IRJET- Rough Hyperideals in Join Hyperlattice

International Research Journal of Engineering and Technology (IRJET) Volume: 07 Issue: 09 | Sep 2020 www.irjet.net

e-ISSN: 2395-0056 p-ISSN: 2395-0072

Rough Hyperideals in Join Hyperlattice S.Ramalakshmi1, B.Nathiya2, S.Nagamarilakshmi3, S.Halwath Kani4 1-4Student,

Department of Mathematics, Thassim Beevi Abdul Kader College for Women, Kilakarai ---------------------------------------------------------------------------***--------------------------------------------------------------------------Abstract: In this paper, we consider a rough hyperideals in join hyperlattice. Moreover, we investigate some theorems and properties for rough hyperideals in join hyperlattice.

Keywords: Rough hyperideals, Hyper congruences, Hyperlattices Introduction: In this section, we introduce the notion of rough hyperideals in hyperlattices and discuss some properties of them.

Given a hyperlattice đ??ż, by đ?‘ƒâˆ—(đ??ż) we will denote the set of all nonempty subsets of đ??ż. If đ?œƒ is an equivalence relation on đ??ż, then, for every đ?‘Ž ∈ đ??ż, [đ?‘Ž]đ?œƒ stands for the equivalence class of đ?‘Ž with the represent đ?œƒ. For any nonempty subset đ??´ of đ??ż, we denote [đ??´]đ?œƒ = {[đ?‘Ž]đ?œƒ| đ?‘Ž ∈ đ??´}. For any đ??´, đ??ľ ∈ đ?‘ƒâˆ—(đ??ż), we denote đ??´đ?œƒĚ…đ??ľ if the following conditions hold: (1) for all đ?‘Žâˆˆ đ??´, ∃đ?‘? ∈ đ??ľ such that đ?‘Žđ?œƒđ?‘?; (2) for all đ?‘‘ ∈ đ??ľ, ∃đ?‘? ∈ đ??´ such that đ?‘?đ?œƒđ?‘‘.

Now, we can introduce the notion of hyper congruences on hyperlattices in the following manner.

Definition 1. Let (đ??ż, , â‹ ) be a hyperlattice. An equivalence relation đ?œƒ on đ??ż is called a hyper congruence on đ??ż if for all a, đ?‘Ž , b, đ?‘? the following implication holds: ađ?œƒđ?‘Ž , and bđ?œƒđ?‘? Ě… imply (a b) đ?œƒĚ…(a

b) and (a

Ě… b) đ?œƒĚ… (a

b).

Obviously, an equivalence relation đ?œƒ on (đ??ż, , â‹ ) is a hyper congruence if and only if for all đ?‘Ž, đ?‘?, x ∈ đ??ż, we have that đ?‘Žđ?œƒđ?‘? implies (a x) đ?œƒĚ… (b x) and (a x)đ?œƒĚ… (b x).

Lemma 2. Let (đ??ż, , â‹ ) be a hyperlattice, and let đ?œƒ be a hyper congruence on đ??ż. For all đ?‘Ž, đ?‘?, ∈ đ??ż, then đ?‘Ž đ?‘? .đ?‘Ž đ?‘? đ?‘Ž đ?‘? .

Proof. Suppose that x ∈ đ?‘Ž That x ∈

đ?‘? , then there exist

đ?‘Ž

and

∈. Since ađ?œƒ , bđ?œƒ , by Definition 1, we have (a implies that there exists y ∈ a

x∈ đ?‘Ž

∈ đ?‘Ž

đ?‘? , which implies đ?‘Ž đ?‘?

đ?‘?

đ?‘?

đ?‘Ž

∈ đ?‘? such

b) đ?œƒ (

).

b such that đ?œƒđ?‘Ś. Therefore, we have đ?‘Ž

đ?‘? . Similarly, we can prove that

đ?‘Ž đ?‘? .

