Soft Incline Algebras

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Soft Incline Algebras

N.O. Alshehri

Department of Mathematics, Faculty of Sciences(Girls) King Abdulaziz University, Jeddah, Saudi Arabia

Abstract:In this paper we introduce the notion of soft incline algebras and filter soft incline algebras, and investigate some of their properties. Moreover, we introduce the notions of prime and maximal soft ideal and quotient of soft incline algebras.

Key words:Soft incline algebras, filter soft incline algebras, prime and maximal soft ideal, quotient of soft incline algebras

INTRODUCTION

Cao et al. [1] introduced the notion of incline algebras in their book: Incline algebra and applications, and was studied by some authors [2-10]. Inclines are a generalization of both Boolean and fuzzy algebras, and a special type of a semiring, and they give a way to combine algebras with ordered structures to express the degree of intensity of binary relations. An incline is a structure which has an associative, commutative addition, and a distributive multiplication such thatx+x=x, x+xy = xfor all x, y. It has both a semiring structure and a poset structure. Inclines can also be used to represent automa and other mathematical systems, in optimization theory, to study inequalities for nonnega- tive matrices of polynomials. Ahn et al. [2] introduced the notion of quotient incline and obtained the structure of incline algebras. They also introduced the notion of prime and maximal ideals in an incline, and studied some relations between them in incline algebras.

Most of the problems in engineering, medical science, economics, environments, and so forth, have various uncertainties. The problems in system identification in- volve characteristics which are essentially non proba- bilistic in nature. In response to this situation Zadeh [11]

introduced fuzzy set theory as an alternative to probabil- ity theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [12]. Molodtsov [13]

initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields including game theory, opera- tions research, Riemann-integration, Perron integration.

At present, work on soft set theory is progressing rapidly.

The algebraic structure of soft set theories has been stud-

ied increasingly in recent years. Aktas and Cagman [14]

defined the notion of soft groups. Feng et al. [15] initiated the study of soft semirings and soft rings were defined by Acar et al. [16]. Jun [17] introduced soft BCK/BCI-algebras. In this paper we introduce the notion of soft incline algebras and filter soft incline algebras and investigate some of their properties. Moreover, we also introduce the notion of prime and maximal soft ideal and quotient of soft incline algebras. For other ter- minologies and notations readers are refereed to [18-24].

REVIEW OF LITERATURE

An incline (algebra) is a setKwith two binary opera- tions denoted by “+” and “∗” satisfying the following axioms for allx, y∈ K,

(a) x+y=y+x,

(b) x+ (y+z) = (x+y) +z, (c) x∗(y∗z) = (x∗y)∗z, (d) x∗(y+z) = (x∗y) + (x∗z), (e) (y+z)∗x= (y∗x) + (z∗x), (f) x+x=x,

(g) x+ (x∗y) =x, (h) y+ (x∗y) =y.

Furthermore, an incline algebraKis said to be commutative ifx∗y=y∗xfor allx, y∈ K. For convenience, we pronounce “+” (resp. “∗”) as addition (resp. multiplication). Every distributive lattice is an incline. An incline is a distributive lattice (as a semiring) if and only ifx∗x=xfor allx∈ K. Note thatx≤y⇔x+y=y for allx, y∈ K. A subincline of an inclineKis a nonempty subsetMofKwhich is closed under addition and multiplication. A subinclineM is said to be an ideal of an inclineK if x ∈ M andy ≤ xtheny ∈ M. An element0 in an incline algebraK is a zero element if x+ 0 = x = 0 +xandx∗0 = 0∗x = 0, for any x∈ K. By a homomorphism of inclines we shall mean a mappingf from an inclineKinto an inclineLsuch that

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f(x+y) =f(x) +f(y)andf(x∗y) =f(x)∗f(y) for allx, y ∈ K. Molodtsov [?] initiated the concept of soft set theory as a new approach for modeling uncertainties. Then Maji et al. [?] expanded this theory to fuzzy soft set theory. Molodtsov [?] defined the notion of soft set in the following way: LetUbe an initial universe and E be a set of parameters. LetP(U)denotes the power set ofU and let A be non-empty subset ofE. A pair (F, A)is called asoft setoverU, whereFis a mapping given byF :A→P(U). In other words, a soft set over Uis a parameterized family of subsets of the universeU. Forx ∈ A,F(x)may be considered as the set ofx−

approximate elements of the soft set(F, A). Clearly, a soft set is not a classical subset ofU.

Definition 1. Let(F, A)and(G, B)be two soft sets over a common universeU.(F, A)is a said to be soft subset of(G, B)if

1. A⊆Band

2. F(x)⊆G(x)for allx∈A.

We write(F, A)⊂(G, B).e

Definition 2. Two soft sets (F, A) and (G, B) over a common universeU are said to be soft equal if(F, A) is a soft subset of(G, B)and(G, B)is a soft subset of (F, A). We write(F, A) = (G, B).

