1.
1|P ag eContents. Definition 1.7[3]: Let U be an initial The concept of lattices. universe set and E be a set of Basic Definitions regarding soft sets parameters. Let P (U) denotes the . power set of U and A be a non-empty Examples showing soft set and subset of E. A pair (F, A) is called a lattices. soft set over U, where F is a mapping Introduction to soft lattices . given by F: AP (U). Certain results and theorems .1. THE CONCEPT OF LATTICES. Definition 1.8[3]: For two soft setsDefinition 1.1[1]: A lattice L is a (F,A) & (G, B) over a commonpartially ordered set in which every pair universe U, we say that (F,A) is a softof elements has the least upper bound subset of (G, B) if;(∨-join) and the greatest lower bound (1). A B, and.(∧-meet). (2). A, F () and G() areIt is denoted by (L, ∨, ∧). identical approximations. ( F ()⊆ G() ) .Definition 1.2[1]: A nonempty subset We write (F, A) ⊆ (G, B).R of L is said to be a sub-lattice of Lif,a, b ∈ R implies a ∨ b, a ∧ b ∈ R. Definition 1.9[3]: Two soft sets (F,A) and (G, B) over a common universe UDefinition 1.3[1]: A nonempty subset I are said to be soft equal if (F, A) is aof L is said to be an ideal of L if, soft subset of (G, B) and (G, B) is a soft(1) a, b ∈ I implies a ∨ b ∈ I, and subset of (F, A).(2) for any a, b ∈ L such that a ≤ b and i.e. (F, A) ⊆ (G, B) and (G, B) ⊆ (F, A).b ∈ I implies a ∈ I Definition 1.10[3]: Union of two softDefinition 1.4[2]: A lattice (L, ∨, ∧) is sets (F1, E1) and (F2, E2) over thesaid to be distributive if; common universe U is the soft set (F3,x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and E3).where E3 = E1 E2, and e E3.x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z);x,y,z∈ L. F1(e) if e E1-E2 F3(e)= F2(e) if e E2-E1Definition 1.5[2]: A lattice (L, ∨, ∧) is F1(e) F2(e) if e E1E2said to be modular if ; for all y in L, Write as, (F1, E1) (F2, E2)= (F3, E3).x ≤ z x ∨ (y ∧ z) = (x ∨ y) ∧ z. Definition 1.11[3]: Intersection of twoDefinition 1.6[2]: A mapping f of a soft sets (F1,E1) and (F2, E2) over alattice (L1, ∨1, ∧1) into a lattice (L2, ∨2, common universe U is the soft set (F3,∧2) is said to be a lattice E3). where E3 = E1 E2 ; and e E3,homomorphism if; F3(e)=F1(e) F2(e)f (a ∨1 b) = f (a) ∨2 f (b) and Write as,(F1, E1) (F2, E2) = (F3, E3).f (a ∧1 b) = f (a) ∧2 f (b) ; a, b in L1. If such a mapping is one-one Definition 1.12[4]: Let (F, A) and (G,and onto, then it is called as a lattice B) be two soft sets over the commonisomorphism. universe U . Then (F, A) AND (G, B)
2.