A hyper congruence relation đ?œƒ on (đ??ż, , â‹ ) is called -complete if đ?‘Ž

đ?‘? =đ?‘Ž

đ?‘?

for all a, b ∈ đ??ż. Similarly, đ?œƒ is called â‹ -complete if

đ?‘Ž đ?‘? đ?‘Ž đ?‘? for all a, b ∈ đ??ż. We call đ?œƒ complete if it is both -complete and â‹ -Complete. Now, we briefly recall the rough set theory in Pawlak’s sense.

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Let đ?œƒ be an equivalence relation on đ??ż, and let đ??´ be a nonempty subset of đ??ż.

Then, the sets đ?œƒ(đ??´) = { ∈ đ??ż | đ??´ } and đ?œƒ(đ??´) = {x ∈ L | ⊆ đ??´} are called, respectively, the upper and lower approximations of đ??´ with respect to đ?œƒ. đ?œƒ(đ??´) = (đ?œƒ (đ??´), đ?œƒĚ… đ??´ ) is called a rough set with respect to đ?œƒ.

Proposition 3. Let đ?œƒ be a hyper congruence on a hyperlattice (đ??ż, ,â‹ ). If đ??´, đ??ľ are two nonempty subsets of đ??ż, then (i)

đ?œƒĚ…(A)

đ?œƒĚ…(B)

đ?œƒĚ… (A B). In particular, if đ?œƒ is a

complete, then đ?œƒĚ…(A)

đ?œƒĚ… (B)

đ?œƒĚ…(A B).

(ii)

đ?œƒĚ…(A)

đ?œƒĚ…(B)

đ?œƒĚ…(Aâ‹ B). In particular, if đ?œƒ is a

complete, then đ?œƒĚ…(A)

đ?œƒĚ…(B)

đ?œƒĚ…(Aâ‹ B).

Proof: Suppose that x∈ đ?œƒĚ… (A) đ?‘Ž

đ?‘?

đ?œƒĚ… (B). There exist 1 ∈ đ?œƒĚ… (A) and 2 ∈ đ?œƒĚ… (đ??ľ) such that ∈ 1

It follows that there exists a, b ∈ L such that a ∈ ⊆ by lemma 2.

On the other hand, since đ?‘Ž (A B), which implies ∈ 1

đ?‘?

2

A B, we obtain đ?‘Ž

∈ đ?‘Ž

and

đ?œƒĚ…(A B). Therefore đ?œƒĚ…(A)

∈ đ?‘?

It follows that a ∈ A and b ∈ Ě… Ě… Ě… Therefore, đ?œƒ(A) đ?œƒ (B) = đ?œƒ(A B).

B. Since đ?œƒ is a hyper congruence on đ??ż, we have

đ?‘? đ?œƒĚ… (B)

If đ?œƒ is complete, let x ∈ đ?œƒĚ…(A B), then (A B) have y ∈ đ?‘Ž đ?‘?. Since đ?œƒ is a complete, we can obtain x ∈ đ?‘Ś Thus, there exists

A and b ∈

2.

đ?œƒĚ…(A B).

. Therefore, there exists y đ?‘Ž đ?‘? = đ?‘Ž đ?‘? .

such that x ∈

(A B), and so for some a ∈ A and b ∈ B, we

. ∈ đ?œƒĚ… (A) and

B. Hence,

∈ đ?œƒĚ… (B), and we have x ∈

đ?œƒĚ…(A)

đ?œƒĚ… (B).

(2) is similar to that of (1).

Proposition 4: Let đ?œƒ be a hyper congruence on a hyperlattice (đ??ż, ,â‹ ) and đ??´, đ??ľ are two nonempty subsets of đ??ż, then (i)

If A and B are two -hyperideals of L, then đ?œƒĚ…(A)

đ?œƒĚ… (B)

đ?œƒĚ…(A B).

(ii)

If A and B are two -hyperideals of L, then đ?œƒĚ…(A)

đ?œƒĚ…(B)

đ?œƒĚ…(A B).

