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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 𝐴 each element 𝑥 is to logic is and BL-algebras. A basic logic algebra (BL-algebra associated with a point-value 𝜇𝐴(𝑥) 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 𝑥. 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 𝑡V(𝑥) and a false-membership function 𝑓V(𝑥) 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 [𝑡V(𝑥), 1 −V 𝑓 (𝑥)] 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∈𝐹 vague filters, and Boolean vague filters of a residuated lattice (SPIF1) , are introduced and some related properties are obtained. (SPIF2) ((𝑥 → 𝑦) ∗ 𝑧) ((𝑦 𝑥)→∈𝐹 and 𝑧∈𝐹imply (𝑥 → 𝑦) 𝑦∈𝐹 for any 2. Preliminaries 𝑥, 𝑦,, 𝑧∈𝐿 (SPIF3) (𝑧 ∗ (𝑥 𝑦)) → ((𝑦 → 𝑥) 𝑥)∈𝐹 We recall some definitions and theorems which will be and 𝑧∈𝐹imply (𝑥 𝑦) →𝑦∈𝐹 for any needed in this paper. 𝑥, 𝑦,. 𝑧∈𝐿 Definition 1 (see [6]). A residuated lattice is an algebraic Proposition 6 𝐹 𝐿 𝐹 (𝐿,∧,∨,∗,→,,0,1) (see [11]). Let be a filter of . is a subpositive structure such that implicativefilterifandonlyif (1) (𝐿,∧,∨,0,1) is a bounded lattice with the least ele- (SPIF4) (𝑥 → 𝑦) ((𝑦 𝑥)→∈𝐹 imply ment 0 and the greatest element 1, (𝑥 → 𝑦) 𝑦∈𝐹 for any 𝑥, 𝑦, ∈𝐿 (2) (𝐿,∗,1)is a monoid; that is, ∗ is associative and 𝑥∗1 = (SPIF5) (𝑥 𝑦) → ((𝑦 → 𝑥) 𝑥)∈𝐹 imply 1∗𝑥=𝑥, (𝑥 𝑦) →𝑦∈𝐹 for any 𝑥, 𝑦. ∈𝐿 (3) 𝑥∗𝑦 ≤𝑧if and only if 𝑥≤𝑦if →𝑧 and only if Definition 7 (see [1]). Let 𝑋 be a nonempty set. A fuzzy set in 𝑦≤𝑥𝑧,forall𝑥, 𝑦,. 𝑧∈𝐿 𝑋 is a mapping 𝜇:𝑋 → [0,1].
In the rest of this paper, we denote the residuated lattice Proposition 8 (see [12]). Let 𝐿 be a residuated lattice. A fuzzy (𝐿, ∧, ∨, ∗, → , , 0,1) by 𝐿, ¬𝑥=𝑥→,and 0 ∼𝑥=𝑥0. set 𝜇 is called a fuzzy filter in 𝐿 if it satisfies