Definition 3. Let(F, A)and(G, B)be two soft sets over a common universeU. The bi(restricted)-intersection of (F, A)and (G, B)is defined as the soft set (H, C) = (F, A)ue(G, B), whereC =A∩B 6= ∅andH(x) = F(x)∩G(x)for allx∈C.

Definition 4. The bi(restricted)-intersection of a nonempty family soft sets {(Fi, Ai) | i ∈ Λ} over a common universe U is defined as the soft set (H, B) = eu_i∈Λ(Fi, Ai) , where B = T

i∈ΛAi and H(x) =T

x∈ΛFi(x)for allx∈B.

Definition 5. The extended intersection of two soft sets (F, A)and(G, B)over a common universeU is defined as the soft set(H, C) = (F, A)∩e (G, B), whereC = A∪Band for allx∈C

H(x) =







F(x) if x∈A−B G(x) if x∈B−A F(x)∩G(x) if x∈A∩B Definition 6. The extended intersection of a nonempty family soft sets{(Fi, A_i)|i ∈ Λ}over a common uni- verseUis defined as the soft set(H, B) =Te

i∈Λ(F_i, A_i) , where B = S

i∈ΛAi and H(x) = T

x∈ΛFi(x), Λ(x) ={i|i∈Ai}for allx∈B.

Definition 7. The restricted intersection of a nonempty family soft sets{(Fi, A_i)|i ∈ Λ}over a common uni- verseUis defined as the soft set(H, B) =Se

i∈Λ(F_i, A_i) , where B = T

i∈ΛAi and H(x) = S

Definition 8. Let(F, A)and(G, B)be two soft sets over a common universeU, then “(F, A)AND(G, B)" denoted by(F, A)∧(G, B)e and defined by(F, A)∧(G, B)=e (H, A×B), where H(x, y) = F(x)∩ G(y) for all (x, y)∈A×B.

Definition 9. The∧- intersection of a nonempty family soft sets {(Fi, A_i) | i ∈ Λ} over a common universe U is defined as the soft set(H, B) = Ve

i∈Λ(F_i, A_i), whereB =Q

i∈ΛAiandH(x) =T

x∈ΛFi(x),Λ(x) = {i|i∈Ai}for allx∈B.

Definition 10. Let (F, A)and (G, B) be two soft sets over a common universeU, then “(F, A)OR(G, B)" denoted by(F, A)∨(G, B)e and defined by(F, A)∨(G, B)=e (H, A×B), where H(x, y) = F(x)∪ G(y) for all (x, y)∈A×B.

Definition 11. The∨- union of a nonempty family soft sets{(Fi, Ai)|i∈Λ}over a common universeUis defined as the soft set(H, B) =We

i∈Λ(Fi, Ai), whereB= Q

i∈ΛA_iandH(x) =S

x∈ΛF_i(x),Λ(x) ={i|i∈A_i} for allx∈B.

Definition 12. Let (F, A)and (G, B) be two soft sets over a common universeU. The Cartesian product of the two soft sets(F, A)and(G, B)is defined as the soft set(H, A×B) = (F, A)×(G, B), wheree H(x, y) = F(x)×G(y)for all(x, y)∈A×B.

Definition 13. The Cartesian product of the nonempty family soft sets{(Fi, A_i)|i ∈ Λ}over a common uni- verseUis defined as the soft set(H, B) =Qe

i∈Λ(F_i, A_i) , where B = Q

i∈ΛAi and H(x) = Q

Definition 14. For a soft set (F, A), the set Supp(F, A) = {x ∈ A | F(x) 6= ∅} is called the support of the soft set(F, A), and the soft set(F, A)is called a non-null if Supp(F, A)6=∅.

SOFT INCLINE ALGEBRAS

Definition 15. Let (F, A) be a soft set over K. Then (F, A)is called a soft incline algebra overKifF(x)is a subincline algebra ofKfor allx∈A.

Example 16. Consider the incline algebra K = {a, b, c, d}and we define the sum “+” and product “∗“

onKas:

+ a b c d

a a b c d

b b b b b c c b c b

d d b b d

∗ a b c d

a a a a a

b a b c d

c a c c a

d a d a d

Let (F, A) be a soft set over K, where A = K and F : A → P(K) is a set-valued function defined by

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F(x) = {y ∈ K : xRy ⇔ x ≤ y}. Then F(a) = {a, b, c, d}, F(b) = {b}, F(c) = {b, c} and F(d) = {b, d} are a subincline ofK. Hence(F, A)is a soft incline algebra overK.

