Other kinds of soft $�eta $ mappings via soft topological ordered spaces EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 1, 2019, 176-193 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Other kinds of soft β mappings via soft topological ordered spaces Tareq M. Al-shami1,2∗, Mohammed E. El-Shafei1, Baravan A. Asaad3,4 1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt 2 Department of Mathematics, Sana’a University, Sana’a, Yemen 3 Department of Computer Science, College of Science, Cihan University-Duhok, Kurdistan Region, Iraq 4 Department of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq Abstract. The authors of [13] formulated a soft topological ordered spaces concept and then they established and studied some ordered mappings [14]. In the present work, we define new ordered mappings via soft topological ordered spaces based on soft β-open sets, namely soft xβ-continuous, soft xβ-open, soft xβ-closed and soft xβ-homeomorphism mappings, for x ∈ {I,D,B}. We give various characterizations of each one of the introduced soft mappings. One of the most important obtained results is that an extended soft topologies notion guarantees the equivalent between the soft mappings initiated herein and their counterparts of mappings on topological ordered spaces. We provide several interesting examples to examine the relationships among these soft mappings. 2010 Mathematics Subject Classifications: 54F05, 54F15 Key Words and Phrases: Soft I(D,B)β-continuous mapping; Soft I(D,B)β-open mapping; Soft I(D,B)β-homeomorphism mapping and soft ordered separation axioms. 1. Introduction In the year 1965, Nachbin [37] started studying the topological ordered spaces concept by defining two independent mathematical structures, on a non-empty set X, namely a topology τ and a partial order relation �. Depending on these two structures, he redefine and reinvestigate some topological concepts such as normal, regular and and completely regular spaces to be normally ordered, regularly ordered and completely regular ordered spaces, respectively, on topological ordered spaces. Later on, McCartan [32] presented the notions of Ti-ordered and strong Ti-ordered spaces (i = 0, 1, 2, 3, 4) and compared them with Ti-spaces. Also, he completely descried Ti-ordered and supplied interesting examples ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3312 Email addresses: tareqalshami83@gmail.com (T. M. Al-shami), meshafei@hotmail.com (M. E. El-Shafei), baravan.asaad@uoz.edu.krd (B. A. Asaad) http://www.ejpam.com 176 c© 2019 EJPAM All rights reserved. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 177 to illustrate the concepts introduced and findings obtained therein. Based on β-open sets [2], Leela and Balasubramanian [28] in 2002, probed new ordered axioms; and Rao and Chudamani [42] in 2012, defined new kinds of continuous and homeomorphism mappings on topological ordered spaces. With regard to the generalizations of topological ordered spaces, we observe that this topic takes two directions, the first one is formulated by generalizing a partial order relation (see, for example, [25], [33], [34], [40]) and the second one is formulated by generalizing a topology (see, for example, [3], [8], [10], [12], [18], [20], [21]). To handle problems and phenomena which suffering from uncertainties and incomplete of data, Molotdsov [36] in 1999, proposed a new mathematical tool, namely soft sets. He pointed out that the previous theories such as probability and fuzzy set theory have diffi- culties which attributed to the inadequacies of their parameterizations tools and show that soft set theory is more suitable for dealing with uncertainties with adequate parameteri- zations. Maji et al. [31] introduced some soft operators such as soft equality relation, soft union and intersection between two soft sets. These soft operators were generalized and studied in several directions in [17], [24], [29], [30] and [41]. Aktas and Cağman [6] were the first who studied soft algebraic structure. They introduced the soft group and soft subgroup notions and concluded their basic properties. In 2010, Acar et al. [4] presented a concept of soft rings and investigated its main features; and in 2013, Shah and Shaheen [45] established the notions of a soft topological group and a soft