Proposition 2 (see [6, 9]). Let 𝐿 be a residuated lattice. Then (fF1) 𝑥≤𝑦imply 𝜇(𝑥) ≤ 𝜇(𝑦), we have the following properties: for all 𝑥, 𝑦,, 𝑧∈𝐿 (fF2) min{𝜇(𝑥), 𝜇(𝑦)} ≤ 𝜇(𝑥 ∗𝑦), (1) 𝑥 → 𝑥=𝑥𝑥=1, for all 𝑥, 𝑦. ∈𝐿 (2) 1 → 𝑥=1𝑥=𝑥, Definition 9 (see [13]). A vague set 𝐴 intheuniverseof (3) 𝑥 → (𝑦 𝑧) = 𝑦, (𝑥→𝑧) discourse 𝑋 is characterized by two membership functions (4) 𝑥≤𝑦if and only if 𝑥→𝑦=1if and only if 𝑥𝑦= given by 1, (i) a true membership function 𝑡𝐴 : 𝑋 → [0, 1], (5) if 𝑥≤𝑦,then𝑥∗𝑧≤𝑦∗𝑧, 𝑧∗𝑥≤𝑧∗𝑦, (ii) a false membership function 𝑓𝐴 : 𝑋 → [0, 1], (6) if 𝑥≤𝑦,then𝑧 → 𝑥≤𝑧, →𝑦 𝑧𝑥≤𝑧𝑦, 𝑡 (𝑥) (7) if 𝑥≤𝑦,then𝑦→𝑧≤𝑥→𝑧, 𝑦𝑧≤𝑥𝑧, where 𝐴 isalowerboundonthegradeofmembershipof 𝑋 derived from the “evidence for 𝑥,” 𝑓𝐴(𝑥) is a lower bound (8) 𝑥≤(𝑥𝑦)→, 𝑦 𝑥≤(𝑥→ 𝑦)𝑦, on the negation of 𝑥 derived from the “evidence against 𝑥,” 𝑡 (𝑥) + 𝑓 (𝑥) ≤ 1 𝑦 → 𝑧≤(𝑥→ 𝑦)(𝑥→ 𝑧)𝑦𝑧≤(𝑥 and 𝐴 𝐴 . (9) , Thus the grade of membership of 𝑥 in the vague set 𝐴 𝑦) (𝑥, 𝑧) is bounded by a subinterval [𝑡𝐴(𝑥), 1 −𝐴 𝑓 (𝑥)] of [0, 1].This 𝑥 𝜇(𝑥) (10) 𝑥 → (𝑦 (𝑥, ∗𝑦))=1 𝑦 (𝑥 → (𝑥 ∗. 𝑦))=1 indicates that if the actual grade of membership of is , then 𝑡𝐴(𝑥) ≤ 𝜇(𝑥)𝐴 ≤1−𝑓 (𝑥).Thevagueset𝐴 is written Definition 3 (see [6, 9]). Let 𝐹 be a nonempty subset of a as 𝐴 = {⟨𝑥,𝐴 [𝑡 (𝑥), 1 −𝐴 𝑓 (𝑥)]⟩ : 𝑥 ,wheretheinterval∈ 𝑋} residuated lattice 𝐿. 𝐹 is called a filter if [𝑡𝐴(𝑥), 1 −𝐴 𝑓 (𝑥)] is called the vague value of 𝑥 in 𝐴, denoted by 𝑉𝐴(𝑥). (F1) 𝑥, 𝑦 ∈𝐹 imply 𝑥∗𝑦∈𝐹, Definition 10 (see [13]). A vague set 𝐴 of a set 𝑋 is called (F2) 𝑥≤𝑦and 𝑥∈𝐹imply 𝑦∈𝐹. (1) the zero vague set of 𝑋 if 𝑡𝐴(𝑥) = 0 and 𝑓𝐴(𝑥) = 1 for Theorem 4 𝐹 (see [6, 9]). Anonemptysubset of a residuated all 𝑥∈𝑋, lattice 𝐿 is a filter if and only if (2) the unit vague set of 𝑋 if 𝑡𝐴(𝑥) = 1 and 𝑓𝐴(𝑥) = 0 for (F3) 1∈𝐹and 𝑥, 𝑥 → 𝑦 ∈𝐹 imply 𝑦∈𝐹,or all 𝑥∈𝑋, (F3 ) 1∈𝐹and 𝑥, 𝑥 𝑦∈𝐹 imply 𝑦∈𝐹. (3) the 𝛼-vague set of 𝑋 if 𝑡𝐴(𝑥) = 𝛼 and 𝑓𝐴(𝑥) = 1 − 𝛼 for all 𝑥∈𝑋,where𝛼 ∈ (0, 1). Definition 5 (see [11]). Let 𝐹 be a nonempty subset of a residuated lattice 𝐿. 𝐹 is called a subpositive implicative filter Let 𝐼[0, 1] denote the family of all closed subintervals of of 𝐿 if [0, 1]. Now we define the refined minimum (briefly, imin) and Journal of Discrete Mathematics 3
an order ≤ on elements 𝐼1 =[𝑎1,𝑏1] and 𝐼2 =[𝑎2,𝑏2] of 𝐼[0, 1] Thus imin{𝑉𝐴(𝑥),𝐴 𝑉 (𝑦)} ≤𝐴 𝑉 (𝑥∗𝑦),forall𝑥, 𝑦.Hence ∈𝐿 as it is a vague filter of 𝐿. The proof of the other cases is similar.