Example 17. Consider the incline algebra K = {a, b, c, d}which is given in example 3.2. ForA = K, letF :A→P(K)be a set-valued function defined by:

F(x) =

a if x=a, b, d,

c if x=c. ∀x∈A

ThenF(a) = F(b) = F(d) = {a} and F(c) = {c}are subincline algebra ofK. Hence(F, A)is a soft incline algebra overK.

Definition 18. Let(F, A)be a soft incline algebra over K. A soft set(g, I)overKis called a soft ideal of(F, A) denoted by(g, I) ˜C(F, A)if it satisfies

(i) I⊂A,

(ii)g(x)CF(x) for allx∈I.

Example 19. Consider the incline algebra K = {a, b, c, d}which is given in example 3.2. Let(F, A)be a soft set overK whereA = K and F : A → P(K) is a set-valued function defined byF(x) = {y ∈ K : xRy ⇔ x∗y ≤ y}. ThenF(a) = F(b) = F(c) = F(d) =Kare a subincline ofK. Hence(F, A)is a soft incline algebra overK. Let we take I = {c, d} ⊂ A and defined a set valued function g : I → P(K) by g(x) = {y ∈ K : xRy ⇔ x+y = x}.Then g(c) ={a, c)CF(c)andg(d) ={a, d}CF(d). Hence (g, I)is a soft ideal of(F, A).

Theorem 20. Let(F, A)be a soft incline algebra over K. IfBis a subset ofA, then(F|B, B)is a soft incline algebra overK.

Proof. Since(F, A)is a soft incline algebra overK, then F(x)is a subincline ofKfor allx∈A. ThereforeF(x) is a subincline ofKfor allx∈B ⊂A. Hence(F|B, B) is a soft incline algebra overK.

The following example shows that there exists a soft set(F, A)overKsuch that

(i)(F, A)is not a soft incline algebra overK.

(ii) there exists a subsetBofAsuch that(F|B, B)is a soft incline algebra overK.

Example 21. Consider the incline algebra K = {0, a, b, c, d,1} and we have define the sum “+” and product “∗“ onKas:

+ 0 a b c d 1

0 0 a b c d 1

a a a a 1 1 1 b b 1 b 1 b 1 c c 1 1 c 1 1 d d 1 b c d 1

1 1 1 1 1 1 1

∗ 0 a b c d 1

0 0 0 0 0 0 0

a 0 0 0 0 0 0 b 0 0 d d d d c 0 0 d d d d d 0 0 d d d d

1 0 0 d d d d

Let(F, A)be a soft set overK, whereA =KandF : A→ P(K)is a set-valued function defined byF(x) = {y ∈ K : xRy ⇔ x ≤ y}.Then(F, A)is not a soft incline algebra overK. Let(F, B)be a soft set overK, whereB = {0, d} ⊂ A. Then(F, B)is a soft incline algebra overK.

Theorem 22. Let(F, A)and(G, B)be two soft incline algebras overK. IfA∩B 6= ∅ , then the intersection (F, A)˜∩(G, B)is a soft incline algebra overK.

Proof. Using Definition 2.3, we can write (F, A)˜∩(G, B) = (H, C), whereC = A ∩ B and H(x) = F(x) or G(x) for all x ∈ C. Note that H : C → P(K) is a mapping, and therefore(H, C) is a soft set overK. Since(F, A)and(G, B)are soft incline algebras overK, it follows thatH(x) =F(x)is a subincline ofK, orH(x) =G(x)is a subincline ofK for allx∈C. Hence(H, C) = (F, A)˜∩(G, B)is a soft incline algebra overK.

Corollary 23. Let(F, A)and(G, A)be two soft incline algebras overK. Then, their intersection(F, A)˜∩(G, A) is a soft incline algebra overK.

Theorem 24. Let(F, A)and(G, A)be two soft incline algebras overK. IfAandBare disjoint, then the union (F, A)˜∪(G, A)is a soft incline algebra overK.

Proof. Using Definition 2.5, we can

write(F, A)˜∪(G, B) = (H, C), whereC =A∪Band for everye∈C,

H(e) =







F(e) if e∈A\B, G(e) if e∈B\A, F(e)∪G(e)if e∈A∩B.

SinceA∩B = ∅, eitherx ∈ A\B or x ∈ B\A for all x ∈ C. If x ∈ A\B, then H(x) = F(x) is a subincline ofK since(F, A)is a soft incline algebra overK. Ifx ∈ B\A, thenH(x) = G(x)is a subincline ofKsince(G, B)is a soft incline algebra overK.

Hence(H, C) = (F, A)˜∪(G, A)is a soft incline algebra overK.

Theorem 25. If(F, A)and(G, B)are soft incline algebras overK, then(F, A)˜∧(G, B)is a soft incline algebra overK.