(1) Let x ∈ đ?œƒĚ…(A B), then there exist a ∈ A and b ∈ B such that (a b) , which implies that there exists t ∈ đ?‘Ž such that x đ?œƒ t. Since A is a -hyperideal of L, we have a b A. It follows that t ∈ A. Hence, we obtain that A= A , which implies x ∈ đ?œƒĚ…(A). In a similar way, we have x ∈ đ?œƒĚ…(B). Thus, x ∈ x x đ?œƒĚ…(A) đ?œƒĚ… (B).

Combining proposition 3, we have đ?œƒĚ…(A)

đ?‘?

đ?œƒĚ… (B) = đ?œƒĚ…(A B).

(2) The proof is similar to that of (1).

Proposition 5: Let đ?œƒ be a hypercongurence relation on a hyperlattice (L, , â‹ ). If A and B are then đ?œƒĚ…(A B) = đ?œƒĚ…(A) đ?œƒĚ… (B).

-hyperideals (â‹ -hyperideals) of L,

Proof: Let x ∈ đ?œƒĚ…(A) đ?œƒĚ…(B), we have A and B . Then , there exist ∈ A and ∈ B such that đ?œƒ and đ?œƒ .It follows from đ?œƒ which is a hyper congruence relation that đ?œƒĚ… x x, which implies that there exists t ∈ such that t đ?œƒ . Since A and B are -hyperideals of L, we have A B. So, t ∈ A B. It follows that (A B) = (A B) , which implies x ∈ đ?œƒĚ…(A B). Hence, đ?œƒĚ…(A) đ?œƒĚ…(B) đ?œƒĚ…(A B). On the other hand, it is clear that đ?œƒĚ…(A B) đ?œƒĚ… (A) đ?œƒĚ…(B). Therefore, đ?œƒĚ…(A B) = đ?œƒĚ…(A) đ?œƒĚ…(B). In a similar way, if A and B are â‹ -hyperideals of L, we can also obtain đ?œƒĚ…(A B) = đ?œƒĚ…(A) đ?œƒĚ…(B).

Next, we will introduce and investigate a new algebraic structure called rough hyperideals in join hyper lattices. Let us begin with introducing the following definitions.

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Definition 6: Let đ?œƒ be a hypercongruence on a hyperlattice (L, , â‹ ), and let A be a non empty subset of L. A is called a lower (an upper)rough sub hyperlattice of L if đ?œƒ (A)(đ?œƒĚ…(A)) is a sub hyperlattice of L. A is called a rough sub hyperlattice of L if A is both a lower rough sub hyperlattice and an upper rough sub hyperlattice of L. Similarly, A is called a lower (an upper) rough -hyperideal of L if (A)(đ?œƒĚ…(A)) is a -hyperideal of L. And we call A as rough -hyperideal of L if A is both a lower rough -hyperideal and an upper rough -hyperideal of L. In a similar way, a rough â‹ -hyperideal of L can be defined.

Example 7: Let L = {a, b, c, d} be the hyperlattice. Let đ?œƒ be a hyper congruence relation on the hyperlattice L with the following equivalent classes: đ?‘Ž = {a, b}, đ?‘? = {c, d}. Considering A = {a, b, c}, we can obtain that (A) = {a, b} đ?œƒĚ…(A) = L. Notice that {a, b} and L are -hyperideals, so A is a rough -hyperideal of L. If A = {b, c, d}, we have that (A) = {c, d} and đ?œƒĚ…(A) = L. we obtain that {c, d} and L are â‹ -hyperideals, so A is a rough â‹ -hyperideal of L. Example 8: In example 7, A = {a, b, c} is a rough -hyperideal of (L,

â‹ ), but A is not a -hyperideal of L.

Conclusion: Hence, we have successfully introduced the Rough hyperideals in join hyperlattice. And we investigated some of their properties.

Reference: [1] https://www.researchgate.net/publication/340109245_Fuzzy_Soft_Hyperideals_In_Join_Hyperlattices [2]https://www.researchgate.net/search?context=publicSearchHeader&q=

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