Vimala and Bharathi
Jamal Academic Research Journal: An Interdisciplinary Special Issue(February 2016), pp: 245-250 ISSN 0973 - 0303
FUZZY l-FILTERS IN LATTICE ORDERED GROUP J. Vimala1* and P. Bharathi2 *
Correspondence:
[email protected] 1 Alagappa University, Karaikudi, TamilNadu, INDIA Full list of author information is available at the end of the article † Equal contributor
Abstract This Paper investigates the concepts of fuzzylfilters in lattice ordered group. We obtain a characterization theorem for fuzzyl-filters. Also some important results and properties on fuzzyl-filters are discussed. Keywords: lattice ordered group, fuzzyl-filter, levell-filter. 2010 AMS Classification: 06B10, 06F15, 06A06, 06D72.
1 Introduction he concept of fuzzy set was introduced in 1965 by l.A.Zadeh[13]. The main problem in fuzzy mathematics is how to carry out the ordinary concepts to the fuzzy case. Rosenfeld[10] applied it to group theory and developed the theory of fuzzy groups. Then several algebraists took interest in the study of fuzzy algebras of various algebraic structures and the concepts of fuzzy sub lattice, fuzzy ideal, fuzzy prime ideal in lattice were introduced by many authors. U.M.Swamy and others [ 12] studies various fuzzy algebraic systems and gave some interesting results on the theory of fuzzy ideals & fuzzy filters of a lattice.
T
The partially ordered algebraic systems play an important role in algebra. Some important concepts in partially ordered systems are lattice ordered groups and lattice ordered rings. These concepts play a major role in many branches of Algebra. So the study of fuzzy set theory of lattice ordered algebraic structures in order to get a better insight in fuzzy algebra is necessary. G.S.V.SathyaSaibaba[11] considered fuzzy lattice ordered group as a mapping from lattice ordered group into a complete lattice and he introduced the notions of l-fuzzy l-ideal of lattice ordered group. Our main aim in this paper is to introduce and study the new sort of fuzzy l-filters and level fuzzy l-filters of lattice ordered group. Also characterization theorem for fuzzy l-filters is established. Some more results related to this topic are also derived.
2 Preliminaries Definition 2.1 A non-empty set G is called a l-group(lattice ordered group) iff (i) (G, +) is a group. (ii) (G, ≤) is a lattice. (iii) x ≤ y implies a + x + b ≤ a + y + b for all a, b, x, y ∈ G. Definition 2.2 A non-empty set G is called a l-group iff (i) (G, +) is a group. (ii) (G, ∨, ∧) is a lattice. (iii) a + (x ∨ y) = (a + x) ∨ (a + y) and a + (x ∧ y) = (a + x) ∧ (a + y) for alla, b, x, y ∈ G. Result 2.3 The above two definitions of l-group are equivalent.
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Definition 2.3 let G be a l-group. A non-empty subset I of G is called an l-ideal of G if (i) I is a subgroup of G. (ii) I is a sub lattice of G (iii) 0 < x < a and a ∈ I ⇒ x ∈ 1 Definition 2.4 A fuzzy set is a pair (X, µ), where X is any non empty set and µ : X → [0, 1]. Definition 2.5 let µbe a fuzzy set on a non empty set X and t ∈ [0, 1]. Then the set µt = {x ∈ X/µ (x) ≥ t} is called the level set ofµ. Definition 2.6 letµbe a fuzzy set on a non empty set X. Then the set {x ∈ X/µ (x) > 0} is called the support of µ and it is denoted by Supp(µ). Definition 2.7 let G be a lattice ordered group. A fuzzy set µ of G is said to be fuzzy l-ideal of G if (i) µ (x − y) ≥ µ (x) ∧ µ (y) (ii) µ (x ∨ y) ≥ µ (x) ∧ µ (y) (iii) µ (x ∧ y) ≥ µ (x) ∧ µ (y) (iv) 0 < x < a ⇒ µ (x) ≥ µ (a) for all x, y, a, b ∈ G. Definition 2.8 let G be a lattice ordered group. For a fuzzy l-ideal µ of G, the set µt = {x ∈ G/µ (x) ≥ t}is an level l-ideal of G for all t ∈ [0, 1]. Proposition 2.1 Every constant function of a lattice ordered group G is a fuzzy `-ideal of G. Proposition 2.2 let G be a lattice ordered group. If µis a fuzzy`-ideal of G then Supp (µ)is an `-ideal of G if Supp 6= φ. Now our main idea is to introduce fuzzyl-filters in lattice ordered groups and discuss their important properties.