topological ring over a group and a ring, respectively. Hida [27] adopted a differen view to define soft topological group which help to make it a natural extension of the usual topological group notion. In the year 2011, Shabir and Naz [44] initiated the concept of soft topological spaces and gave its fundamental notions such as soft open and soft closed sets, soft neighborhoods, soft interior and soft closure points. They also probed soft separation axioms and examined their properties. Min [35] gave deeper explanation for soft regular spaces and corrected some errors in [44]. Later on, desire of obtaining a deeper understanding of soft topology prompted interested researchers to carry out many studies on soft topological notions and their features. In 2012, Rong [43] investigated the countability axioms of soft topological spaces and and studied the possibility of carry over the results of countability axioms via general topology to the soft topology setting. Aygünoǧlu and Aygün [16] introduced and studied a soft compactness concept; and Hida [26] gave two types of soft compactness and pointed out the relationships between them. The authors of [5] and [1] introduced the notions of soft β-open sets and soft β-separations axioms, respectively. They examined which results related to β-open sets and βTi-spaces from the topological spaces remain valid in the context of soft topological spaces. Recently, Al-shami et al. [13] introduced a concept of soft topological ordered spaces and established the notions of p-soft Ti-ordered spaces (i = 0, 1, 2, 3, 4) depending on to- tally non belong relations, which introduced in [23], and monotone soft neighborhoods. Also, they [14] defined newly ordered mappings via topological ordered spaces and ob- tained interesting results. Al-shami and Kočinac [15] verified the equivalence between the enriched and extended soft topologies and concluded many findings related to soft mappings and soft axioms. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 178 We aim in this study to propose and investigate newly ordered mappings on soft topological ordered spaces, namely soft xβ-continuous, soft xβ-open, soft xβ-closed and soft xβ-homeomorphism mappings, for x ∈ {I,D,B}. The examples which illustrate the relationships among these soft mappings are given and the conditions which guarantee the equivalent between soft xβ-open and soft xβ-closed mappings are discussed, for x ∈ {I,D,B}. Also, the various characterizations of each one of the initiated soft mappings are investigated and the interrelations between these soft mappings and their counterparts of mappings in topological ordered spaces are studied amply. 2. Preliminaries In what follows, we mention the definitions and results related to soft set, soft topolog- ical spaces and ordered spaces that will be needed in investigating the concepts introduced and results obtained herein. Definition 1. [36] A notation GE is said to be a soft set over X if G is a mapping of a set of parameters E into 2X and it is written as a set of ordered pairs GE = {(e,G(e)) : e ∈ E and G(e) ∈ 2X}. For x ∈ X and a soft set GE over X, we say that x ∈ GE if x ∈ G(e), for each e ∈ E and x 6∈ GE if x 6∈ G(e), for some e ∈ E. Definition 2. [31] A soft set GE over X is called a null soft set, denoting by Φ̃, if G(e) = ∅, for each e ∈ E; and it is called an absolute soft set, denoting by X̃, if G(e) = X, for each e ∈ E. Definition 3. [7] The relative complement of a soft set GE is denoted by GcE, where Gc : E → 2X is a mapping defined by Gc(e) = X \G(e), for each e ∈ E. In this connection, it is worth noting that x 6∈ GE does not imply that x ∈ GcE. Definition 4. [44] A soft topology on a non-empty set X is a collection τ of soft sets over X under a parameters set E satisfying the following axioms: (i) X̃ and ∅̃ belong to τ . (ii) τ is closed under finite soft intersection. (iii) τ is closed under arbitrary soft union. The triple (X, τ,E) is called a soft topological space. Every member of τ is called a soft open set and its relative complement is called soft closed. Proposition 1. [44] Let (X, τ,E) be a soft topological space. Then τe = {G(e) : GE ∈ τ} defines a topology on X, for each e ∈ E. Definition 5. [38] Consider (X, τ,E) is a soft topological space and τe is a topology on X as in the above proposition. Then τ? = {GE : G(e) ∈ τe, for each e ∈ E} is a soft topology on X finer than τ . In [15], the authors termed τ? an extended soft topology. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 179 Definition 6. [46] Consider f : X → Y and φ : A → B are two mappings and let fφ : S(XA) → S(YB) be a soft mapping. Let GK and HL be soft subsets of S(XA) and S(YB), respectively. Then (i) fφ(GK) = (fφ(G))B is a soft subset of S(YB) such that fφ(G)(b) = { ⋃ a∈φ−1(b) ⋂ K f(G(a)) : φ−1(b) ⋂ K 6= ∅ ∅ : φ−1(b) ⋂ K = ∅ for each b ∈ B. (ii) f−1φ (HL) = (f−1φ (H))A is a soft subset of S(XA) such that f−1φ (H)(a) = { f−1(H(φ(a))) : φ(a) ∈ L ∅ : φ(a) 6∈ L for each a ∈ A. Remark 1. Henceforth, a soft mapping fφ : S(XA) → S(YB) implies that a mapping f of the universe set X into the universe set Y and a mapping φ of the set of parameters A into the set of parameters B Definition 7. [46] A soft mapping fφ : S(XA) → S(YB) is said to be injective (resp. surjective, bijective) if f and φ are injective (resp. surjective, bijective). Proposition 2. [46] Consider fφ : S(XA)→ S(YB) is a soft mapping and let GA and HB be two soft subsets of S(XA) and S(YB), respectively. Then we have the following results: (i) GA⊆̃f−1φ fφ(GA) and the equality relation holds if fφ is injective. (ii) fφf −1 φ (HB)⊆̃HB and the equality relation holds if fφ is surjective. Definition 8. [5] A soft subset HE of (X, τ,E) is said to be soft β-open if HE⊆̃cl(int(cl(HE))). And its relative complement is said to be soft β-closed. Definition 9. ([5], [44]) For a soft subset HE of (X, τ,E), we define the following four operators: (i) int(HE) (resp. intβ(HE)) is the largest soft open (resp. soft β-open) set contained in HE . (ii) cl(HE) (resp. clβ(HE)) is the smallest soft closed (resp. soft β-closed) set containing HE . Definition 10. [5] A soft mapping fφ : (X, τ,A)→ (Y, θ,B) is said to be: (i) Soft β-continuous if the inverse image of each soft open subset of (Y, θ,B) is a soft β-open subset of (X, τ,A). T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 180 (ii) Soft β-open (resp. soft β-closed) if the image of each soft open (resp. soft closed) subset of (X, τ,A) is a soft β-open (resp. soft β-closed) subset of (Y, θ,B). (iii) Soft β-homeomorphism if it is bijective, soft β-continuous and soft β-open. Definition 11. [19, 38] A soft set PE over X is called soft point if there exists e ∈ E and there exists x ∈ X such that P (e) = {x} and P (a) = ∅, for each a ∈ E \ {e}. A soft point will be shortly denoted by P xe and we say that P xe ∈ GE, if x ∈ G(e). Definition 12. [13] 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 13. [13] We define an increasing soft operator i : (SS(XE),�)→ (SS(XE),� ) and a decreasing soft operator d : (SS(XE),�)→ (SS(XE),�) as follows, for each soft subset GE of SS(XE) (i) i(GE) = (iG)E, where iG is a mapping of E into X given by iG(e) = i(G(e)) = {x ∈ X : y � x, for some y ∈ G(e)}. (ii) d(GE) = (dG)E, where dG is a mapping of E into X given by dG(e) = d(G(e)) = {x ∈ X : x � y, for some y ∈ G(e)}. Definition 14. [13] A soft subset GE of a partially ordered soft set (X,E,�) is said to be increasing (resp. decreasing) if GE = i(GE)(resp. GE = d(GE)). Theorem 1. [13] If a soft mapping fφ : (S(XA),�1) → (S(YB),�2) is increasing, then the inverse image of each increasing (resp. decreasing) soft subset of Ỹ is an increasing (resp. a decreasing) soft subset of X̃. Definition 15. [13] A quadrable system (X, τ,E,�) is said to be a soft topological ordered space, where (X, τ,E) is a soft topological space and (X,E,�) is a partially ordered soft set. Henceforth, the two notations (X, τ,E,�1) and (Y, θ, F,�2) stand for soft topological ordered spaces. Definition 16. [42] A mapping (X, τ,�1)→ (Y, θ,�2) is said to be: (i) I (resp. D, B) β-continuous if the inverse image of each open set is I (resp. D, B) β-open. (ii) I (resp. D, B) β-open if the image of each open set is I (resp. D, B) β-open. (iii) I (resp. D, B) β-closed if the image of each open set is I (resp. D, B) β-closed. (iv) I (resp. D, B) β-homeomorphism if it is bijective, I (resp. D, B) β-continuous and I (resp. D, B) β-open. Definition 17. [14] The composition of two soft mappings fφ : (X, τ,E,�1)→ (Y, θ, F,�2 ) and gλ : (Y, θ, F,�2) → (Z, υ,K,�3) is a soft mapping fφ ◦ gλ : (X, τ,E,�1) → (Z, υ,K,�3) and is given