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Research Article Vague Filters of Residuated Lattices View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Crossref Hindawi Publishing Corporation Journal of Discrete Mathematics Volume 2014, Article ID 120342, 9 pages http://dx.doi.org/10.1155/2014/120342 Research Article Vague Filters of Residuated Lattices Shokoofeh Ghorbani Department of Mathematics, 22 Bahman Boulevard, Kerman 76169-133, Iran Correspondence should be addressed to Shokoofeh Ghorbani; [email protected] Received 2 May 2014; Revised 7 August 2014; Accepted 13 August 2014; Published 10 September 2014 AcademicEditor:HongJ.Lai Copyright © 2014 Shokoofeh Ghorbani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 and we 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. 1. Introduction of vague ideal in pseudo MV-algebras and Broumand Saeid [4] introduced the notion of vague BCK/BCI-algebras. In the classical set, there are only two possibilities for any The concept of residuated lattices was introduced by Ward elements: in or not in the set. Hence the values of elements and Dilworth [5] as a generalization of the structure of the in a set are only one of 0 and 1. Therefore, this theory cannot set of ideals of a ring. These algebras are a common structure handle the data with ambiguity and uncertainty. among algebras associated with logical systems (see [6–9]). Zadeh introduced fuzzy set theory in 1965 [1]tohandle The residuated lattices have interesting algebraic and logical such ambiguity and uncertainty by generalizing the notion properties. The main example of residuated lattices related of membership in a set. In a fuzzy set each element is to logic is and BL-algebras. A basic logic algebra (BL-algebra associated with a point-value () selected from the unit for short) is an important class of logical algebras introduced interval [0, 1], which is termed the grade of membership in by Hajek [10]inordertoprovideanalgebraicproofofthe the set. This membership degree contains the evidences for completeness of “Basic Logic” (BL for short). Continuous t- both supporting and opposing . norm based fuzzy logics have been widely applied to fuzzy A number of generalizations of Zadeh’s fuzzy set theory mathematics, artificial intelligence, and other areas. The filter are intuitionistic fuzzy theory, L-fuzzy theory, and vague theory plays an important role in studying these logical theory. Gau and Buehrer proposed the concept of vague algebras and many authors discussed the notion of filters of set in 1993 [2], by replacing the value of an element in a these logical algebras (see [11, 12]). From a logical point of setwithasubintervalof[0, 1].Namely,atruemembership view, a filter corresponds to a set of provable formulas. function V() and a false-membership function V() are In this paper, the concept of vague sets is applied to resid- used to describe the boundaries of membership degree. uated lattices. In Section 2, some basic definitions and results These two boundaries form a subinterval [V(), 1 −V ()] are explained. In Section 3,weintroducethenotionofvague of [0, 1]. The vague set theory improves description of the filters of a residuated lattice and investigate some related objective real world, becoming a promising tool to deal with properties. Also, we obtain the characterizations of vague inexact, uncertain, or vague knowledge. Many researchers filters. In Section 4, the smallest vague filters containing a haveappliedthistheorytomanysituations,suchasfuzzy given vague set are established and we obtain the distributive control, decision-making, knowledge discovery, and fault sublattice of the set of all vague filters of a residuated lattice. diagnosis. Recently in [3], Jun and Park introduced the notion In Section 5, notions of vague filters, subpositive implicative 2 Journal of Discrete Mathematics 1∈ vague filters, and Boolean vague filters of a residuated lattice (SPIF1) , are introduced and some related properties are obtained. (SPIF2) (( → ) ∗ ) (( )→∈ and ∈imply ( → ) ∈ for any 2. Preliminaries , ,, ∈ (SPIF3) ( ∗ ( )) → (( → ) )∈ We recall some definitions and theorems which will be and ∈imply ( ) →∈ for any needed in this paper. , ,. ∈ Definition 1 (see [6]). A residuated lattice is an algebraic Proposition 6 (,∧,∨,∗,→,,0,1) (see [11]). Let be a filter of . is a subpositive structure such that implicativefilterifandonlyif (1) (,∧,∨,0,1) is a bounded