Proof. By means of Definition 2.9, we know that (F, A)˜∧(G, B) = (H, A × B), where H(x, y) = F(x) ∩ G(y) for all (x, y) ∈ A ×B.

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Since F(x) and G(y)are subinclines of K, the intersection F(x)∩G(y)is also a subincline ofK. Hence H(x, y)is a subincline ofKfor all(x, y)∈A×B, and therefore(F, A)˜∧(G, B) = (H, A×B)is a soft incline algebra overK.

Definition 26. A soft incline algebra(F, A)over K is said to be trivial (resp., whole) ifF(x) = {0} (resp., F(x) =K) for allx∈A.

Example 27. consider the incline algebra K = {0, a, b,1}and we have define the sum “+” and product “∗“ onKas:

+ 0 a b 1

0 0 a b 1

a a a b 1

b b b b 1

1 1 1 1 1

∗ 0 a b 1

0 0 0 0 0

a 0 a a a

b 0 a b b

1 0 a b 1

Let(F, A)be a soft set overK, whereA = KandF : A →P(K)is a set-valued function defined byF(x) = {y ∈ K : xRy ⇔ x∗y ≤y}.ThenF(0) = F(a) = F(b) =F(1) =Kare a subincline ofK. Hence(F, A) is a whole soft incline algebra overK.

Definition 28. Let(F, A)and(G, B)be two soft incline algebras overK. Then(F, A)is called a soft subalgebra of(G, B), denoted by(F, A) ˜<(G, B), if it satisfies:

(i)A⊂B,

(ii)F(x)is a subincline ofG(x)for allx∈A.

Example 29. Consider the incline algebra K = {0, a, b,1}which is given in example 3.13. ForA =K, letF : A → P(K)be a set-valued function defined by F(x) = {y ∈ K : xRy ⇔ x 6 y}.Then F(0) = F(1) = K, F(a) = {a, b,1}, andF(b) = {b,1} are subincline ofK.Hence(F, A)is a soft incline algebra over K. Let B = {0, a, b} be a subset of A and let G : B → P(K) be a set-valued function defined by G(x) ={y ∈ K:xRy ⇐⇒x+y=x}for allx∈B.

ThenG(0) = {0}, G(a) = {0, a,}, G(b) = {0, a, b}

andG(1) = Kare subincline ofF(0), F(a), F(b)and F(1), respectively .Hence(G, B)is a soft subalgebra of (F, A).

Theorem 30. Let(F, A)and(G, A)be two soft incline algebras overK.

(i) IfF(x)⊂G(x)for allx∈A, then(F, A) ˜<(G, A).

(ii) IfB={0}and(H, B),(F, K)are soft incline algebras overK, then(H, B) ˜<(F, K).

Proof. Straightforward.

Theorem 31. Let(F, A)be a soft incline algebra over Kand let(G₁, B₁)and(G₂, B₂)be soft subinclines of (F, A). Then

(i)(G1, B1)˜∩(G2, B2) ˜<(F, A).

(ii)B1∩B2=∅ ⇒(G1, B1)˜∪(G2, B2) ˜<(F, A).

Proof. (i) Using Definition 2.3, we can write (G1, B1)˜∩(G2, B2) = (G, B), whereB = B1 ∩B2

and G(x) = G₁(x) orG₂(x) for all x ∈ B.

Obviously,B ⊂A. Letx∈B. Thenx∈B1andx∈B2

. Ifx∈B1, thenG(x) =G1(x)is a subincline ofF(x) since(G₁, B₁) ˜<(F, A). Ifx∈B₂, thenG(x) =G₂(x) is a subincline ofF(x)since(G2, B2) ˜<(F, A). Hence (G1, B1)∩(G2, B2) = (G, B)<(F, A).

(ii) Assume that B₁ ∩ B₂ = ∅. By definition 2.5 we can write (G1, B1)˜∪(G2, B2)

= (G, B) where B = B1 ∪ B2 and G(x) =







G1(x) if x∈B1\B2, G2(x) if x∈B2\B1, G₁(x)∪G₂(x)if x∈B₁∩B₂.

∀x ∈ B Since (Gi, Bi) ˜<(F, A) fori = 1,2, B = B1 ∪ B2 ⊂ A and Gi(x) is a subincline of F(x) for all x ∈ Bi

, i = 1,2.Since B₁ ∩ B₂ = ∅ , G(x) is a subincline of F(x) for all x ∈ B. Therefore (G1, B1)˜∪(G2, B2) = (G, B) ˜<(F, A).

Definition 32. Let (F, A) be a soft set over K. Then (F, A) is called an idealistic soft incline algebra over KifF(x)is an ideal ofKfor allx∈A.

