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Contents. 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 sets
Definition 1.1[1]: A lattice L is a (F,A) & (G, B) over a common
partially ordered set in which every pair universe U, we say that (F,A) is a soft
of 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() are
It 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 L
if,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 U
Definition 1.3[1]: A nonempty subset I are said to be soft equal if (F, A) is a
of 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 soft
Definition 1.4[2]: A lattice (L, ∨, ∧) is sets (F1, E1) and (F2, E2) over the
said 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-E1
Definition 1.5[2]: A lattice (L, ∨, ∧) is F1(e) F2(e) if e E1E2
said 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 two
Definition 1.6[2]: A mapping f of a soft sets (F1,E1) and (F2, E2) over a
lattice (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 common
isomorphism.
universe U . Then (F, A) AND (G, B)](https://image.slidesharecdn.com/reportsem1-111103072844-phpapp01/85/Report-sem-1-1-320.jpg)
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denoted 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) ⋂ Φ
Definition 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) = and
defined 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 soft
Definition 1.15 [4]: A soft set (F, A) set over L. Then (F, A) is said to be a
over 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 is
Definition 1.16 [4]: A soft set (F, A) a map-ping, X is a universe set, then M
over 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 L
clearly, 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), the
following 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 L
3. (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)).](https://image.slidesharecdn.com/reportsem1-111103072844-phpapp01/85/Report-sem-1-2-320.jpg)
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Therefore, F (x) is a sub-lattice of L, for
all 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). Take
F(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 can
In 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 ≠ ∅ and
a 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) if
c, d, 1}.
1. K ⊆ A,
Define,
2. H (x) is a sub-lattice of F (x), x ∈ K
F (x) = { y ∈ L : x R y⇔ x ∧ y = 0}.
.](https://image.slidesharecdn.com/reportsem1-111103072844-phpapp01/85/Report-sem-1-3-320.jpg)
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Example 2.4.[2]. Consider the lattice L A) be a soft sub-lattice of (H, B). If f is
as shown in Figure 3. Let A = {0, a, b, a homomorphism from L1 to L2, then
c, d, 1}. (f (F ), A) is a soft sub-lattice of (f
Let 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 lattice
F (d) = {0, b, c, d}, F (1) = {0, a, b, homomorphism if, f is a lattice
homomorphism from L1 onto L2, g is a
c, 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) by
H (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-lattice
of 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 lattice
Definition 2.5[2]. Let (F, A) be a soft
homomorphism from L1 onto L2 and g
lattice over L1. Let f be a lattice
homomorphism 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 lattice
B) be soft lattices over L1 such that (F, homomorphic to (H, B).](https://image.slidesharecdn.com/reportsem1-111103072844-phpapp01/85/Report-sem-1-4-320.jpg)
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Definition 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-lattice
is a distributive sub-lattice of L, for all of L, for all x ∈ A. Therefore, (F, A) is
x ∈ A. a soft distributive lattice over L. But L
is not a distributive lattice.
Example 2.6[2]. Consider the lattice L
as 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 soft
Then 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 a
L, for all x ∈ A.
soft modular lattice over L if F (x) is a
Therefore, (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 L
Proposition 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 a
distributive 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 modular
this 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 the
as shown in Figure 2. Let A = {a, b, above theorem is not true.
c, d}. That is, if (F, A) is a soft
Define, modular lattice over L then L need not
F(x)= {y ∈ L : x R y ⇔ x ∧ y = y}. be a modular lattice. The following
example illustrates this remark.](https://image.slidesharecdn.com/reportsem1-111103072844-phpapp01/85/Report-sem-1-5-320.jpg)
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Example 2.9[2]. Consider the lattice L
as 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 function
Here F (x) is a modular sub-lattice of L, G by
for 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 lattice
of (F, A). Then (H, B) is a soft ideal of (F, A).
modular lattice over L.
(2) Let (F, A) be a soft modular lattice
over L1. Let (H, B) be a soft lattice
homo- morphic image of (F, A) over
L2. Then (H, B) is a soft modular lattice
over L2.