lattice with the least ele- (SPIF4) ( → ) (( )→∈ imply ment 0 and the greatest element 1, ( → ) ∈ for any , , ∈ (2) (,∗,1)is a monoid; that is, ∗ is associative and ∗1 = (SPIF5) ( ) → (( → ) )∈ imply 1∗=, ( ) →∈ for any , . ∈ (3) ∗ ≤if and only if ≤if → and only if Definition 7 (see [1]). Let be a nonempty set. A fuzzy set in ≤,forall, ,. ∈ is a mapping : → [0,1]. In the rest of this paper, we denote the residuated lattice Proposition 8 (see [12]). Let be a residuated lattice. A fuzzy (, ∧, ∨, ∗, → , , 0,1) by , ¬=→,and 0 ∼=0. set is called a fuzzy filter in if it satisfies Proposition 2 (see [6, 9]). Let be a residuated lattice. Then (fF1) ≤imply () ≤ (), we have the following properties: for all , ,, ∈ (fF2) min{(), ()} ≤ ( ∗), (1) → ==1, for all , . ∈ (2) 1 → =1=, Definition 9 (see [13]). A vague set intheuniverseof (3) → ( ) = , (→) discourse is characterized by two membership functions (4) ≤if and only if →=1if and only if = given by 1, (i) a true membership function : → [0, 1], (5) if ≤,then∗≤∗, ∗≤∗, (ii) a false membership function : → [0, ,1] (6) if ≤,then → ≤, → ≤, () (7) if ≤,then→≤→, ≤, where isalowerboundonthegradeofmembershipof derived from the “evidence for ,” () is a lower bound (8) ≤()→, ≤(→ ), on the negation of derived from the “evidence against ,” () + () ≤ 1 → ≤(→ )(→ )≤( and . (9) , Thus the grade of membership of in the vague set ) (, ) is bounded by a subinterval [(), 1 − ()] of [0, 1].This () (10) → ( (, ∗))=1 ( → ( ∗. ))=1 indicates that if the actual grade of membership of is , then () ≤ () ≤1− ().Thevagueset is written Definition 3 (see [6, 9]). Let be a nonempty subset of a as = {⟨, [ (), 1 − ()]⟩ : ,wheretheinterval∈ } residuated lattice . is called a filter if [(), 1 − ()] is called the vague value of in , denoted by (). (F1) , ∈ imply ∗∈, Definition 10 (see [13]). A vague set of a set is called (F2) ≤and ∈imply ∈. (1) the zero vague set of if () = 0 and () = 1 for Theorem 4 (see [6, 9]). Anonemptysubset of a residuated all ∈, lattice is a filter if and only if (2) the unit vague set of if () = 1 and () = 0 for (F3) 1∈and , → ∈ imply ∈,or all ∈, (F3 ) 1∈and , ∈ imply ∈. (3) the -vague set of if () = and () = 1 − for all ∈,where ∈ (0, 1). Definition 5 (see [11]). Let be a nonempty subset of a residuated lattice . is called a subpositive implicative filter Let [0, 1] denote the family of all closed subintervals of of if [0, 1]. Now we define the refined minimum (briefly, imin) and Journal of Discrete Mathematics 3 an order ≤ on elements 1 =[1,1] and 2 =[2,2] of [0, 1] Thus imin{(), ()} ≤ (∗),forall, .Hence ∈ as it is a vague filter of . The proof of the other cases is similar. imin (1,2)=[min {1,2},min {1,2}], (1) Lemma 15. 1 ≤2 iff 1 ≤2,1 ≤2. Let be a vague set of a residuated lattice such that it satisfies (VF1). Then the following are equivalent: Similarly we can define ≥, =,andimax.Thentheconceptof imin and imax could be extended to define iinf and isup of (VF4) imin{VA(x), VA(x → y)} ≤ VA(y), [0, 1] infinite number of elements of . It is a known fact that (VF5) imin{VA(x), VA(x y)} ≤ VA(y), = {[0, 1], , ,≤} iinf isup is a lattice with universal bounds , ∈ zero vague set and unit vague set. For , ∈ [0, 1] we now for all . define (, )-cut and -cut of a vague set. Proof. Suppose that satisfies (VF4). We have ≤ ( ) → () ≤ (( ) →) Definition 11 (see [13]). Let be a vague set of a universe .By(VF1), .Wegetthat with the true-membership function and false-membership imin { () , ( )} function .The(, )-cut of the vague set is a crisp subset (,) of the set given by ≤ imin { (( ) → ), ( )} (4) ={∈: () ≥ [, ]}, (,) ≤ () . (2) where ≤. Conversely, if satisfies (VF5), then the inequality ≤(→ ) analogously entails (VF4). Clearly (0,0) =.The(, )-cuts are also called vague-cuts of the vague set . Theorem 16. Let be a vague set of a residuated lattice . Then is a vague filter if and only if it satisfies (VF4) and Definition 12 (see [13]). The -cut of the vague set is a crisp = subset of the set given by (,).Equivalently,we (VF6) () ≤ (1),forall∈. can define the -cut as ={∈:() ≥ .} Proof. Let be a vague filter of .Since≤1,then() ≤ (1) ∈ ∗(→ )≤ 3. Vague Filters of Residuated Lattices ,forall by (VF1). We have .By (VF1) and (VF2) we get In this section, we define the notion of vague filters of { () , ( → )} residuated lattices and obtain some related results. imin (5) ≤ (∗(→))≤ () . Definition 13. Avagueset
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