Theorem 33.If(F, A)and(G, B)are any idealistic soft incline algebras overKandB ⊆Athen(G, B)is also a soft ideal of(F, A).

Proof. Is immediately follows from the definition of soft ideal of incline algebra.

Definition 34. A soft subincline(G, B)of a soft incline (F, A)is said to be a softK-ideal of(F, A)if:

(i)A⊂B,

(ii)G(x)is aK-ideal ofF(x).

Theorem 35(4.21). Let(G, B)be a soft subincline of a soft incline(F, A). Then(G, B)is a soft ideal of(F, A) if and only if(G, B)is aK-ideal of(F, A).

Proof. Let(G, B)be a soft ideal of(F, A), and letx∈ F(x)andy, z ∈G(x)such thatx+y ∈ G(x). Since x+x=x, x+z=x+ (x+y) =x+y=zand hence x6z.Hencex∈G(x), i.e.,G(x)is aK-ideal ofF(x).

Therefore(G, B)is a softK-ideal of(F, A).Conversely, assume thaty ∈F(x)andx∈G(x)withy 6x.Then y+x=x.SinceG(x)is aK-ideal ofF(x), y∈G(x), proving thatG(x)is an ideal ofF(x).therefore(G, B) is aK-ideal of(F, A).

Definition 36. Let K, L be two incline algebras and f : K → L a mapping of incline algebras. If (F, A) and(G, B)are soft sets overKandLrespectively, then (f(F), A)is a soft set overLwheref(F) :A→P(L) is defined by f(F)(x) = f(F(x))for allx ∈ A and (f⁻¹(G), B)is a soft set overKwheref⁻¹(G) :B → P(K) is defined by f⁻¹(G)(y) = f⁻¹(G(y))for all y∈B.

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Lemma 37. Letf :K →Lbe an onto homomorphism of incline algebras.

(i) If (F, A) is a soft incline algebra over K, then (f(F), A)is a soft incline algebra overL.

(ii) If (G, B) is a soft incline algebra over L, then (f⁻¹(G), B)is a soft incline algebra overKif it is non- null.

Proof. (i) For every x ∈ A, we havef(F)(x) = f(F(x))is a subincline ofLsinceF(x)is a subincline ofKand its homomorphic image is also a subincline of L. Hence(f(F), A)is a soft incline algebra overL.

(ii) It is easy to see that Supp(f⁻¹(G), B) ⊆ Supp(G, B). Lety∈Supp(f⁻¹(G), B). ThenG(y)6=

∅. Since the nonempty setG(y)is a subincline ofL, its homomorphic inverse imagef⁻¹(G(y))is also a subincline ofK. Hencef⁻¹(G(y))is a subincline ofLfor all y ∈ Supp(f⁻¹(G), B). That is,(f⁻¹(G), B)is a soft incline algebra overK.

Theorem 38. Letf :K →Lbe a homomorphism of incline algebras. Let(F, A)and(G, B)be two soft incline algebras overKand Lrespectively.

(i) IfF(x) = ker(f)for allx ∈ A, then(f(F), A)is the trivial soft incline algebra overL.

(ii) Iff is onto and(F, A)is whole, then(f(F), A)is the whole soft incline algebra overL.

(iii) IfG(y) =f(K)for ally ∈B, then(f⁻¹(G), B)is the whole soft incline algebra overK.

(iv) If f is injective and (G, B) is trivial, then (f⁻¹(G), B)is the trivial soft incline algebra overK.

Proof. (i) Assume thatF(x) = ker(f)for allx ∈ A.

Then,f(F)(x) =f(F(x)) ={e₂}for allx∈Awhere e2∈L. Hence(f(F), A)is the trivial soft incline algebra overLby Lemma 3.23 and Definition 3.12.

(ii) Suppose thatf is onto and(F, A)is whole. Then, F(x) =Kfor allx∈A, and sof(F)(x) =f(F(x)) = f(K) = Lfor allx∈ A. It follows from Lemma 3.23 and Definition 3.12 that(f(F), A)is the whole soft incline algebra overL.

(iii) Assume thatG(y) = f(K)for ally ∈ B. Then, f⁻¹(G)(y) = f⁻¹(G(y)) = f⁻¹(f(K)) = K for all y ∈ B. Hence, (f⁻¹(G), B) is the whole soft incline algebra overKby Lemma 3.23 and Definition 3.12.

(iv) Suppose that f is injective and(G, B)is trivial.

Then,G(y) ={e2}for ally∈B, and sof⁻¹(G)(y) = f⁻¹(G(y)) = f⁻¹({e2}) = ker(f) = {e1} for all y ∈ B wheree₁ ∈ K and e₂ ∈ L. It follows from Lemma 3.23 and Definition 3.12 that(f⁻¹(G), B)is the trivial soft incline algebra overK.