3 Fuzzy l-filters in Commutative l-group Definition 3.1 let G be a lattice ordered group. A non-empty subset F of G is called an l-filter of G if (i) F is a subgroup of G. (ii) F is a sublattice of G (iii) 0 < x < a and x ? F implies a ? F. Definition 3.2 let G be a lattice ordered group. A fuzzy set µ of G is said to be fuzzy l-filter of G if (i) µ (x − y) ≥ µ (x) ∧ µ (y) (ii) µ (x ∨ y) ≥ µ (x) ∧ µ (y) (iii) µ (x ∧ y) ≥ µ (x) ∧ µ (y) (iv) 0 < x < a ⇒ µ (x) ≤ µ (a) for all x, y,a ∈ G. Definition 3.3 let µ1 and µ2 be any two fuzzy `-filters of a lattice ordered group G. Then µ1 is said to be contained in µ2 denoted by µ1 ⊆ µ2 if µ1 (x) ≤ µ2 (x)for all x ∈ G. If µ1 (x) = µ2 (x)for all x ∈ Gthen µ1 and µ2 are said to be equal and we can writeµ1 = µ2 . Definition 3.4 The union of two fuzzy `-filters µ1 and µ2 of a lattice ordered group G S S denoted by (µ1 µ2 ) is a fuzzy subset of G defined by(µ1 µ2 ) (x) = max {µ1 (x) , µ2 (x)} for allx ∈ GThe intersection of two fuzzy `-filters µ1 and µ2 of a lattice ordered group G T T denoted by (µ1 µ2 ) is a fuzzy subset of G defined by (µ1 µ2 ) (x) = min {µ1 (x) , µ2 (x)} for all x ∈ G. Proposition 3.1 let µ1 and µ2 be any two fuzzy`-filters of a lattice ordered group G. If S T µ1 ⊆ µ2 then µ1 µ2 = µ2 and µ1 µ2 = µ1 . Proof: Assume thatµ1 ⊆ µ2 and let x ∈ G.Then µ1 (x) ≤ µ2 (x)for all x ∈ G. S We have (µ1 µ2 ) = max {µ1 (x) , µ2 (x)}= µ2 (x) ⇒ µ1
[
µ2 = µ2
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Also µ1
T
µ2 (x) = min {µ1 (x) , µ2 (x)}= µ1 (x)
⇒ µ1
\
µ2 = µ1 .
Proposition 3.2 letµ1 and µ2 be any two fuzzy `-filters of an lattice ordered group G. S T Thenµ1 µ2 ⊇ µ1 µ2 . Proof : lex ∈ G be arbitrary. S Then µ1 µ2 = max {µ1 (x) , µ2 (x)} ≥ min {µ1 (x) , µ2 (x)} \ = µ1 µ2 (x) [ \ ⇒ µ1 µ2 (x) ≥ µ1 µ2 (x) ⇒ µ1
[
µ2 ⊇ µ1
\
µ2 .
Definition 3.7 If µ1 and µ2 are any two fuzzy `-filters of a lattice ordered group G the join of µ1 and µ2 is defined by (µ1 ∨ µ2 ) (x) = sup {min {µ1 (y) , µ2 (z)}} , where x, y, z ∈ G. x=y∨z
The meet of µ1 and µ2 is defined by (µ1 ∧ µ2 ) (x) = sup {min {µ1 (y) , µ2 (z)}} , where x=y∧z
x, y, z ∈ G. Proposition 3.3 If µ1 and µ2 are any two fuzzy l-filters of a lattice ordered group G then T µ1 ∧ µ2 = µ1 µ2 . Proof: By definition, µ1 ∧ µ2 (x) = sup {min {µ1 (y) , µ2 (z)}} x=y∧z
\ ≥ min {µ1 (y) , µ2 (z)} = µ1 µ2 (x)
\ ⇒ (µ1 ∧ µ2 ) (x) ≥ µ1 µ2 (x) T Again (µ1 µ2 ) (x) = min {µ1 (x) , µ2 (x)} let x = a ∨ b. We have a ∨ b ≥ a and a ∨ b ≥ b. ⇒ x ≥ a and x ≥ b. Since µ1 is a fuzzy l-filter, µ1 (x) ≥ µ1 (a) and µ2 is a fuzzy l-filter ⇒ µ2 (x) ≥ µ2 (b) \ ⇒ µ1 (a) ∧ µ2 (b) ≤ µ1 (x) ∧ µ2 (x) = µ1 µ2 (x)
