α-level Fuzzy Soft Sets

In this paper, based on soft lattices, with the help of fuzzy level (cut) set, α-level fuzzy soft sets and α-level fuzzy soft lattices are defined, and the structure and characteristics of our definitions are explained with examples, at the same time, their differences and relations are compared with classic soft set.

And (P, ) is a partial order set. Definition 1.4. (Davey & Priestley, 1999) Let P be a none-empty ordered set. (i) If x ∨ y and x ∧ y exist for all x, y ∈ P, then P is called a lattice; (ii) If ∨S and ∧S exist for all S ⊆ P, then P is called a complete lattice.
Denotation (i) x y means x is less than y, then there is a line between x and y, and x is below of y in their diagram. (ii) x y mean x and y is non-comparability and no line between them. (iii) In the following, we will use the operators ∨ and ∧, for fuzzy set, x ∨ y = max{x, y} and x ∧ y = min{x, y}; for lattice, x ∨ y = sup{x, y}, and x ∧ y = in f {x, y}.
Definition 1.5. (Maji, Biswas, & Roy, 2001) Let U be an initial universe set and E be a parameters set. Let F (U) be the power set of all fuzzy subsets of U, A ⊂ E. Then a pair (F, A) is called a fuzzy soft set over U, where F : A → F (U) is a mapping.
Example 1.1. Suppose a fuzzy soft set (F, E) describes attractiveness of the shirts with respect to the given parameters, which the authors are going to wear. U = {x 1 , x 2 , x 3 , x 4 , x 5 } which is the set of all shirts under consideration. F (U) is the collection of all fuzzy subsets of U, E = {e 1 = color f ul , e 2 = bright , e 3 = cheap , e 4 = warm }. Let F(e 1 ) = { x 1 0.5 , x 2 0.9 , x 3 0 , x 4 0 , x 5 0 } = x 1 0.5 + x 2 0.9 + x 3 0 + x 4 0 + x 5 0 which means the membership x 1 belongs to the colorful shirt is 0.5 (simply denoted as m(F(e 1 ))(x 1 ) = 0.5) , the membership x 2 belongs to the colorful shirt is 0.9 (i.e. m(F(e 1 ))(x 2 ) = 0.9), and so on. Similarly, for fuzzy soft set (F, B), we use the symbol m(F, B)(x) = {m(F(e))(x)|x ∈ U, e ∈ E} represents the membership of object x belongs to (F, B).
We can also represent the fuzzy soft set (F, E) using the following Table 1. x 1 x 2 x 3 x 4 x 5 e 1 0.5 0.9 0 0 0 e 2 1.0 0.8 0.7 0 0 e 3 0 0 0 0.6 0 e 4 0 1.0 0 0 0.3 Definition 1.6. (Maji, Biswas, & Roy, 2001) Let U be an initial universe set and E be a parameters set. Let F (U) be the power set of all fuzzy subsets of U, A, B ⊂ E, (F, A) and (G, B) be two fuzzy soft sets over U.
For example, in the above example, , e 1 ∈ ¬E(e 1 E).
(iv) The fuzzy soft union of (F, A) and (G, B) is the fuzzy soft set (H, C), where C = A ∪ B, and ∀e ∈ C, denoted as The fuzzy soft intersection of (F, A) and (G, B) is the fuzzy soft set (H, C) is denoted as (F, A) (G, B) and is defined as (F, A) (G, B) = (H, C), where C = A ∩ B, and ∀e ∈ C, H(e) = F(e) ∩ G(e).
Definition 1.7. (Fu, 2010) Let triplet M = (F, E, L), where L is a complete lattice, E is an attributes set, and F : E → L is a mapping, that is, ∀e ∈ E, F(e) ⊆ L and F(e) is a sublattice of L, then the pair (e, F(e)) is a soft lattice over L, simply speaking, the triple M is called the soft lattice.
Remark 1.1. (i) The operations between soft lattices are similar to those between soft sets and fuzzy soft sets, and they are not repeated. We will not describe them here. (ii) In the above definition, the condition that L is a complete lattice is too strong. In literature Shao, & Qin, (2012), as long as the soft lattice defined by the author is a lattice, but if there is no completeness, there is no guarantee that the soft operation between the soft lattices can meet the closeness. That is to say, the completeness of L is the condition to ensure the closeness of soft lattice operation.  between the soft lattices can meet the closeness. That is to say, the completeness of L is the condition to ensure the closeness of soft lattice operation. Proof: Proof: 2. α-LEVEL FUZZY SOFT SETS Definition 2.1. (Maji, Biswas, & Roy, 2001;Ahmad & Kharal, 2009) Let U be an initial universal set and E be a set of parameters. F(U ) be the class of all fuzzy subsets of U . Let A ⊆ E, the pair(F, A) is called a fuzzy soft set over U , where F : A → F(U ). That is, ∀e ∈ A, F (e) is a fuzzy subset of U .   Proof. B 1 ) and (F 2 , B 2 ) are two lattices.

α-level Fuzzy Soft Sets
Definition 2.1. (Maji, Biswas, & Roy, 2001;Ahmad & Kharal, 2009) Let U be an initial universal set and E be a set of parameters. F (U) be the class of all fuzzy subsets of U. Let A ⊆ E, the pair(F, A) is called a fuzzy soft set over U, where F : A → F (U). That is, ∀e ∈ A, F(e) is a fuzzy subset of U.
Definition 2.2. (Shao & Qin, 2012) Let(F, A) be a fuzzy soft set over L, (F, A) is called a fuzzy soft lattice if F(e) is a fuzzy sublattice of L for each e ∈ A.
Remark 2.1. In reference Shao & Qin (2012), the authors defined the fuzzy soft lattice as the above definition and discussed their properties. The fuzzy soft lattice given in this definition was a very abstract definition. The author did not give an example to describe their conclusion. In this paper, we will define fuzzy soft lattice from another point of view.

α-level Fuzzy Soft Lattices
Definition 3.1. Let (L, M, F) be a fuzzy soft formal context in which the universe L is a lattice. (F, B) is a fuzzy soft set over L, define the preference order on L as: ∀h, k ∈ L, h k if and only if m(F(e)) h m(F(e)) k , ∀ e ∈ B, clearly, is a partial order, and stipulate: if h k, then h is in the below of k in the lattice structure.
(i) If ∀e ∈ B, ∃α ∈ [0, 1], such that F(e) is a fuzzy subset of L, (F(e)) α is a sublattice of L, then (F, B) is a α-level fuzzy soft lattice over L.
(ii) If ∀e ∈ B, ∀α ∈ [0, 1], such that F(e) is a fuzzy subset of L, (F(e)) α is a sublattice of L, then (F, B) is simply called a fuzzy soft lattice over L.
Suppose that B 1 = {e 1 , e 2 , e 4 } ⊆ E which means the cost of the given clothes, B 2 ={e 3 ,e 4 ,e 5 } ⊆ E which represents the attractiveness of the given clothes, (F 1 , B 1 ), (F 2 , B 2 ) are fuzzy soft sets, in which, .3 + k 0.7 , · · · , we simply represent them using the Table 3. Let α take the different values, the α-level sets of fuzzy soft sets (F 1 , B 1 ) and (F 2 , B 2 ) are represented in the Table 4.

The Relationship
Result 4.2. α-level fuzzy soft lattices are the special case of α-level fuzzy soft sets which embody two aspects.
Firstly, the universe L includes the partial (preference) order and forms the lattice structure.
The Example 3.1 and 3.2 can help us to understand this result.