Theorem 39. Let f : K → L be a homomorphism of incline algebras and let (F, A) and (G, B) be soft incline algebras over K. Then (F, A) ˜<(G, B) =⇒ (f(F), A) ˜<(f(G), B).

Proof. Assume that (F, A) ˜<(G, B). Let x ∈ A.

Then A ⊂ B and F(x) is a subincline of G(x).

Since f is a homomorphism, f(F)(x) = f(F(x)) is a subincline of f(G(x)) = f(G)(x)and, therefore, (f(F), A) ˜<(f(G), B).

Theorem 40. Letf :K →Lbe an onto homomorphism of incline algebras and(F, A), (G, B)two soft incline algebras overKandL, respectively.

(i) The soft function(f, IA)from(F, A)to(H, A)is a soft homomorphism fromK toL, whereIA : A → A is the identity mapping and the set-valued functionH : A→P(L)is defined byH(x) =f(F(x))for allx∈A.

(ii) Iff :K →Lis an isomorphism, then the soft function(f⁻¹, I_B)from(G, B)to(S, B)is a soft homomorphism fromLtoK, whereIB : B → B is the identity mapping and the set-valued functionS : B →P(K)is defined byS(x) =f⁻¹(G(x))for allx∈B.

Proposition 41. Let K1, K2 and K3 be incline algebras and(F, A),(G, B)and (H,C) soft incline algebras overK1,K2, andK3respectively. Let the soft function (f, g)from (F, A) to(G, B)be a soft homomorphism fromK1toK2, and the soft function (f0, g0)from(G, B) to(H, C)a soft homomorphism fromK2toK3.Then the soft function(f0 ◦f, g0 ◦g)from(F, A)to(H, C)is a soft homomorphism fromK1toK3.

Definition 42. Let(F, A)and(H, B)be two soft incline algebras overKandL, respectively. Then the cartesian product of soft incline algebras(F, A)and(H, B)is denoted by(F, A) ˜×(H, B) = (U, A×B)andUis defined asU(a, b) =F(a)×H(b)for all(a, b)∈A×B.

Theorem 43. Let(F, A)and(H, B)be two soft incline algebras over K and L, respectively. Then the cartesian product (F, A) ˜×(H, B) is a soft incline algebra over K₁ ×K₂ and (F, A) ˜×(H, B)is soft isomorphic to(H, B) ˜×(F, A).

Proof. First part is straightforward. We will prove next part. Now we show that(f, g) : (F, A) ˜×(H, B) → (H, B) ˜×(F, A) is a soft isomorphism That is(f, g) : (U, A×B)→(W, B×A)is a soft isomorphism where W(b, a)is defined asW(b, a) =H(b)×F(a). Actually, we prove three conditions.

1. We show thatf :K1× K2→ K2× K1is an isomorphism. Letf be a function defined byf(r, s) = (s, r).

Then obviouslyf is an isomorphism.

2. Now we show thatg:A×B→B×Ais a bijective mapping. The mappingg is defined byg(a, b) = (b, a)

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then obviouslyg is a bijective mapping. 3.

f(U(a, b)) = f(F(a)×H(b))

= f({(r, s) :r∈F(a), s∈H(b)})

= {(s, r) :s∈H(b), r∈F(a)}

= H(b)×F(a)

= W(b, a)

= W(g(a, b))

for all (a, b) ∈ A ×B. This implies that (f, g) : (F, A) ˜×(H, B) → (H, B) ˜×(F, A) is a soft isomorphism. Hence,(F, A) ˜×(H, B)'(H, B) ˜×(F, A).

Definition 44. Let(F, A)and(G, B)be two soft incline algebras over K such that A∩B 6= ∅ . Then their restricted intersection is denoted by(F, A)e(G, B) = (H, C)where(H, C)is defined asH(c) =F(c)∩G(c) for allc∈C=A∩B.

Theorem 45. Let (F, A) and (H, B) be two soft incline algebras over K. Then their restricted intersection (F, A)e(H, B)is a soft incline algebra over K.

Proof. Suppose that(F, A)e(G, B) = (H, C),we show that for all c ∈ C, H(c) is a subincline of K. Since F(c), G(c)are subinclin ofK.ThenH(c) =F(c)∩G(c) is a subincline ofK. Hence(H, C) = (F, A)e(G, B) is a soft incline algebra overK.

Definition 46. The extended intersection of two soft incline algebras(F, A)and(G, B)over a incline algebra Kis the soft set(H, C), whereC =A∪B, and for all e∈C.

H(e) =







F(e) if e∈A\B, G(e) if e∈B\A, F(e)∩G(e)if e∈A∩B.

We write (H, C) = (F, A)uE(G, B).