by (fφ ◦ gλ)(P xe ) = fφ(gλ(P xe )). T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 181 3. Soft I(D,B)β-continuity In this section, the notions of I(D,B)β-continuity at soft point, ordinary point and on the universe set are given and studied. Each one of the introduced soft mappings are characterized and some examples are provided to show the relationships among them. Definition 18. A soft subset HE of (X, τ,E,�1) is said to be: (i) Soft I (resp. Soft D, Soft B) β-open if it is soft β-open and increasing (resp. decreasing, balancing). (ii) Soft I (resp. Soft D, Soft B) β-closed if it is soft β-closed and increasing (resp. decreasing, balancing). Definition 19. A soft mapping fφ : (X, τ,E,�1)→ (Y, θ, F,�2) is called: (i) Soft I (resp. Soft D, Soft B) β-continuous at P xe ∈ X̃ if for each soft open set HF con- taining fφ(P xe ), there exists a soft I (resp. soft D, soft B) β-open set GE containing P xe such that fφ(GE)⊆̃HF . (ii) Soft I (resp. Soft D, Soft B) β-continuous at x ∈ X if it is soft I (resp. soft D, soft B) β-continuous at each P xe . (iii) Soft I (resp. Soft D, Soft B) β-continuous if it is soft I (resp. soft D, soft B) β-continuous at each x ∈ X. Theorem 2. A soft mapping fφ : (X, τ,E,�1)→ (Y, θ, F,�2) is soft I (resp. soft D, soft B) β-continuous if and only if the inverse image of each soft open subset of Ỹ is a soft I (resp. soft D, soft B) β-open subset of X̃. Proof. We prove the theorem in the case of fφ is soft Dβ-continuous and the other cases can be achieved similarly. Necessity: Let GF be a soft open subset of Ỹ , Then we have the following two cases: (i) Either f−1φ (GF ) = ∅̃. (ii) Or f−1(GF ) 6= ∅̃. By choosing P xe ∈ X such that P xe ∈ f−1φ (GF ), we obtain fφ(P xe ) ∈ GF . So there exists a soft Dβ-open set HE containing P xe such that fφ(HE)⊆̃GF . Since P xe is chosen arbitrary, then f−1φ (GF ) = ⋃̃ Pxe ∈f −1 φ (GF ) HE . From the two cases above, we conclude that f−1φ (GF ) is a soft Dβ-open subset of X̃. Sufficiency: Let GF be a soft open subset of Ỹ containing fφ(P xe ). Then P xe ∈ f−1φ (GF ). By hypothesis, f−1φ (GF ) is a soft Dβ-open set. Since fφ(f−1φ (GF ))⊆̃GF , then fφ is a soft Dβ-continuous mapping at P xe ∈ X and since P xe is chosen arbitrary, then fφ is a soft Dβ-continuous mapping. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 182 Remark 2. From Definition (19), we can note the following: (i) Every soft I (D, B) β-continuous mapping is always soft β-continuous. (ii) Every soft Bβ-continuous mapping is soft Iβ-continuous or soft Dβ-continuous. The two examples below elucidates that the converse of the two results of the remark above need not be true in general. Example 1. Let the two parameters sets A = {12 , 1 4}, B = {13 , 1 5} and the two universe sets X = {m,n, r, s}, Y = {u, v, w}. Consider a mapping φ : A → B is defined as, φ(12) = 1 3 and φ(14) = 1 5 , and a mapping f : X → Y is defined as, f(m) = u, f(n) = v and f(r) = f(s) = w. We define a partial order relation on X as �= 4 ⋃ {(m,n), (n, r), (m, r)} and we define two soft topologies τ and θ on X and Y , respectively, as τ = {∅̃, X̃, FA, GA} and θ = {∅̃, Ỹ , HB}, where FA = {(12 , {m,n, s}), ( 1 4 , {m, r})}, GA = {(12 , ∅), ( 1 4 , {r})} and HB = {(13 , {u}), ( 1 5 , {w})}. Since f−1φ (HB) = {(12 , {m}), ( 1 4 , {r, s})} is a soft β-open set, then fφ : S(XA) → S(YB) is a soft β-continuous mapping. On the other hand, f−1φ (HB) is neither a soft Dβ-open nor a soft Iβ-open set. Hence fφ is not soft I (soft D, soft B) β-continuous. Example 2. In Example above, if we only replace the partial order relation by �= 4 ⋃ {(m,n)}(resp. �= 4 ⋃ {(n, r)}), then the soft mapping fφ is soft D-continuous (resp. soft I-continuous), but is not soft B-continuous. Definition 20. For a soft subset HE of (X, τ,E,�), we define the following six operators: (i) H iβo E (resp. Hdβo E , Hbβo E ) is the largest soft I (resp. soft D, soft B) β-open set contained in HE . (ii) H iβcl E (resp. Hdβcl E , Hbβcl E ) is the smallest soft I (resp. soft D, soft B) β-closed set containing HE . Lemma 1. For any soft subset HE of (X, τ,E,�), the following statements hold: (i) (Hdβcl E )c = (Hc E)iβo. (ii) (H iβcl E )c = (Hc E)dβo. (iii) (Hbβcl E )c = (Hc E)bβo. Proof. (i) (Hdβcl E )c = { ⋃̃ FE : FE is a soft Dβ-closed set containing HE}c = ⋂̃ {F cE : F cE is a soft Iβ-open set