2|P ag edenoted by (F, A) ∧ (G, B) and is 6. (F, E) ⋂ ((G, E) ⋃ (H, E)) = ((F, E)defined by; ⋂ (G, E)) ⋃ ((F, E) ⋂(H, E)).(F, A) ∧ (G, B) = (H, A × B), where, 7. (F, E) ⋃ ((F, E) ⋂(G, E)) = (F, E).H(α, β) = F (α) ∩ G(β), (α, β) ∈A×B. 8. (F, E) ⋂ ((F, E) ⋃(G, E)) = (F, E). 9. (F, E) ⋃ Φ = (F, E) and (F, E) ⋂ ΦDefinition 1.13[4]: Let (F, A) and (G, = Φ.B) be two soft sets over the common 10. (F, E) ⋃ A = A and (F, E) ⋂ A = (F,universe U. Then (F, A) OR (G, B) E).denoted by (F, A) ∨ (G, B) and is 11. (F, E) ⋃(F c, E) = anddefined by; (F, E) ⋂(F c, E) = Φ(F, A) ∨ (G, B) = (H, A × B) where,H ((α, β)) = F (α) ∪ G(β),(α, β)∈A×B. 2. INTROCTION TO S.LATTICES.Definition 1.14[4]: The complement of L- Lattice, A- any nonempty set.a soft set (F, A) is denoted by (F, A)C A set-valued function F: A → ℘(L)and is defined by,(F, A)C = (FC, ךA); can be defined as, where FC: ךA→P(U) is a F (x) = {y ∈ L/ x R y}.mapping given by FC(ךα)=U- F(α); The pair (F, A) is a soft set over L.α ךA. Definition 2.1[6]: Let (F, A) be a softDefinition 1.15 [4]: A soft set (F, A) set over L. Then (F, A) is said to be aover U is said to be a null soft set soft lattice over L if F (x) is a sub-denoted by ∅ lattice of L, for all x ∈ A. if, A, F ( ) (null set). Let triplet M = (f, X, L) ,where L is a complete lattice , f : X → L isDefinition 1.16 [4]: A soft set (F, A) a map-ping, X is a universe set, then Mover U is said to be absolute soft set is called the soft lattice .denoted by A if A, F ( ) U. The set of all soft lattices over Lclearly, A c & c A is denoted by Sl (L).Lemma 1.1[2]:The operation ” ∪ ” in Definition 2.2[6]: A soft lattice (f, X,S(U ) is idempotent and associative. L) is said to be null soft lattice, if ∀ǫ ∈ X, f(ǫ) = 0.Lemma 1.2[2]: In SE (U), thefollowing holds Definition 2.3[6]: A soft lattice (f, X,1. (F, E) ⋃ (F, E) = (F, E) and L) is said to be absolute soft lattice, if (F, E) ⋂ (F, E) = (F, E). ∀ǫ ∈X, f(ǫ) = 1.2. (F, E) ⋃ (G, E) = (G, E) ⋃ (F, E) & (F, E) ⋂ (G, E) = (G, E) ⋂(F, E). Example 2.1[2]: Consider the lattice L3. (F, E) ⋃ ((G, E) ⋃ (H, E)) = ((F, E) as shown in Figure. Let A = {a, b, d}. ⋃ (G, E)) ⋃ (H, E). F (x) = {y∈ L: x R y ⇔ x ∨y = 1}.4. (F, E) ⋂ ((G, E) ⋂ (H, E)) = ((F, E) ⋂ (G, E)) ⋂ (H, E). Then F (a) = {1, f },5. (F, E) ⋃ ((G, E) ⋂ (H, E)) = ((F, E) F (b) = {1, e}, F (d) = {1, c, e, f }. ⋃ (G, E)) ⋂ ((F, E) ⋃(H, E)).
3.
3|P ag eTherefore, F (x) is a sub-lattice of L, forall x ∈ A. Hence (F, A) ∈ Sl(L). Figure 3 Figure 1 Then F (0) = {0, a, b, c, d, 1}, F (a) = {0, b, c, d}, F (b) = {0, a, c}, F (c) = {0, a, b}, F (d) = { 0, a}, F (1) = {0}.Example 2.2[2]. Consider the lattice L Here, F (b) = {0, a, c},as shown in Figure. Let A = {0, a, b, c, F (c) = {0, a, b} are not sub-d, 1}. lattices of L.Define, Therefore, (F, A) ∉ Sl(L). TakeF(x)={y ∈ L: x R y ⇔ x ∨ y = 1}. B = {0, a, d, 1} ⊂ A.Then F (0) = {1}, F (a) = {1, Then FB (0) = {0, a, b, c, d, 1},b, c, d}, F (b) = {1, a, c, d}, FB (a) = {0, b, c, d}, FB (d) = {0, a}, FB (1) = {0}. Therefore, (F /B, B) ∈F (c) = {1, a, b, d}, F (d) = Sl(L).{1, a, b, c}, F (1) = {0, a,b, c, d, 1}. Proposition 2.1[2]. Every lattice canIn F (a), b, c ∈ F (a). be considered as a soft lattice.But b ∧ c = 0 ∉ F (a). Proposition 2.2[2]. Let (F, A), (H, B) Therefore, F (a) is not ∈ Sl(L) be such that A ∩ B ≠ ∅ anda sub-lattice of L. F (e) ∩ H (e) ≠ φ for all e ∈ A ∩ B. Similarly, F (b), F (c), Then their intersection;F (d) are not sub-lattices of L. (F, A) ∩ (H, B) ∈ Sl(L).Hence, (F, A) ∉ Sl(L). Proposition 2.3.[2]. Let (F, A), (G, B) ∈ Sl(L). If A ∩ B = ∅, then, (F, A) ∪ (G, B) ∈ Sl(L). Figure 2 Proposition 2.4.[23]. Let (F, A), (G, B) ∈ Sl(L) be such that F (x) ∩ G(y) = φ, for all x ∈ A, y ∈ B. Then (F, A) ⋀ (G, B) ∈ Sl(L). Definition 2.4.[3]. Let (F, A) and (H,Example 2.3 [2]. Consider the lattice K) be two soft lattices over L. Then (H,L as shown in Figure. Let A = {0, a, b, K) is a soft sub-lattice of (F, A) ifc, d, 1}. 1. K ⊆ A,Define, 2. H (x) is a sub-lattice of F (x), x ∈ KF (x) = { y ∈ L : x R y⇔ x ∧ y = 0}. .
