Vague Filters of Residuated Lattices

Notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are investigated. The characterizations of (subpositive implicative, Boolean) vague filters is obtained. We prove that the set of all vague filters of a residuated lattice forms a complete lattice andwe find its distributive sublattices.The relation among subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vague filters are equivalent to Boolean vague filters.


Introduction
In the classical set, there are only two possibilities for any elements: in or not in the set.Hence the values of elements in a set are only one of 0 and 1.Therefore, this theory cannot handle the data with ambiguity and uncertainty.
Zadeh introduced fuzzy set theory in 1965 [1] to handle such ambiguity and uncertainty by generalizing the notion of membership in a set.In a fuzzy set  each element  is associated with a point-value   () selected from the unit interval [0, 1], which is termed the grade of membership in the set.This membership degree contains the evidences for both supporting and opposing .
A number of generalizations of Zadeh's fuzzy set theory are intuitionistic fuzzy theory, L-fuzzy theory, and vague theory.Gau and Buehrer proposed the concept of vague set in 1993 [2], by replacing the value of an element in a set with a subinterval of [0, 1].Namely, a true membership function  V () and a false-membership function  V () are used to describe the boundaries of membership degree.These two boundaries form a subinterval [ V (), 1 −  V ()] of [0, 1].The vague set theory improves description of the objective real world, becoming a promising tool to deal with inexact, uncertain, or vague knowledge.Many researchers have applied this theory to many situations, such as fuzzy control, decision-making, knowledge discovery, and fault diagnosis.Recently in [3], Jun and Park introduced the notion of vague ideal in pseudo MV-algebras and Broumand Saeid [4] introduced the notion of vague BCK/BCI-algebras.
The concept of residuated lattices was introduced by Ward and Dilworth [5] as a generalization of the structure of the set of ideals of a ring.These algebras are a common structure among algebras associated with logical systems (see [6][7][8][9]).The residuated lattices have interesting algebraic and logical properties.The main example of residuated lattices related to logic is and BL-algebras.A basic logic algebra (BL-algebra for short) is an important class of logical algebras introduced by Hajek [10] in order to provide an algebraic proof of the completeness of "Basic Logic" (BL for short).Continuous tnorm based fuzzy logics have been widely applied to fuzzy mathematics, artificial intelligence, and other areas.The filter theory plays an important role in studying these logical algebras and many authors discussed the notion of filters of these logical algebras (see [11,12]).From a logical point of view, a filter corresponds to a set of provable formulas.
In this paper, the concept of vague sets is applied to residuated lattices.In Section 2, some basic definitions and results are explained.In Section 3, we introduce the notion of vague filters of a residuated lattice and investigate some related properties.Also, we obtain the characterizations of vague filters.In Section 4, the smallest vague filters containing a given vague set are established and we obtain the distributive sublattice of the set of all vague filters of a residuated lattice.In Section 5, notions of vague filters, subpositive implicative

Preliminaries
We recall some definitions and theorems which will be needed in this paper.
Let [0, 1] denote the family of all closed subintervals of [0, 1].Now we define the refined minimum (briefly, imin) and an order ≤ on elements (1) Similarly we can define ≥, =, and imax.Then the concept of imin and imax could be extended to define iinf and isup of infinite number of elements of [0, 1].It is a known fact that  = {[0, 1], iinf, isup, ≤} is a lattice with universal bounds zero vague set and unit vague set.For ,  ∈ [0, 1] we now define (, )-cut and -cut of a vague set.
Definition 11 (see [13]).Let  be a vague set of a universe  with the true-membership function   and false-membership function   .The (, )-cut of the vague set  is a crisp subset  (,) of the set  given by where  ≤ . ( Clearly  (0,0) = .The (, )-cuts are also called vague-cuts of the vague set .

