IRJET- Soft Hyperfilters in Hyperlattices

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1390
SOFT HYPERFILTERS IN HYPERLATTICES
R. Punitha1, S. Fathima Bazeela2
1Head Department of Mathematics, Thassim Beevi Abdul Kader College for Women, Kilakarai, Tamil Nadu, India.
2Student, Thassim Beevi Abdul Kader College for Women, Kilakarai, Tamil Nadu, India.
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Abstract - Firstly, hyper filters of hyperlatticesareintroduced
and several interesting examples of them are given. Secondly,
soft hyper filters are proposed, which are generalizations of
hyper filter sand soft hyper filters in hyperlattices. Finally,
under the soft homomorphism of hyperlattices, the image and
pre-image of soft hyper filters are studied.
Key Words: Filters, Softset, Hyperfilters and
Softhyperfilters
1. INTRODUCTION
The concept of hyperstructure was introduced in 1934 by a
French mathematician, Marty [1]. In a classical algebraic
structure, the composition of two elements is an element,
while in an algebraic hyperstructure, the compositionoftwo
elements is a set. There appeared many components of
hyperalgebras such as hypergroups in [2], hyperrings in [3]
etc. As soft set theory, the theory of fuzzy softsetsturnedout
to have applications. Roy and Maji [4] presented some
applications of this notion to decision-making problems, we
introduce hyperfilters soft hyperfilters in hyperlattices, and
study some properties of them
Definition 1.1:
Let X be a universe set and E be a set of parameters. Let
P(X) be the power set of X and A ⊆ E. A pair (F, A) is called a
soft set over X, where A is a subset of the set of parameters E
and F : A → P(X) is a set-valued mapping.
Example 1.2:
Let I = [0, 1] and E be all convenient parameter sets for
the universe X. Let X denote the set of all fuzzy sets on X and
A ⊆ E.
Definition 1.3:
A pair (f, A) is called a fuzzy soft set over X, where A is a
subset of the set of parameters E and f : A → I X is a mapping.
That is, for all a ∈ A, f(a) = fa : X → I is a fuzzy set on X.
Definition 1.4:
Let (L,≤) be a non empty partial ordered set and ∨ : L×L
→ ρ(L)∗ be a hyperoperation, where ρ(L) is a power set of L
and ρ(L)∗ = ρ(L){∅} and ∧ : L×L→L be an operation. Then
(L,∨,∧) is a hyperlattice if for all a,b,c ∈L,
(i)a ∈ a∨a,a∧a = a;
(ii)a∨b = b∨a,a∧b = b∧a;
(iii)(a∨b)∨c = a∨(b∨c);(a∧b)∧c = a∧(b∧c);
(iv) a ∈ [a∧(a∨b)]∩[a∨(a∧b)];
(v) a ∈ a∨b ⇒ a∧b = b;
where for all non empty subsets A and B of L, A∧B = {a∧b |a
∈ A,b ∈ B} and A∨B =∪{a∨b |a ∈ A,b ∈ B}.
Definition 1.5:
Let (L,∨,∧) is a hyperlattice. A Partial ordering relation≤
is defined on L by x ≤ y f and only if x ∧ y = x and x ∨ y = y .
Definition 1.6:
A Nonempty subset F of a Hyperlattice L is called a Filter
of L if (i)a ∧ b ∈ F and a∨x ∈ F
(ii a ∈ F and a ≤ b then b ∈ F
Properties of filters 1.7:
A Filter F of L is a subset F ⊆ L with the following
properties:
(i) 1 ∈ F
(ii) a ∈ F and a ≤ b , b ∈ L then b ∈ F
(iii)If a , b ∈ F then ab ∈ F
Definition 1.8:
Let (L,∨,∧) is a hyperlattice. For any x ∈ L the set {
x∈ L |a ≤ x} is a filter ,which is called as a principal filter
generated by a.
2. SOFT HYPERFILTERS IN HYPERLATTICES
In this section, we will introduce soft hyperfilters in
hyperlattices and give several interesting examples of them.
Definition 2.1:
Let L be a nonempty set and P∗(L) be the set of all
nonempty subsets of L. A hyperoperation on L is a map ◦ :
L×L → P∗(L), which associates a nonempty subset a◦b with
any pair (a,b) of elements of L×L. The couple (L,◦) is called a
hypergroupoid.