imin (𝐼1,𝐼2)=[min {𝑎1,𝑎2},min {𝑏1,𝑏2}], (1) Lemma 15. 𝐴 𝐿 𝐼1 ≤𝐼2 iff 𝑎1 ≤𝑎2,𝑏1 ≤𝑏2. 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), 𝐼[0, 1] infinite number of elements of . It is a known fact that (VF5) imin{VA(x), VA(x y)} ≤ VA(y), 𝐿 = {𝐼[0, 1], , ,≤} iinf isup is a lattice with universal bounds 𝑥, 𝑦 ∈𝐿 zero vague set and unit vague set. For 𝛼, 𝛽 ∈ [0, 1] we now for all . define (𝛼, 𝛽)-cut and 𝛼-cut of a vague set. Proof. Suppose that 𝐴 satisfies (VF4). We have 𝑥 ≤ (𝑥 𝑦) → 𝑦 𝑉 (𝑥) ≤ 𝑉 ((𝑥 𝑦) →𝑦) Definition 11 (see [13]). Let 𝐴 be a vague set of a universe 𝑋 .By(VF1), 𝐴 𝐴 .Wegetthat 𝑡 with the true-membership function 𝐴 and false-membership imin {𝑉𝐴 (𝑥) ,𝑉𝐴 (𝑥 𝑦)} function 𝑓𝐴.The(𝛼, 𝛽)-cut of the vague set 𝐴 is a crisp subset 𝐴 𝑋 (𝛼,𝛽) of the set given by ≤ imin {𝑉𝐴 ((𝑥 𝑦) →𝐴 𝑦),𝑉 (𝑥 𝑦)} (4) 𝐴 ={𝑥∈𝑋:𝑉 (𝑥) ≥ [𝛼, 𝛽]}, (𝛼,𝛽) 𝐴 ≤𝑉𝐴 (𝑦) . (2) where 𝛼≤𝛽. Conversely, if 𝐴 satisfies (VF5), then the inequality 𝑥≤(𝑥→ 𝑦) 𝑦 analogously entails (VF4). Clearly 𝐴(0,0) =𝑋.The(𝛼, 𝛽)-cuts are also called vague-cuts 𝐴 of the vague set . Theorem 16. Let 𝐴 be a vague set of a residuated lattice 𝐿. Then 𝐴 is a vague filter if and only if it satisfies (VF4) and Definition 12 (see [13]). The 𝛼-cut of the vague set 𝐴 is a crisp 𝐴 𝑋 𝐴 =𝐴 subset 𝛼 of the set given by 𝛼 (𝛼,𝛼).Equivalently,we (VF6) 𝑉𝐴(𝑥) ≤𝐴 𝑉 (1),forall𝑥∈𝐿. can define the 𝛼-cut as 𝐴𝛼 ={𝑥∈𝑋:𝑡𝐴(𝑥) ≥ .𝛼} Proof. Let 𝐴 be a vague filter of 𝐿.Since𝑥≤1,then𝑉𝐴(𝑥) ≤ 𝑉 (1) 𝑥∈𝐿 𝑥∗(𝑥→ 𝑦)≤𝑦 3. Vague Filters of Residuated Lattices 𝐴 ,forall by (VF1). We have .By (VF1) and (VF2) we get In this section, we define the notion of vague filters of {𝑉 (𝑥) ,𝑉 (𝑥 → 𝑦)} residuated lattices and obtain some related results. imin 𝐴 𝐴 (5) ≤𝑉 (𝑥∗(𝑥→𝑦))≤𝑉 (𝑦) . Definition 13. Avagueset𝐴 of a residuated lattice 𝐿 is called 𝐴 𝐴 𝐿 a vague filter of if the following conditions hold: Hence (VF4) hold. Conversely, let 𝐴 satisfy (VF4) and (VF6). Since 𝑥≤𝑦, (VF1) if 𝑥≤𝑦,then𝑉𝐴(𝑥) ≤𝐴 𝑉 (𝑦),forall𝑥, 𝑦, ∈𝐿 then 𝑥→𝑦=1; (VF2) imin{𝑉𝐴(𝑥),𝐴 𝑉 (𝑦)} ≤𝐴 𝑉 (𝑥∗𝑦),forall𝑥, 𝑦 ∈ 𝐿. 𝑉𝐴 (𝑦) ≥ imin {𝑉𝐴 (𝑥 → 𝑦)𝐴 ,𝑉 (𝑥)} (6) That is, = imin {𝑉𝐴 (1) ,𝑉𝐴 (𝑥)}=𝑉𝐴 (𝑥) . 𝑥≤𝑦 𝑡 (𝑥) ≤ 𝑡 (𝑦) 1−𝑓 (𝑥) ≤ 1−𝑓 (𝑦) (1) implies 𝐴 𝐴 and 𝐴 𝐴 , 𝑉 𝑥, 𝑦 ∈𝐿 Hence 𝐴 isorder-preservingand(VF1)holds.By for all , Proposition 2 part (10), 𝑥 → (𝑦 (𝑥.By ∗𝑦))=1 (2) min{𝑡𝐴(𝑥),𝐴 𝑡 (𝑦)} ≤𝐴 𝑡 (𝑥∗𝑦)and min{1 − 𝑓𝐴(𝑥), 1 − (VF4), we have 𝑓𝐴(𝑦)} ≤ 1 −𝐴 𝑓 (𝑥 ∗ 𝑦),forall𝑥, 𝑦. ∈𝐿 𝑉𝐴 (𝑦 (𝑥 ∗𝑦)) In the following proposition, we will show that the vague filters of a residuated lattice exist. ≥ imin {𝑉𝐴 (𝑥) ,𝑉𝐴 (𝑥 → (𝑦 (𝑥 ∗𝑦)))}=𝑉 𝐴 (𝑥) . (7) Proposition 14. Unit vague set and 𝛼-vague set of a residuated lattice 𝐿 are vague filters of 𝐿. By Lemma 15, 𝑉 (𝑥 ∗ 𝑦) ≥ {𝑉 (𝑦 (𝑥 ∗ 𝑦)),𝑉 (𝑦)} Proof. Let 𝐴 be a 𝛼-vague set of 𝐿.Itisclearthat𝐴 satisfies 𝐴 imin 𝐴 𝐴 (VF1). For 𝑥, 𝑦 ∈𝐿 we have (8) ≥ imin {𝑉𝐴 (𝑥) ,𝑉𝐴 (𝑦)}. 𝑡𝐴 (𝑥 ∗ 𝑦) = 𝛼= min {𝛼, 𝛼} = min {𝑡𝐴 (𝑥) ,𝑡𝐴 (𝑦)}, Hence (VF2) holds. 1−𝑓𝐴 (𝑥∗𝑦) Theorem 17. Let 𝐴 be a vague set of a residuated lattice 𝐿. =𝛼=min {𝛼, 𝛼} = min {1 − 𝑓𝐴 (𝑥) ,1−𝑓𝐴 (𝑦)}. Then 𝐴 is a vague filter if and only if it satisfies (VF5) and (3) (VF6). 4 Journal of Discrete Mathematics
Definition 18. Avagueset𝐴 of a residuated lattice 𝐿 is called (1) Suppose that 𝑥≤𝑦and 𝑥∈𝐴(𝛼,𝛽).By(VF1),𝑉𝐴(𝑦) ≥ a vague lattice filter of 𝐿 if it satisfies (VF1) and 𝑉𝐴(𝑥) ≥ [𝛼, 𝛽]. Therefor 𝑦∈𝐴(𝛼,𝛽). 𝑥, 𝑦 ∈𝐴 𝑉 (𝑥), 𝑉 (𝑦) ≥ (VLF2) imin{𝑉𝐴(𝑥),𝐴 𝑉 (𝑦)} ≤𝐴 𝑉 (𝑥 ∧ 𝑦),forall (2) Suppose that (𝛼,𝛽).Then 𝐴 𝐴 [𝛼, 𝛽] 𝑉 (𝑥∗𝑦) ≥ 𝑥, 𝑦. ∈𝐿 .By(VF2),wehave 𝐴 imin{𝑉𝐴(𝑥),𝐴 𝑉 (𝑦)} ≥ [𝛼, 𝛽].Hence𝑥∗𝑦(𝛼,𝛽) ∈𝐴 Proposition 19. 