Definition 2.9[2]. Let (F, A) be a soft
lattice over L. A soft set (G, B) over L
is 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 shown
in Figure. Let A = {0, a, b, c, 1}.
Define,
F (x) = {y ∈ L : x R y ⇔ x ∧ y =
y}.](https://image.slidesharecdn.com/reportsem1-111103072844-phpapp01/85/Report-sem-1-6-320.jpg)
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- 1. 1|P ag e Contents. 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 sets Definition 1.1[1]: A lattice L is a (F,A) & (G, B) over a common partially ordered set in which every pair universe U, we say that (F,A) is a soft of 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() are It 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 L if,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 U Definition 1.3[1]: A nonempty subset I are said to be soft equal if (F, A) is a of 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 soft Definition 1.4[2]: A lattice (L, ∨, ∧) is sets (F1, E1) and (F2, E2) over the said 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-E1 Definition 1.5[2]: A lattice (L, ∨, ∧) is F1(e) F2(e) if e E1E2 said 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 two Definition 1.6[2]: A mapping f of a soft sets (F1,E1) and (F2, E2) over a lattice (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 common isomorphism. universe U . Then (F, A) AND (G, B)
- 2. 2|P ag e denoted 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) ⋂ Φ Definition 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) = and defined 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 soft Definition 1.15 [4]: A soft set (F, A) set over L. Then (F, A) is said to be a over 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 is Definition 1.16 [4]: A soft set (F, A) a map-ping, X is a universe set, then M over 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 L clearly, 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), the following 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 L 3. (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 e Therefore, F (x) is a sub-lattice of L, for all 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). Take F(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 can In 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 ≠ ∅ and a 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) if c, d, 1}. 1. K ⊆ A, Define, 2. H (x) is a sub-lattice of F (x), x ∈ K F (x) = { y ∈ L : x R y⇔ x ∧ y = 0}. .
- 4. 4|P ag e Example 2.4.[2]. Consider the lattice L A) be a soft sub-lattice of (H, B). If f is as shown in Figure 3. Let A = {0, a, b, a homomorphism from L1 to L2, then c, d, 1}. (f (F ), A) is a soft sub-lattice of (f Let 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 lattice F (d) = {0, b, c, d}, F (1) = {0, a, b, homomorphism if, f is a lattice homomorphism from L1 onto L2, g is a c, 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) by H (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-lattice of 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 lattice Definition 2.5[2]. Let (F, A) be a soft homomorphism from L1 onto L2 and g lattice over L1. Let f be a lattice homomorphism 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 lattice B) be soft lattices over L1 such that (F, homomorphic to (H, B).
- 5. 5|P ag e Definition 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-lattice is a distributive sub-lattice of L, for all of L, for all x ∈ A. Therefore, (F, A) is x ∈ A. a soft distributive lattice over L. But L is not a distributive lattice. Example 2.6[2]. Consider the lattice L as 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 soft Then 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 a L, for all x ∈ A. soft modular lattice over L if F (x) is a Therefore, (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 L Proposition 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 a distributive 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 modular this 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 the as shown in Figure 2. Let A = {a, b, above theorem is not true. c, d}. That is, if (F, A) is a soft Define, modular lattice over L then L need not F(x)= {y ∈ L : x R y ⇔ x ∧ y = y}. be a modular lattice. The following example illustrates this remark.
- 6. 6|P ag e Example 2.9[2]. Consider the lattice L as 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 function Here F (x) is a modular sub-lattice of L, G by for 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 lattice of (F, A). Then (H, B) is a soft ideal of (F, A). modular lattice over L. (2) Let (F, A) be a soft modular lattice over L1. Let (H, B) be a soft lattice homo- morphic image of (F, A) over L2. Then (H, B) is a soft modular lattice over L2. Definition 2.9[2]. Let (F, A) be a soft lattice over L. A soft set (G, B) over L is 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 shown in Figure. Let A = {0, a, b, c, 1}. Define, F (x) = {y ∈ L : x R y ⇔ x ∧ y = y}.