4.
4|P ag eExample 2.4.[2]. Consider the lattice L A) be a soft sub-lattice of (H, B). If f isas shown in Figure 3. Let A = {0, a, b, a homomorphism from L1 to L2, thenc, d, 1}. (f (F ), A) is a soft sub-lattice of (fLet K = {0, b, c, d}. (H ), B).Define, F : A → ℘(L) by Definition 2.6[2]. Let (F, A) and (H, B)F (x) = {y ∈ L : x R y ⇔ x ∨ y = x}. be two soft lattices over L1 and L2 respectively.Then F (0) = {0}, F (a) = {0, a}, Let f : L1 → L2 and g : A → B.F (b) = {0, b}, F (c) = {0, c}, Then (f, g) is said to be a soft latticeF (d) = {0, b, c, d}, F (1) = {0, a, b, homomorphism if, f is a lattice homomorphism from L1 onto L2, g is ac, d, 1}. mapping from A onto B, Define the set-valued function f (F (x)) = H (g(x)), for all x ∈ A. Then (F, A) ∼ (H, B). H : K → P (L) byH (x) = {y ∈ L : x R y ⇔ x ∨ y = x, x Example 2.5[2]. Consider the lattice L1∈ K }. and L2 as shown in Figures . Then H (0) = {0}, H (b) = {0, b},H (c) = {0, c}, H (d) = {0, b, c, d}. Therefore, (F, A), (H, K ) ∈ Sl(L).Here K ⊆ A and H (x) is a sub-latticeof F (x), for all x ∈ K . Therefore, (H, K ) is a soft sub- Let A = {0, a, b, 1} and B = {0 ′, 1′}.lattice of (F, A). Define, F (x) = {y ∈ L1: x R y ⇔ x ∨ y= x, x ∈ A}.Proposition 2.5[2]. Let (F, A), (H, A) Then, (F, A) ∈ Sl(L1). Define the∈ Sl(L). Then (F, A) is a soft sub- set-valued function H by;lattice of (H, A) if and only if F (x) ⊆ H (x) = {y ∈ L2 : x R y ⇔ x ∨ y =H (x), for all x ∈ A. x, x ∈ B}.Corollary 2.1[2]. Every soft lattice is a Then (H, B) ∈ Sl(L2).soft sub-lattice of itself. Define f : L1 → L2 by f (0) = 0′, That is, if (F, A) is a soft lattice f (a) = 0′, f (b) = 1′, f (1) = 1′.over L, then (F, A) is a soft sub-lattice Define g : A → B by g(0) = 0′,of (F, A). g(a) = 0′, g(b) = 1′, g(1) = 1′. Then f is a latticeDefinition 2.5[2]. Let (F, A) be a soft homomorphism from L1 onto L2 and glattice over L1. Let f be a latticehomomorphism from L1 to L2. Then is a mapping from A onto B. (f (F ))(x) = f (F (x)), for all x ∈ A. Also f (F (x)) = H (g(x)), for all x ∈ A.Proposition 2.6[2]. Let (F, A) and (H, Hence (F, A) is a soft latticeB) be soft lattices over L1 such that (F, homomorphic to (H, B).
5.