contained in Hc E} = (Hc E)iβo. By analogy with (i), one can prove (ii) and (iii). T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 183 Theorem 3. The following five properties of a soft mapping fφ : (X, τ,E �1)→ (Y, θ, F,�2 ) are equivalent: (i) fφ is soft Iβ-continuous; (ii) f−1φ (LF ) is a soft Dβ-closed subset of X̃, for each soft closed subset LF of Ỹ ; (iii) (f−1φ (MF ))dβcl⊆̃f−1φ (cl(MF )), for every MF ⊆̃Ỹ ; (iv) fφ(Ndβcl E )⊆̃cl(fφ(NE)), for every NE⊆̃X̃; (v) f−1φ (int(MF ))⊆̃(f−1φ (MF ))iβo, for every MF ⊆̃Ỹ . Proof. (i) ⇒ (ii) : Consider LF is a soft closed subset of Ỹ . By hypothesis, f−1φ (LcF ) is a soft Iβ-open subset of X̃ and by the fact that f−1φ (LcF ) = (f−1φ (LF ))c, we obtain f−1φ (LF ) is soft Dβ-closed as required. (ii) ⇒ (iii) : It follows from (ii) that f−1φ (cl(ME)) is a soft Dβ-closed subset of X̃, for every MF ⊆̃Ỹ . So (f−1φ (MF ))dβcl⊆̃(f−1φ (cl(MF ))dβcl = f−1φ (cl(MF )). (iii) ⇒ (iv) : From the fact that Ndβcl E ⊆̃(f−1φ (fφ(NE))dβcl and from (iii), we have (f−1φ (fφ(NE))dβcl ⊆̃f−1φ (cl(fφ(NE)). This implies that fφ(Ndβcl E )⊆̃cl(fφ(NE)). (iv) ⇒ (v) : For any soft subset MF of Ỹ , we obtain from Lemma (1) that fφ(X̃ − (f−1φ (NE))iβo) = fφ(((f−1φ (NE))c)dβcl). It follows from (iv), that fφ(((f−1φ (NE))c)dβcl) ⊆̃cl(fφ(f−1φ (NE))c) = cl(fφ(f−1φ (N c E)))⊆̃cl(Ỹ − NE) = Ỹ − int(NE). Therefore (X̃ − (f−1φ (NE))iβo)⊆̃f−1φ (Ỹ−int(NE)) = X̃−f−1φ (int(NE)). Thus f−1φ (int(NE))⊆̃(f−1φ (NE))iβo. (v) ⇒ (i): Consider MF is a soft open subset of Ỹ . Then f−1φ (MF ) = f−1φ (int(MF ))⊆̃ (f−1φ (MF ))iβo. So (f−1φ (MF ))iβo = f−1φ (MF ) and this means that f−1φ (MF ) is a soft Iβ-open subset of X̃. Hence the desired result is proved. Theorem 4. The following five properties of a soft mapping fφ : (X, τ,E �1)→ (Y, θ, F,�2 ) are equivalent: (i) fφ is soft Dβ-continuous (resp. soft Bβ-continuous); (ii) f−1φ (LF ) is a soft Iβ-closed (resp. soft Bβ-closed) subset of X̃, for each soft closed subset LF of Ỹ ; (iii) (f−1φ (MF ))iβcl⊆̃f−1φ (cl(MF ))( resp. (f−1φ (MF ))bβcl⊆̃f−1φ (cl(MF )), for every MF ⊆̃Ỹ ; (iv) fφ(N iβcl E )⊆̃cl(fφ(NE))( resp. fφ(N bβcl E )⊆̃cl(fφ(NE)), for every NE⊆̃X̃; (v) f−1φ (int(MF ))⊆̃(f−1φ (MF ))dβo( resp. f−1φ (int(MF ))⊆̃(f−1φ (MF ))bβo, for every MF ⊆̃Ỹ . Proof. The proof is similar to that of Theorem (3). T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 184 Theorem 5. Let τ? be an extended soft topology on X. Then a soft mapping gφ : (X, τ?, E,�1) → (Y, θ, F,�2) is soft I (resp. soft D, soft B) β-continuous If and only if a mapping g : (X, τ?e ,�1)→ (Y, θφ(e),�2) is I (resp. D, B) β-continuous. Proof. Necessity: Let U be an open subset of (Y, θφ(e),�2). Then there exists a soft open subset GF of (Y, θ, F,�2) such that G(φ(e)) = U . Since gφ is a soft I (resp. soft D, soft B) β-continuous mapping, then g−1φ (GF ) is a soft I (resp. soft D, soft B) β-open set. From Definition (6), it follows that a soft subset g−1φ (GF ) = (g−1φ (G))E of (X, τ,E,�1) is given by g−1φ (G)(e) = g−1(G(φ(e))), for each e ∈ E. By hypothesis, τ? is an extended soft topology on X, we obtain a subset g−1(G(φ(e))) = g−1(U) of (X, τe,�1) is I (resp. D, B) β-open. Hence a mapping g is I (resp. D, B) β-continuous. Sufficiency: Let GF be a soft open subset of (Y, θ, F,�2). Then from Definition (6), it follows that a soft subset g−1φ (GF ) = (g−1φ (G))E of (X, τ?, E,�1) is given by g−1φ (G)(e) = g−1(G(φ(e))), for each e ∈ E. Since a mapping g is I (resp. D, B) β-continuous, then a subset g−1(G(φ(e))) of (X, τ?e ,�1) is I (resp. D, B) β-open. By hypothesis, τ? is an extended soft topology on X, we obtain g−1φ (GF ) is a soft I (resp. soft D, soft B) β- open subset of (X, τ?, E,�1). Hence a soft mapping gφ is soft I (resp. soft D, soft B) β-continuous. Proposition 3. Let a surjective soft mapping fφ : (X, τ,E �1) → (Y, θ, F,�2) be soft Bβ-continuous. Then: (i) If �1 is linearly order, then θ is the soft indiscrete topology. (ii) If θ is the soft discrete topology, then �1 is an equality relation. 