𝐴 Let be a vague filter of a residuated lattice and then 𝐴(𝛼,𝛽) is a filter of 𝐿. 𝐿;then𝐴 is a vague lattice filter. Conversely, suppose that, for all 𝛼, 𝛽 ∈ [0,,thesets 1] 𝐴(𝛼,𝛽) In the following example, we will show that the converse is either empty or a filter of 𝐿.Let𝑡𝐴(𝑥) =1 𝛼 , 𝑡𝐴(𝑦) =2 𝛼 , of Proposition 19 may not be true in general. 1−𝑓𝐴(𝑥) =1 𝛽 ,and1−𝑓𝐴(𝑦) =2 𝛽 .Put𝛼=min{𝛼1,𝛼2} and 𝛽=min{1 − 𝛽1,1−𝛽2}.Then𝑡𝐴(𝑥),𝐴 𝑡 (𝑦) ≥ 𝛼 and 1− {0,𝑎,𝑏,𝑐,1} 0<𝑎<𝑏,𝑐<1 Example 20. Let such that , 𝑓𝐴(𝑥), 1 −𝐴 𝑓 (𝑦) ≥.Hence 𝛽 𝑉𝐴(𝑦),𝐴 𝑉 (𝑥) ≥ [𝛼, 𝛽];thatis, 𝑏 𝑐 ∗ → where and areincomparable.Definetheoperations , , 𝑥, 𝑦 ∈𝐴(𝛼,𝛽).So𝐴(𝛼,𝛽) =0̸ and, by assumption, 𝐴(𝛼,𝛽) is a and as follows: filter of 𝐿.Weobtainthat𝑥∗𝑦∈𝐴(𝛼,𝛽);thatis, ∗ 0𝑎𝑏𝑐1 𝑉 (𝑥∗𝑦)≥[𝛼,𝛽] 0 00000 𝐴 𝑎 000𝑎𝑎 =[ {𝛼 ,𝛼 }, {1 − 𝛽 ,1−𝛽 }] 𝑏 0𝑎𝑏𝑎𝑏 min 1 2 min 1 2 (12) 𝑐 000𝑐𝑐 = imin {[𝑡𝐴 (𝑥) ,1−𝑓𝐴 (𝑥)][𝑡𝐴 (𝑦) , 1 −𝐴 𝑓 (𝑦)]} 1 0𝑎𝑏𝑐1 = imin {𝑉𝐴 (𝑥) ,𝑉𝐴 (𝑦)}. → 0𝑎𝑏𝑐1 0 11111 Hence (VF2) holds. 𝑎 𝑐1111 Suppose that 𝑥≤𝑦, 𝑡𝐴(𝑥) = 𝛼,and1−𝑓𝐴(𝑥) =.Hence 𝛽 (9) 𝑏 𝑐𝑐1𝑐1 𝑥∈𝐴(𝛼,𝛽). By assumption 𝐴(𝛼,𝛽) isafilter.Wegetthat𝑦∈ 𝑐 0𝑏𝑏11 𝐴(𝛼,𝛽);thatis,𝑉𝐴(𝑦) ≥ [𝛼, 𝛽]𝐴 =[𝑡 (𝑥), 1 −𝐴 𝑓 (𝑥)] =𝐴 𝑉 (𝑥). 1 0𝑎𝑏𝑐1 Hence 𝐴 is a vague filter of 𝐿. 0𝑎𝑏𝑐 1 The filters like 𝐴(𝛼,𝛽) are also called vague-cuts filters of 𝐿. 0 11111 𝑎 𝑏1111 Corollary 22. Let 𝐴 be a vague filter of a residuated lattice 𝐿. 𝑏 0𝑐1𝑐1 Then for all 𝛼∈[0,1],theset𝐴𝛼 is either empty or a filter of 𝑐 𝑏𝑏𝑏11 𝐿. 1 0𝑎𝑏𝑐1. Corollary 23. Any filter 𝐹 of a residuated lattice 𝐿 is a vague- Then 𝐿 is a residuated lattice. Let 𝐴 be the vague set defined cut filter of some vague filter of 𝐿. as follows: Proof. For all 𝑥∈𝐿, define 𝐴={⟨0, [0.1, 0.2]⟩ , ⟨𝑎, [0.3, 0.4]⟩ , ⟨𝑏, [0.3, 0.4]⟩ , (10) ⟨𝑐, [0.3, 0.8]⟩ , ⟨0, [0.5, 0.9]⟩} . 