Theorem 47. Let (F, A) and (H, B) be two soft incline algebras over K. Then their extended intersection (F, A)uE(G, B)is a soft incline algebra over K.

Proof. Since(F, A)and(H, B)are soft incline algebras over K, their extended intersection overK is a soft set (L, C), whereC=A∪Band for allc∈C, it is defined as

L(c) =







F(c) if c∈A\B, H(c) if c∈B\A, F(c)∩H(c)if c∈A∩B.

From here we have,F(c)is a subincline ofKfor allc∈A\B. This implies thatL(c)is a subincline ofK for allc ∈ A\B.

Similarly,L(c) = H(c)is a subincline ofKfor allc ∈ B\AandL(c) =F(c)∩H(c)is a subincline ofKfor allc∈A∩B. Thus(L, C) = (F, A)u_E(G, B)is a soft incline algebra over K.

Definition 48. The restricted union(H, C)of two soft incline algebras(F, A)and(G, B)overKis defined as

the soft set(H, C) = (F, A)∪R (G, B), where C = A∩BandH(c) =F(c)∪G(c)for allc∈C.

Theorem 49. Let(F, A)and(H, B)be two soft incline algebras over K. Then their restricted union is a soft incline algebra over Kif and only if eitherF(x)⊂H(x) orH(x)⊂ F(x)for allx∈A∩B.

Proof. Let (F, A)∪R (H, B) = (L, C) be restricted union of two soft sets overK whereA∩B = C and Lis defined as L(x) = F(x)∪H(x)for allx ∈ C.

Let(F, A)and(H, B)be two soft incline algebras over K. Suppose either F(x) ⊂ H(x)or H(x) ⊂ F(x) for all x ∈ C. Then F(x) ∪ H(x) = H(x) or F(x)∪H(x) = F(x)for allx ∈ C. SinceH(x)and F(x)are subincline ofKfor allx∈C, this implies that F(x)∪H(x)is a subincline ofKfor allx ∈C. Thus L(x) =F(x)∪H(x)is a subincline ofKfor allx∈C.

Hence(L, C) = (F, A)∪R(H, A)is a soft incline algebras overK.

The proof of converse part is obvious.

Definition 50. Let (F, A) and (G, B)be any two soft incline algebras overK. Then extended union(H, C) of two soft incline algebras (F, A)and (G, B) is denoted as the soft set (H, C) = (F, A) ∪E (G, B) where C = A ∪ B and H is defined as H(e) =







Theorem 51. Let(F, A)and(H, B)be two soft incline algebras overK. Then their extended union is a soft incline algebra overKif and only if eitherF(x)⊂H(x)or H(x)⊂F(x)for allx∈A∩B.

Proof. Let(F, A)and(H, B)be any two soft incline algebras overK. Then their extended union(L, C)is denoted as the soft set(L, C) = (F, A)∪E(H, B)where C = A ∪B and L(x) is defined as, for all x ∈ C L(x) =







F(x) if x∈A\B, H(x) if x∈B\A, F(x)∪H(x)if x∈A∩B.

Suppose ei- therH(x)⊂F(x)orF(x)⊂H(x)for allx∈A∩B.

ThenF(x)∪H(x) = L(x)for allx ∈ A∩B. Since eachF(x)is a subincline ofKfor allx ∈ Aand each H(x) is a subincline of K for all x ∈ B. Hence, F(x)∪H(x)is a subincline ofK for allx ∈ A∩B.

Thus,L(x) =F(x)∪H(x)is a subincline ofKfor all x∈A∩B. Ifx∈A\BthenL(x) =F(x)is a subincline ofK. Similarly,L(x) =G(x)is a subincline ofK for allx∈ B\A. So (L, C) = (F, A)∪_E (H, B)is a soft incline algebra overKfor allx∈C.

The proof of converse part is obvious.

Definition 52. The restricted product(H, C)of two soft incline algebras(F, A)and(G, B)overKis denoted by the soft set(H, C) = (F, A)ˆ◦(G, B)whereC =A∩B

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andH is a set valued function fromC toP(K)and is defined asH(c) =F(c)G(c)for allc∈C. The soft set (H, C)is called the restricted soft product of(F, A)and (G, B)overK.

Definition 53. Let(F, A)be a soft set over a incline al- gebraK. Then inverse of(F, A)is denoted by(F, A)⁻¹ and is defined as follows(F, A)⁻¹ = {(F(a))⁻¹ : a ∈ A}, where(F(a))⁻¹is called the inverse ofF(a)and is defined as(F(a))⁻¹={x⁻¹:x∈F(a)}

Theorem 54. Let (F, A) and (G, B) be any two soft sets over K. Then ((F, A)ˆ◦(G, B))⁻¹ = (G, B)⁻¹ˆ◦(F, A)⁻¹.