Definition 2.2:
Let L be a nonempty set endowed with two
hyperoperations ”⊗” and ”⊕”. The triple (L,⊗,⊕)iscalleda
hyperlattice if the following relations hold: for all a,b,c ∈ L,
(1) a ∈ a⊗a, a ∈ a⊕a;
(2) a⊗b = b⊗a, a⊕b = b⊕a;
(3) (a⊗b)⊗c = a⊗(b⊗c), (a⊕b)⊕c = a⊕(b⊕c)
(4) a ∈ a⊗(a⊕b), a ∈ a⊕(a⊗b).
Definition 2.3:
Let (L,⊗,⊕) be a hyperlattice and A be a non-empty
subset of L. A is called a ⊕-hyperfilter of L if for all a,b ∈ A
and x ∈ L,
(i) a⊕b ⊆ A and a⊗x ⊆ A
(ii) a ∈ A and a ≤ b then b ∈ A
Definition 2.4:
Let (L,⊗,⊕) be a hyperlattice and A be a non-empty
subset of L. A is called a ⊗-hyperfilter of L if for all a,b ∈ A
and x ∈ L,
(i) a⊗b ⊆ A and a⊕x ⊆ A
(ii) a ∈ A and a ≤ b then b ∈ A
We now introduce theorems of hyperfilters.
THEOREM 2.5:
Any hyperfilter A of a Hyperlattice L satisfies,If a A
and a ≤ b then b A
Proof:
Given (L, ) be a hyperlattice and A is a hyperflter of
L .Assume that for all a A and a≤b .
Take (ab)= 1 A imples ab A implies ab A so that b A
when a A.Hence proved.
Theorem 2.6:
In a hyperlattice(L, ) ,Every filter is a hyperfilter
Proof:
Given (L, ) be a hyperlattice and A be any filter of L
.Let a,b A .Take b(a b)=ba bb=ba 1=ba≥a
Implies that b(a b)≥a and b(a b) A implies that a b
A ;similarly when x L implies a x A(By the previous
theorem) For all a A and a≤b implies thar b A.
From the above two result A is a hyperfilter .Hence
proved
Definition 2.7:
Let (L, ) be a hyperfilter and (F,A) be a softset over
L,(F,A) is called a soft hyperfilter over L,if F(x) is
hyperfilter of L for all x sup (F,A)
Definition 2.8:
Let (L, ) be a hyperfilter and (F,A) be a softset over L,
(F,A) is called a soft hyperfilter over L,if F(x) is
hyperfilter of L for all x sup (F,A)
Example2.9:
Let µ be a fuzzy hyperfilter of a hyperlattice(L, , ),the
fuzzyset of µ satisfies the following condition:
For all x ,y L (i) µ(z) ≥ µ(x)∧µ(y)
(ii) ⋀ µ(z) ≥ µ(x)∨µ(y)
Clearly µ is a fuzzy hyperfilter of L.if and only if for
all t [0,1] with ≠ 0 .Let = { x L |µ(x) ≥t } is a
hyperfilter of L. and F(t) = { x L |µ(x) ≥t } for all t
[0,1] and F(t) is a hyperfilter of L.
Note:
Every fuzzy hyperflter( hyperilter)canbeintrepreted
as soft hyperfilter( hyperfilter)
Theorem 2.10:
If (L, ) is a hyperlattice and (F,L) denote a softset
over L,then the following conditions hold:
(i) (F,L) is a soft hyperfilter of L
(ii) (F’ , L) is a soft hyperfilter of L
Proof:
(i) By hypothesis, (L, ) be a hyperlattice .so
clealy (L,∧,∨)be a lattice.Now definetwohyperoperationson
L .For all a , b L therefore,a b = { x L | a ∨ x = b ∨x = a
∨b } and a b = { x L | x ≤ a ∧b }
For all a L ,define a principal filter generated by a ,I(a) = {x
L | x ≤ a } = a. Hence I(a) is a hyperfilter of the

hyperlattice L. Now define a map F: L P(L) by , F(a) = I(a)=
a for all a L(By definition of Softset)then (F,L) becomes a
soft hyperfilter over L.
(ii) By hypothesis, (L, ) be a hyperlattice .so clealy
(L,∧,∨)be a lattice.Now define two hyperoperationsonL.For
all a , b L a ⊗ b = { x L | a ∨ b ≤ x } and a b = { x L |
a∧x= b∧x=a∧b}
For all a L , define a principal filter generated by a ,F(a) =
{ x L | x ≥ a } = a. Hence F(a) is a hyperfilter of the
hyperlattice L. Now define a map F’: L P(L) by ,F’(a) = F(a)
= a for all a L(By definition of Softset)then (F’,L) becomes
a soft hyperfilter over L.
Hence proved.
3. CONCLUSIONS
In this paper, we apply the notion of soft sets to the theoryof
hyperlattices. We introduce hyperfilters and soft
hyperfilters, and study some properties of them. This study
is just at the begining and it can be continuated in many
directions:
(1) To do some further work on the properties of soft
hyperfilters, which may be useful to characterize the
structure of hyperlattices;
(2) To study the construction the quotient hyperlattices in
the mean of soft structures and soft hyperfilters theorems
of hyperlattices;
(3) To apply the soft set theory of hyperlattices to some
applied fields, such as decision making, data analysis and
forecasting and so on.
REFERENCES
[1] F. Marty, Sur une generalization de la notion de groupe,
in: 8th Congress Math.Scandinaves,Stockholm,1934,pp. 45-
49.
[2] J. Jantosciak, Transposition hypergroups:
noncommutative join spaces, Journal of Algebra,187(1977)
97-119.
[3] R. Rosaria, Hyperaffine planes over hyperrings, Discrete
Mathematics, 155 (1996) 215-223.
[4] A.R. Roy, P.K. Maji, A fuzzy soft set theoretic approach to
decision making problems, J. Comput. Appl. Math., 203
(2007) 412-418.

IRJET- Soft Hyperfilters in Hyperlattices

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