Vague Filters of Residuated Lattices
In this section, we define the notion of vague filters of residuated lattices and obtain some related results.
Definition 13.A vague set  of a residuated lattice  is called a vague filter of  if the following conditions hold: In the following proposition, we will show that the vague filters of a residuated lattice exist.Proposition 14. Unit vague set and -vague set of a residuated lattice  are vague filters of .
Proof.Let  be a -vague set of .It is clear that  satisfies (VF1).For ,  ∈  we have Thus imin{  (),   ()} ≤   ( * ), for all ,  ∈ .Hence it is a vague filter of .The proof of the other cases is similar.
Theorem 16.Let  be a vague set of a residuated lattice .
It is routine to verify that  is a vague lattice filter, but it is not a vague filter of .
Let  be the vague set defined as follows: Then  is a vague filter of .
In the following theorems, we will study the relationship between filters and vague filters of a residuated lattice.
Theorem 21.Let  be a vague set of a residuated lattice .Then  is a vague filter of  if and only if, for all ,  ∈ [0, 1], the set  (,) is either empty or a filter of , where  ≤ .
The filters like  (,) are also called vague-cuts filters of .
Corollary 22.Let  be a vague filter of a residuated lattice .Then for all  ∈ [0, 1], the set   is either empty or a filter of .

Corollary 23. Any filter 𝐹 of a residuated lattice 𝐿 is a vaguecut filter of some vague filter of 𝐿.
Proof.For all  ∈ , define In the following proposition, we will show that vague filters of a residuated lattice are a generalization of fuzzy filters.
Proposition 24.Let  be a vague set of a residuated lattice .Then  is a vague filter of  if and only if the fuzzy sets   and 1 −   are fuzzy filters in .

Lattice of Vague Filters
Let  and  be vague sets of a residuated lattice .If   () ≤   (), for all  ∈ , then we say  containing  and denote it by  ⪯ .
The unite vague set of a residuated lattice  is a vague filter of  containing any vague filter of .For a vague set, we can define the intersection of two vague sets  and  of a residuated lattice  by  ∩ () = imin{  (),   ()}, for all  ∈ .It is easy to prove that the intersection of any family of vague filters of  is a vague filter of .Hence we can define a vague filter generated by a vague set as follows.
Definition 25.Let  be a vague set of a residuated lattice .A vague filter  is called a vague filter generated by , if it satisfies (i)  ⪯ , (ii)  ⪯  that implies  ⪯ , for all vague filters  of .
Theorem 27.Let  be a vague set of a residuated lattice .
Then the vague value of  in ⟨⟩ is for all  ∈ .
Proof.First, we will prove that ⟨⟩ is a vague set of : Now, we will show that ⟨⟩ is a vague filter of .
(VF1) Suppose that ,  ∈  is arbitrary.For every  > 0, we can take  1 , . . .,   , Therefore ⟨⟩ is a vague filter of  by Theorem 16.It is clear that  ⪯ ⟨⟩.Now, let  be a vague filter of  such that  ⪯ .
Then we have for all  ∈ .Hence ⟨⟩ ⪯  and then ⟨⟩ is a vague filter generated by .
We denote the set of all vague filters of a residuated lattice  by VF().
Let  and  be vague sets of a residuated lattice .Then  *  is defined as follows: for all  ∈ .
Theorem 29.Let  and  be vague sets of a residuated lattice .Then  *  is a vague set of .
Theorem 30.Let  and  be vague filters of a residuated lattice .Then  *  is a vague filter of .
Proof.By Theorem 29,  *  is a vague set of .We will prove that  *  is a vague filter of .

Conclusion
In this paper, we introduced the notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice and investigated some related properties.We studied the relationship between filters and vague filters of a residuated lattice and showed that vague filters of a residuated lattice is a generalization of fuzzy filters.We proved that the set of all vague filters of a residuated lattice forms a complete lattice and obtained some characterizations of subpositive implicative vague filter.The extension theorem of subpositive implicative vague filters was obtained.Finally, it was proved that subpositive implicative vague filters are equivalent to Boolean vague filters.
*  ≤  if and only if  ≤  →  if and only if  ≤   , for all , ,  ∈ .