Proof. Suppose that the inverse of restricted soft product of(F, A)and(G, B)denoted by((F, A)ˆ◦(G, B))⁻¹

= (H, C)is defined asH(c) = (F(c)G(c))⁻¹ for all c ∈ C and (G, B)⁻¹ˆ◦(F, A)⁻¹ = (L, C) and is defined asL(c) = (G(c))⁻¹(F(c))⁻¹ for all c ∈ C. But then(F(c)G(c))⁻¹= (G(c))⁻¹(F(c))⁻¹for allc∈C.

This implies that L(c) = H(c)for all c ∈ C. Thus ((F, A)ˆ◦(G, B))⁻¹= (G, B)⁻¹ˆ◦(F, A)⁻¹.

Definition 55. A soft incline algebra(F, A)over K is said to be abelian soft incline algebra over K if each F(α)is an abelian subincline ofKfor allα∈A.

Example 56. Consider the incline algebra K = {0, a, b, c, d,1} in example 3.7. For A = {0, d}, let F : A → P(K)be a set-valued function defined by F(x) = {y ∈ K, xRy ⇔ x ≤ y}.Then(F, A)is an abelian soft incline algebra overK.

Definition 57. Let(F, A)be a soft incline algebra over K and(H, B)be a soft subincline of(F, A). Then we say that(H, B)is an abelian soft subaincline of(F, A) ifH(x)is an abelian subincline ofF(x)for allx∈B.

Theorem 58. Let(F, A)be an abelian soft incline algebra overKand(G, B)be a soft incline algebra overK.

Then their restricted intersection(F, A)e(G, B)is an abelian soft incline algebra overKfor allc∈A∩B.

Proof. Suppose(F, A)e(G, B) = (L, C)whereL(c) = F(c)∩G(c)for allc ∈C = A∩B. ThenL(c)is an abelian subincline ofK or allc ∈ C.Hence(L, C) = (F, A)e(G, B)is an abelian soft incline algebra over K.

Definition 59. Let(F, A)be a soft set overK. Then (F, A) is called a filter soft incline algebra overK if F(x)is a filter ofKfor allx∈ A. Let us illustrate this definition using the following examples.

Example 60. Let(F, A)be a soft set overK which is given in Example 4.2. Then it is easy to verify that each F(x)is a filter ofKfor allx∈A.Hence(F, A)is a filter soft incline algebra overK.

Theorem 61. Let (F, A) and (G, B)be two filter soft incline algebras overK. IfA∩B 6= ∅ , then the intersection(F, A)˜∩(G, B)is a filter soft incline algebra overK.

Proof. Using Definition 2.3, we can write (F, A)˜∩(G, B) = (H, C), whereC = A ∩ B and H(x) = F(x) or G(x) for all x ∈ C. Note that H :C→P(K)is a mapping, and therefore(H, C)is a soft set overK. Since (F, A)and(G, B)are filter soft incline algebras overK, it follows thatH(x) = F(x) is a filter ofK, orH(x) =G(x)is a filter ofK for all x∈C. Hence(H, C) = (F, A)˜∩(G, B)is a filter soft incline algebra overK.

Corollary 62. Let (F, A) and (G, A) be two filter soft incline algebras over K. Then, their intersection (F, A)˜∩(G, A)is a filter soft incline algebra overK.

Theorem 63. Let (F, A)and (G, A) be two filter soft incline algebras overK. IfAand B are disjoint, then the union(F, A)˜∪(G, A)is a filter soft incline algebra overK.

Proof. Using Definition 2.3, we can

write(F, A)˜∪(G, B) = (H, C), whereC =A∪Band for everye∈C,

H(e) =







Since A∩B = ∅ , either x ∈ A\B or x ∈ B\A for all x ∈ C. If x ∈ A\B, then H(x) = F(x) is a filter ofKsince(F, A)is a filter soft incline algebra over K. If x ∈ B\A, then H(x) = G(x) is a filter of K since (G, B) is a filter soft incline algebra over K.

Hence(H, C) = (F, A)˜∪(G, A)is a filter soft incline algebra overK.

Theorem 64. If(F, A)and(G, B)are filter soft incline algebras overK, then(F, A)˜∧(G, B)is a filter soft incline algebra overK.

Proof. By means of Definition 2.12, we know that (F, A)˜∧(G, B) = (H, A × B), where H(x, y) = F(x) ∩ G(y) for all (x, y) ∈ A ×B.

SinceF(x) andG(y)are filters of K, the intersection F(x)∩ G(y) is also a filter of K. Hence H(x, y) is a filter of K for all (x, y) ∈ A ×B, and therefore (F, A)˜∧(G, B) = (H, A×B) is a filter soft incline algebra overK.

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