4. Soft I(D,B)β-openness and soft I(D,B)β-closedness In this section, the concepts of soft I(D,B)-open and soft I(D,B)-closed mappings are introduced and two examples are provided to elucidate the relationships among them. Then the equivalent conditions for each one of these soft mappings are discussed and some results related to them are initiated. Definition 21. A soft mapping fφ : (X, τ,E,�1)→ (Y, τ, F,�2) is called: (i) Soft I (resp. Soft D, Soft B) β-open if the image of every soft open subset of X̃ is a soft I (resp. soft D, soft B) β-open subset of Ỹ . (ii) Soft I (resp. Soft D, Soft B) β-closed if the image of every soft closed subset of X̃ is a soft I (resp. soft D, soft B) β-closed subset of Ỹ . Remark 3. From Definition (21), we can note the following: (i) Every soft I (D, B) β-open mapping is soft β-open. (ii) Every soft I (D, B) β-closed mapping is soft β-closed. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 185 (iii) Every soft Bβ-open (resp. soft Bβ-closed) mapping is soft Iβ-open or soft Dβ-open (resp. soft Iβ-closed or soft Dβ-closed). We construct the following two examples to show that the converse of the three state- ments of remark above fails. Example 3. Let the two soft topological spaces (X, τ,A), (Y, θ,B) and the two mappings f : X → Y , φ : A→ B be the same as in Example (1). Consider a partial order relation on Y as �= 4 ⋃ {(u,w), (w, v), (u, v)}. Then one can easily noted that fφ : S(XA)→ S(YB) is soft β-open and soft β-closed mapping. Because fφ(GA) = {(13 , ∅), ( 1 5 , {w})} is neither a soft Dβ-open nor a soft Iβ-open set, then fφ is not a soft I (soft D, soft B) β-open mapping and because fφ(F cA) = {(13 , {w}), ( 1 5 , {v, w})} is neither a soft Dβ-closed nor a soft Iβ-closed set, then fφ is not a soft I (soft D, soft B) β-closed mapping. Example 4. In Example above, if we only replace the partial order relation by �= 4 ⋃ {(u,w)}(resp. �= 4 ⋃ {(w, v)}), then the soft mapping fφ is soft Iβ-open and soft Iβ-closed (resp. soft Dβ-open and soft Dβ-closed), but is not soft Bβ-open and soft Bβ- closed. Theorem 6. The following three properties of a soft mapping fφ : (X, τ,E �1) → (Y, θ, F,�2) are equivalent: (i) fφ is soft Iβ-open; (ii) int(f−1φ (MF ))⊆̃f−1φ (M iβo F ), for every MF ⊆̃Ỹ ; (iii) fφ(int(NE))⊆̃(fφ(NE))iβo, for every NE⊆̃X̃. Proof. (i)⇒ (ii): Given a soft subsetMF of Ỹ , it is obvious that int(f−1φ (MF )) is a soft open subset of X̃. Then, by hypothesis, it follows that fφ(int(f−1φ (MF ))) is a soft Iβ-open subset of Ỹ . Since fφ(int(f−1φ (MF )))⊆̃fφ(f−1φ (MF ))⊆̃MF , then int(f−1φ (MF ))⊆̃f−1φ (M iβo F ). (ii)⇒ (iii): Given a soft subsetNE of X̃, from (ii), we obtain int(f−1φ (fφ(NE)))⊆̃f−1φ ((fφ(NE))iβo). Since int(NE)⊆̃f−1φ (fφ(int(f−1φ (fφ(NE)))))⊆̃f−1φ ((fφ(NE))iβo), then fφ(int(NE))⊆̃(fφ(NE))iβo as required. (iii)⇒ (i): Let GE be a soft open subset of X̃. Then fφ(int(GE)) = fφ(GE)⊆̃(fφ(GE))iβo. Hence fφ is a soft Iβ-open mapping. In a similar manner, one can prove the following theorem. Theorem 7. The following three properties of a soft mapping fφ : (X, τ,E �1) → (Y, θ, F,�2) are equivalent: (i) fφ is soft Dβ-open (resp. soft Bβ-open); (ii) int(f−1φ (MF ))⊆̃f−1φ (Mdβo F )( resp. int(f−1φ (MF ))⊆̃f−1φ (M bβo F )), for every MF ⊆̃Ỹ ; (iii) fφ(int(NE))⊆̃(fφ(NE))dβo( resp. fφ(int(NE))⊆̃(fφ(NE))bβo), for every NE⊆̃X̃. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 186 Theorem 8. The following three statements hold for a soft mapping fφ : (X, τ,E �1)→ (Y, θ, F,�2): (i) fφ is soft Iβ-closed if and only if (fφ(GE))iβcl⊆̃fφ(cl(GE)), for every GE⊆̃X̃. (ii) fφ is soft Dβ-closed if and only if (fφ(GE))dβcl⊆̃fφ(cl(GE)), for every GE⊆̃X̃. (iii) fφ is soft Bβ-closed if and only if (fφ(GE))bβcl⊆̃fφ(cl(GE)), for every GE⊆̃X̃. Proof. We only prove the first statement and the others follow similar lines. Necessity: Since fφ is soft Iβ-closed, then fφ(cl(GE)) is a soft Iβ-closed subset of Ỹ and since fφ(GE)⊆̃fφ(cl(GE)), then (fφ(GE))iβcl⊆̃fφ(cl(GE)). Sufficiency: ConsiderHE is a soft closed subset of X̃. Then fφ(HE)⊆̃(fφ(HE))iβcl⊆̃fφ(cl(HE)) = fφ(HE). Therefore fφ(HE) = (fφ(HE))iβcl. This means that fφ(HE) is a soft Iβ-closed set. Hence the proof is complete. Theorem 9. The following three statements hold for a bijective soft mapping fφ : (X, τ,E �1 )→ (Y, θ, F,�2): (i) fφ is soft I (resp. soft D, soft B) β-open if and only if fφ is soft D (resp. soft D, soft B) β-closed. (ii) fφ is soft I (resp. soft D, soft B) β-open if and only if f−1φ is soft I (resp. soft D, soft B) β-continuous. (iii) fφ is soft D (resp. soft I, soft B) β-closed if and only if f−1φ is soft I (resp. soft D, soft B) β-continuous. Proof. For the sake of brevity, we only give proofs of