }^{-1}(f_{\varphi }(G_L))=G_L$ , then $f_{\varphi }^{-1}(P_{\beta }^y)\in G_L$ . This implies that $P_{\beta }^y\in f_{\varphi }(G_L)$ . But this contradicts that $P_{\beta }^y\notin f_{\varphi }(G_L)$ . Hence the soft set $f_{\varphi }(G_L)$ is increasing.

A similar proof is given for the case between parentheses.   □

Definition 4.1

A quadrable system $(X,\tau ,E,⪯)$ is said to be a soft topological ordered space, where $(X,\tau ,E)$ is a soft topological space and $(X,E,⪯)$ is a partially ordered soft set.

Henceforth, we use the abbreviation STOS in a place of soft topological ordered space.

Definition 4.2

A soft subset $W_E$ of an STOS $(X,\tau ,E,⪯)$ is said to be:

1. Increasing soft neighborhood of $x\in X$ if $W_E$ is soft neighborhood of $x\in X$ and increasing.

2. Decreasing soft neighborhood of $x\in X$ if $W_E$ is soft neighborhood of $x\in X$ and decreasing.

Definition 4.3

For two soft subsets $G_E$ and $H_E$ of an STOS $(X,\tau ,E,⪯)$ and $x\in X$ , we say that:

1. $G_E$ containing x provided that $x\in G_E$ .

2. $G_E$ containing $H_E$ provided that $H_E\stackrel{\sim }{\subseteq }{G}_E$ .

3. $G_E$ is a soft neighborhood of $H_E$ provided that there exists a soft open set $F_E$ such that $H_E\stackrel{\sim }{\subseteq }{F}_E\stackrel{\sim }{\subseteq }{G}_E$ .

Definition 4.4

An STOS $(X,\tau ,E,⪯)$ is said to be:

1. Lower p-soft $T_1$ -ordered if for every distinct points $x⋠y$ in X, there exists an increasing soft neighborhood $W_E$ of x such that .

2. Upper p-soft $T_1$ -ordered if for every distinct points $x⋠y$ in X, there exists a decreasing soft neighborhood $W_E$ of y such that .

3. p-soft $T_0$ -ordered if it is lower soft $T_1$ -ordered or upper soft $T_1$ -ordered.

4. p-soft $T_1$ -ordered if it is lower soft $T_1$ -ordered and upper soft $T_1$ -ordered.

5. p-soft $T_2$ -ordered if for every distinct points $x⋠y$ in X, there exist disjoint soft neighborhoods $W_E$ and $V_E$ of x and y, respectively, such that $W_E$ is increasing and $V_E$ is decreasing.

Proposition 4.5

Every p-soft $T_i$ -ordered space $(X,\tau ,⪯,E)$ is p-soft $T_{i-1}$ -ordered, for $i=1,2$ .

Proof

It is obtained immediately from the above definition.   □

Example 4.6

Let $E=\{e_1,e_2\}$ be a set of parameters, $⪯=▵\bigcup \{(x,y),(x,z)\}$ be a partial order relation on $X=\{x,y,z\}$ and $\tau =\{\stackrel{\sim }{\varnothing },\stackrel{\sim }{X},G_{1_E},G_{2_E},G_{3_E}\}$ be a soft topology on X. The soft sets $G_{1_E},G_{2_E}$ and $G_{3_E}$ are defined as follows:

$G_{1_E}=\{(e_1,\{y\}),(e_2,\{y\})\}$ ,

$G_{2_E}=\{(e_1,\{z\}),(e_2,\{z\})\}$ ,

$G_{3_E}=\{(e_1,\{y,z\}),(e_2,\{y,z\})\}$ .

Then $(X,\tau ,⪯,E)$ is a lower p-soft $T_1$ -ordered space. So it is p-soft $T_0$ -ordered. On the other hand, there does not exist a soft open set containing x and does not contain y or z. Thus $(X,\tau ,⪯,E)$ is not p-soft $T_1$ -ordered.

Example 4.7

Let $E=\{e_1,e_2\}$ be a set of parameters, $⪯=▵\bigcup \{(1,x):x\in \mathcal{R}\}$ be a partial order relation on the set of real numbers $\mathcal{R}$ and $\tau =\{\stackrel{\sim }{\varnothing },G_E\stackrel{\sim }{\subseteq }\stackrel{\sim }{\mathcal{R}}:G_E^c$ is finite $\}$ be a soft topology on $\mathcal{R}$ . Obviously, $(\mathcal{R},\tau ,⪯,E)$ is p-soft $T_1$ -ordered, but is not p-soft $T_2$ -ordered.

Theorem 4.8

Let $(X,\tau ,E,⪯)$ be an STOS. Then the following three statements are equivalent:

1. $(X,\tau ,E,⪯)$ is upper (resp. lower) p-soft $T_1$ -ordered;

2. For all $x,y\in X$ such that $x⋠y$ , there is a soft open set $G_E$ containing $y(\mathit{resp}.x)$ in which $x⋠a(\mathit{resp}.a⋠y)$ for every $a⋐G_E$ ;

3. For all $x\in X,{(i(x))}_E(\mathit{resp}.{(d(x))}_E)$ is soft closed.