\ ⇒ µ1 µ2 (x) ≥ min {µ1 (a) , µ2 (b)} \ ⇒ µ1 µ2 (x) ≥ sup {min {µ1 (a) , µ2 (b)}} = (µ1 ∧ µ2 ) (x)
(1)
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\ (2) ⇒ µ1 µ2 (x) ≥ (µ1 ∧ µ2 ) (x) T From (3) and (2) µ1 ∧ µ2 = µ1 µ2 . Proposition 3.4 If µ is a fuzzy l-filter of a lattice ordered group G then µα : G ?[0,1] defined by µα (x) = (µ(x))α ( α ? [0,1] ) is also a fuzzy l-filter. Proof µα (x-y) = (µ(x-y))α = (µ(x) λ µ(y))α = (µ(x)) α λ (µ(y)) α = µα (x) λ µ α (y) ? µα (x-y) = µα (x) λ µ α (y) Similarly µα( x λ y) = µα (x) λ µ α (y) and µα (x V y) = µα (x) λ µ α (y). Now let x, y ? G such that x = y. Since µ is a fuzzy l-filter µ(x) = µ(y). ?µα (y) = (µ(y)) α = (µ(x)) α = µα (x) . ? µα (x) = µα (y). Hence µα is also a fuzzy l-filter. Proposition 3.5 (Characterization Theorem for fuzzy l-filters) let G be a lattice ordered group. A fuzzy set µof G is a fuzzy l−filter of G if and only if the set µt = {x ∈ G/µ (x) ≥ t}is an l-filter of G for all t ∈ [0, 1]. Proof : let µbe a fuzzy l-filter of a lattice ordered group G and let µt = {x ∈ G/µ (x) ≥ t} To prove µt is an l-filter of G. let x, y ∈ µt be arbitrary. ⇒ µ (x) ≥ t and µ (y) ≥ t − −− Now µ (x − y) ≥ min {µ (x) , µ (y)} ≥ t by (3) ⇒ µ (x − y) ≥ t
⇒ x − y ∈ µt . Similarly µ (x ∨ y) ≥ min {µ (x) , µ (y)} ≥ t and µ (x ∧ y) ≥ min {µ (x) , µ (y)} ≥ t
⇒ x ∨ y, x ∧ y ∈ µt let x ∈ µt such that 0 < x < a x ∈ µt ⇒ µ (x) ≥ t
x < a ⇒ t ≤ µ (x) ≤ µ (a) ⇒ µ (a) ≥ t ⇒ a ∈ µt Hence µt is an l-filter of G. Conversely letµt is an l-filter of G. To prove µ is a fuzzy l-filter of G, let min {µ (a) , µ (b)} = r Either µ (a) = r, µ (b) = r
(3)
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i.e., Either µ (b) ≥ µ (a) or µ (a) ≥ µ (b) ⇒ µ (a) ≥ r and µ (b) ≥ r
⇒ a, b ∈ µr ⇒ a − b, a ∧ b, a ∨ b ∈ µr (Since µt is an l-filter) ⇒ µ (a − b) ≥ r, µ (a ∧ b) ≥ r and µ (a ∨ b) ≥ r let a, x ∈ G such that 0 < x < a and let µ (x) = t. ⇒ x ∈ µt ⇒ a ∈ µt (Since µt is an l-filter). ⇒ µ (a) ≥ t = µ (x) ⇒ µ (x) ≤ µ (a). Hence µt is a fuzzy l-filter of G. Definition 3.5 let G be a lattice ordered group. For a fuzzy l-filter µ of G, the set µt = {x ∈ G/µ (x) ≥ t}is an level l-filter of G for all t ∈ [0, 1].
Proposition 3.6 If µ is a fuzzy l-filter of a lattice ordered group G and t,s ? [0,1] then µ t = µ s iff t=s. Proof If t=s then clearly µt = µs . Conversely let µt = µs . Since t ? [0,1] ,there exist some x ? G such that µ(x) = t. ? x ? µt . ?x ? µt = µs . ? x ? µs . ?µ (x) =s ?t=s Similarly s = t. Hence t=s. Author details 1 Alagappa University, Karaikudi, TamilNadu, INDIA. Dt., TamilNadu, INDIA.
2
Sri Sarada Niketan College for Women, Amaravathipudur,Sivagangai
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