5|P ag eDefinition 2.7[2]. Let (F, A) be a soft Then F (a) = {0, a}, F (b) = {0, b},set over L. Then (F, A) is said to be a F (c) = {0, c}, F (d) = {0, d}.soft distributive lattice over L if F (x) Here F (x) is a distributive Sub-latticeis a distributive sub-lattice of L, for all of L, for all x ∈ A. Therefore, (F, A) isx ∈ A. a soft distributive lattice over L. But L is not a distributive lattice.Example 2.6[2]. Consider the lattice Las shown in Figure . Let A = {a, b, c}. Theorem 2.2[2].Define F (x) ={y ∈ L : x R y ⇔ x ∨ y (1) Let (F, A) be a soft distributive=x}. lattice over L and (H, B) be a soft sub- lattice of (F, A). Then (H, B) is a soft distributive lattice over L. (2) Let (F, A) be a soft distributive lattice over L1. Let (H, B) be a soft lattice homomorphic image of (F, A) over L2. Then (H, B) is a softThen F (a) = {0, a}, F (b) = {0, a, b}, distributive lattice over L2.F (c) = {0, a, b, c}. Here F (x) is a distributive sub-lattice of Definition 2.8[2]. Let (F, A) be a soft set over L. Then (F, A) is said to be aL, for all x ∈ A. soft modular lattice over L if F (x) is aTherefore, (F, A) is a soft distributive modular sub-lattice of L, for all x ∈ A.lattice over L. Example 2.8[2]. Consider the lattice LProposition 2.7[2]. Let L be a as shown in figure.distributive lattice. Then every (F, A) ∈ Let A = {0, a, b, c, 1}.Define, F (x) =Sl (L) is a soft distributive lattice over {y ∈ L : x R y ⇔ x ∧ y = 0}.L. Then, F (0) = L, F (a) = {0, b, c, f }, F (b) = {0, a, c, e}, F (c) = {0, a, b, d},Remark 2.1[2]. The converse of the F (1) = {0}.above theorem is not true. Here F (x) is a modular sub-lattice of That is, if (F, A) is a soft L, for all x ∈ A. Therefore, (F, A) is adistributive lattice over L then L need soft modular lattice over L.not be a distributive lattice. The following example illustrates Proposition 2.8[2]. Let L be a modularthis remark. lattice. Then every (F, A) ∈ Sl (L) is a soft modular lattice over L.Example 2.7[2]. Consider the lattice L Remark 2.2[2]. The converse of theas shown in Figure 2. Let A = {a, b, above theorem is not true.c, d}. That is, if (F, A) is a softDefine, modular lattice over L then L need notF(x)= {y ∈ L : x R y ⇔ x ∧ y = y}. be a modular lattice. The following example illustrates this remark.
6.
6|P ag eExample 2.9[2]. Consider the lattice Las shown in Figure. Let A = {a, b, c, Then, F (0) = {0},d}. F (a) = {0, a}, F (b) = {0, a, b},Define, F (c) = {0, c}, F (1) = {0, a, b, c, 1}.F(x) = {y ∈ L : x R y ⇔ x ∧ y = y}. Therefore, (F, A) ∈ Sl(L).Then F (a) = {0, a}, F (b) = {0, b}, Let B = {a, b, c}.F (c) = {0, c}, F (d) = {0, b, c, d}. Define the set-valued functionHere F (x) is a modular sub-lattice of L, G byfor all x ∈ A. Therefore, (F, A) is a G(x) = {y ∈ L : xRy ⇔ x∧y = y}.soft modular lattice over L. But L is not Then G(a) = {0, a}, G(b) = {0, a, b},a modular lattice. G(c) = {0, c}. Therefore, (G, B) is a soft set over L. Here B ⊂ A.Theorem 2.4[2]. Also G(x) is an ideal of F (x), (1) Let (F, A) be a soft modular lattice for all x ∈ B.over L and (H, B) be a soft sub-lattice Therefore, (G, B) is a soft latticeof (F, A). Then (H, B) is a soft ideal of (F, A).modular lattice over L.(2) Let (F, A) be a soft modular latticeover L1. Let (H, B) be a soft latticehomo- morphic image of (F, A) overL2. Then (H, B) is a soft modular latticeover L2.Definition 2.9[2]. Let (F, A) be a softlattice over L. A soft set (G, B) over Lis called a soft lattice ideal of (F, A) if…..1. B ⊆ A, 2. G(x) is an ideal of F (x), for all x ∈ B.Example 2.10[2]. Consider L as shownin Figure. Let A = {0, a, b, c, 1}.Define,F (x) = {y ∈ L : x R y ⇔ x ∧ y =y}.
Be the first to comment