cases outside the parenthesis for the three statements above and the cases between parenthesis can be made similarly. (i) To prove the necessary condition, let HE be a soft closed subset of X̃ and consider fφ is a soft Iβ-open mapping. Then Hc E is soft open and fφ(Hc E) is soft Iβ-open. It follows from the bijectiveness of fφ, that fφ(Hc E) = [fφ(HE)]c. This automatically implies that fφ(HE) is soft Dβ-closed. Thus fφ is a soft Dβ-closed mapping. In a similar manner, we can prove the sufficiency condition. (ii) Necessity: Let GE be a soft open subset of X̃ and consider fφ is a soft Iβ-open mapping. Then fφ(GE) is soft Iβ-open. It follows from the bijectiveness of fφ, that fφ(GE) = (f−1φ )−1(GE). This automatically implies that (f−1φ )−1(GE) is soft Iβ- open. Thus f−1φ is a soft Iβ-continuous mapping. In a similar manner, we can prove the sufficiency condition. (iii) The proof of this statement comes immediately from (i) and (ii) above. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 187 Theorem 10. Let θ? be an extended soft topology on Y and φ is an injective mapping. Then a soft mapping gφ : (X, τ,E,�1) → (Y, θ?, F,�2) is soft I (resp. soft D, soft B) β-open if and only if a mapping g : (X, τe,�1)→ (Y, θ?φ(e),�2) is I (resp. D, B) β-open. Proof. To prove the necessary part, let U be an open subset of (X, τe,�1) and φ(e) = f . Then there exists a soft open subset GE of (X, τ,E,�1) such that G(e) = U . Since gφ is a soft I (resp. soft D, soft B) β-open mapping, then gφ(GE) is a soft I (resp. soft D, soft B) β-open set. From Definition (6), it follows that a soft subset gφ(GE) = (gφ(G))F of (Y, θ, F,�2) is given by gφ(G)(f) = ⋃ e∈φ−1(f) g(G(e)), for each f ∈ F . By hypothesis, θ? is an extended soft topology on Y , a subset ⋃ e∈φ−1(f) g(G(e)) = g(U) of (Y, θφ(e),�2) is I (resp. D, B) β-open. Hence a mapping g is I (resp. D, B) β-open. To prove the sufficient part, let GE be a soft open subset of (X, τ,E,�1). Then from Definition (6), it follows that a soft subset gφ(GE) = (gφ(G))F of (Y, θ?, F,�2) is given by gφ(G)(f) = ⋃ e∈φ−1(f) g(G(e)), for each f ∈ F . Since a mapping g is I (resp. D, B) β-open, then a subset ⋃ e∈φ−1(f) g(G(e)) of (Y, θ?φ(e),�2) is I (resp. D, B) β-open. By hypothesis, θ? is an extended soft topology on Y , gφ(GE) is a soft I (resp. soft D, soft B) β-open subset of (Y, θ?, F,�2). Hence a soft mapping gφ is soft I (resp. soft D, soft B) β-open. The result above is restated in the case of a soft I (resp. soft D, soft B) β-closed mapping and one can prove them similarly. So the proof will be omitted. Theorem 11. Let θ? be an extended soft topology on Y and φ is an injective mapping. Then a soft mapping gφ : (X, τ,E,�1) → (Y, θ?, F,�2) is soft I (resp. soft D, soft B) β- closed if and only if a mapping g : (X, τe,�1)→ (Y, θ?φ(e),�2) is I (resp. D, B) β-closed. Proposition 4. Consider τ is not the indiscrete topology on X. If an injective soft mapping fφ : (X, τ,E �1) → (Y, θ, F,�2) is soft Bβ-open or soft Bβ-closed, then �2 is not linearly ordered. Proposition 5. Let fφ : (X, τ,E,�1)→ (Y, θ, F,�2) and gλ : (Y, θ, F,�2)→ (Z, υ,K,�3 ) be two soft mappings. Then then following properties hold, for x ∈ {I,D,B}. (i) If fφ is a soft xβ-continuous mapping and gλ is a soft continuous mapping, then gλ◦fφ is a soft x-continuous mapping. (ii) If fφ is a soft open (resp. soft closed) mapping and gλ is a soft xβ-open (resp. xβ- closed) mapping, then gλ ◦ fφ is a soft x-open (resp. xβ-closed) mapping. (iii) If gλ ◦ fφ is a soft x-open mapping and fφ is surjective soft continuous, then gλ is a soft x-open mapping. (iv) If gλ ◦ fφ is a soft closed mapping and gλ is an injective soft x-continuous mapping, then fφ is a soft y-closed mapping, where (x, y) ∈ {(I,D), (D, I), (B,B)}. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 188 5. Soft I(D,B)β-homeomorphism The concepts of soft I(D,B)-homeomorphism mappings are established and their main properties are discussed. Illustrative examples are provided to show the relationships among them. Definition 22. A bijective soft mapping gφ : (X, τ,E,�1) → (Y, θ, F,�2) is called soft I (resp. soft D, soft B) β-homeomorphism if it is soft Iβ-continuous and soft Iβ-open (resp. soft Dβ-continuous and soft Dβ-open, soft Bβ-continuous and soft Bβ-open). Remark 4. From Definition (22), we can note the following: (i) Every soft I (soft D, soft B) β-homeomorphism mapping is soft β-homeomorphism. (ii) Every soft Bβ-homeomorphism mapping is soft Iβ-homeomorphism or soft Dβ-homeomorphism. The two items of the remark above are not conversely as the following examples show. Example 5. Let X = {u, v, w, x, y, z} be an universe set and A = {a1, a2} be a parameters set. Consider φ : A→ A and f : X → X are both identity mappings. We define two partial order relations on X and Y , respectively, as �1= 4 ⋃ {(w, v)} and �2= 4 ⋃ {(z, x)} and we define two soft topologies τ and θ on X and Y , respectively, as τ = {∅̃, X̃, FA, GA, HA} and θ = {∅̃, Ỹ , LA}, where FA = {(a1, X), (a2, {w, z})}, GA = {(a1, {u, v}), (a2, X)}, HA = {(a1, {u, v}), (a2, {w, z})} and LA = {(a1, {u, z}), (a2, {v})}. Then one can readily check that a soft mapping fφ : S(XA) → S(YB) is soft β-homeomorphism. On the other hand, fφ(FA) = FA is not a soft Iβ-open set and f−1φ (LA) = LA is not a soft Dβ-open set. Hence fφ is not soft I (soft D, soft B) β-homeomorphism. Example 6. In Example above, if we only replace the partial order relation �1 by �= 4 ⋃ {(w, x)}, then the soft mapping fφ is soft D-homeomorphism, but is not soft B- homeomorphism. Also, if we only replace the partial order relation �2 by �= 4 ⋃ {(y, z)}, then the soft mapping fφ is soft I-homeomorphism, but is not soft B-homeomorphism. Theorem 12. Consider fφ : (X, τ,E,�1) → (Y, θ, F,�2) is a bijective soft mapping and let (γ, λ) ∈ {(Iβ, dβcl), (Dβ, iβcl), (Bβ, bβcl)}. Then fφ is soft γ-homeomorphism if and only if (fφ(GE))λ = fφ(cl(GE)) = cl(fφ(GE)) = fφ(GλE), for every GE⊆̃X̃. Proof. We make a proof for the theorem in the case of (γ, λ) = (Iβ, dβcl) and the other follow similar line. Necessity: The property fφ is a soft Iβ-homeomorphism mapping implies that fφ(GdβclE )⊆̃ cl(fφ(GE)) and (fφ(GE))dβcl⊆̃fφ(cl(GE)), for every GE⊆̃X̃. So fφ(cl(GE))⊆̃fφ(GdβclE ) ⊆̃cl(fφ(GE))⊆̃(fφ(GE))dβcl and cl(fφ(GE))⊆̃(fφ(GE))dβcl⊆̃fφ(cl(GE))⊆̃fφ(GdβclE ). By the preceding two inclusion relations, we obtain the required equality relation. Sufficiency: The equality relation (fφ(GE))dβcl = fφ(cl(GE)) = cl(fφ(GE)) = fφ(GdβclE ) implies that fφ(GdβclE )⊆̃cl(fφ(GE)) and (fφ(GE))dβcl⊆̃fφ(cl(GE)). So fφ is soft Iβ-continuous and soft Dβ-closed mapping. Hence the desired result is proved. T. M. Al-shami, M. E. El-Shafei, B. A. Asaad / Eur. J. Pure Appl. Math, 12 (1) (2019), 176-193 189 Theorem 13. If a bijective soft mapping fφ : (X, τ,E,�1) → (Y, θ, F,�2) is soft Iβ- continuous (resp. soft Dβ-continuous, soft Bβ-continuous), Then the following three state- ments are equivalent: (i) fφ is soft Iβ-homeomorphism (resp. soft Dβ-homeomorphism, soft Bβ-homeomorphism); (ii) f−1φ is soft Iβ-continuous (resp. soft Dβ-continuous, soft Bβ-continuous); (iii) fφ is soft Dβ-closed (resp. soft Iβ-closed, soft Bβ-closed). Proof. (i)⇒ (ii) Since fφ is a soft Iβ-homeomorphism (resp. soft Dβ-homeomorphism , soft Bβ-homeomorphism) mapping, then fφ is soft Iβ-open (resp. soft Dβ-open , soft Bβ-open). It follows from item (ii) of Theorem (9), that f−1φ is soft Iβ-continuous (resp. soft Dβ-continuous, soft Bβ-continuous). (ii)⇒ (iii) The proof follows from item (iii) of Theorem (9). (iii) ⇒ (i) It sufficient to prove that fφ is a soft Iβ-open (resp. soft Dβ-open, soft Bβ- open) mapping. This follows from item (i) of Theorem (9). Theorem 14. Let τ? and θ? be extended soft topologies on X and Y , respectively. Then a soft mapping gφ : (X, τ?, E,�1) → (Y, θ?, F,�2) is soft I (resp. soft D, soft B) β- homeomorphism if and only if a mapping g : (X, τ?e ,�1)→ (Y, θ?φ(e),�2) is I (resp. D, B) β-homeomorphism. Proof. The proof is obtained immediately from Theorem (5) and Theorem (10) Proposition 6. Let a soft mapping fφ : (X, τ,E �1)→ (Y, θ, F,�2) be soft Bβ-homeomorphism. Then: (i) If �1 and �2 are linearly order, then τ and θ are the soft indiscrete topologies. (ii) If τ and θ are the soft discrete topologies, then �1 and �2 are equality relations. Conclusion In [13], the authors have initiated the concept of soft topological ordered spaces as an extended of the soft topological spaces notion and have defined soft ordered separa- tion axioms. Then they [14] have introduced several types of ordered mappings and have established main features. 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