1 if 𝑥∈𝐹 1−𝑓𝐴 (𝑥) =𝑡𝐴 (𝑥) ={ (13) 𝛼 otherwise. It is routine to verify that 𝐴 is a vague lattice filter, but it is not avaguefilterof𝐿. Hence we have 𝑉𝐴(𝑥) = [1, 1],if𝑥∈𝐹and 𝑉𝐴(𝑥) = [𝛼, 𝛼], Let 𝐵 be the vague set defined as follows: otherwise, where 𝛼 ∈ (0, 1).Itisclearthat𝐹=𝐴(1,1).Let 𝑥, 𝑦.If ∈𝐿 𝑥, 𝑦,then ∈𝐹 𝑥∗𝑦∈𝐹.Wehave𝑉𝐴(𝑥 ∗ 𝑦) = 𝐵={⟨0, [0.2, 0.5]⟩ , ⟨𝑎, [0.2, 0.5]⟩ , ⟨𝑏, [0.2, 0.5]⟩ , [1, 1] = imin{𝑉𝐴(𝑥),𝐴 𝑉 (𝑦)}.Ifoneof𝑥 or 𝑦 does not belong (11) 𝐹 𝑉 (𝑥) 𝑉 (𝑦) [𝛼, 𝛼] ⟨𝑐, [0.4, 0.7]⟩ , ⟨0, [0.7, 0.7]⟩} . to ,thenoneof 𝐴 and 𝐴 is equal to . Therefore 𝑉𝐴(𝑥∗𝑦) ≥ [𝛼,𝛼] = imin{𝑉𝐴(𝑥),𝐴 𝑉 (𝑦)}.Supposethat𝑥≤𝑦. 𝑥∈𝐹 𝑦∈𝐹 𝑉 (𝑥) = 𝑉 (𝑦) 𝑥∉𝐹 Then 𝐵 is a vague filter of 𝐿. If ,then .Hence 𝐴 𝐴 .If ,then 𝑉𝐴(𝑥) = [𝛼, 𝛼].Wehave𝑉𝐴(𝑥) ≤𝐴 𝑉 (𝑦).Hence𝐴 is a vague In the following theorems, we will study the relationship filter of 𝐿. between filters and vague filters of a residuated lattice. In the following proposition, we will show that vague Theorem 21. Let 𝐴 be a vague set of a residuated lattice 𝐿. filters of a residuated lattice are a generalization of fuzzy Then 𝐴 is a vague filter of 𝐿 if and only if, for all 𝛼, 𝛽 ∈ [0,, 1] filters. the set 𝐴(𝛼,𝛽) is either empty or a filter of 𝐿,where𝛼≤𝛽. Proposition 24. Let 𝐴 be a vague set of a residuated lattice 𝐿. Proof. Suppose that 𝐴 is a vague filter of 𝐿 and 𝛼, 𝛽 ∈ [0,. 1] Then 𝐴 is a vague filter of 𝐿 if and only if the fuzzy sets 𝑡𝐴 and Let 𝐴(𝛼,𝛽) =0̸ .Wewillshowthat𝐴(𝛼,𝛽) is a filter of 𝐿. 1−𝑓𝐴 are fuzzy filters in 𝐿. Journal of Discrete Mathematics 5
4. Lattice of Vague Filters Now, we will show that ⟨𝐴⟩ is a vague filter of 𝐿. (VF1) Suppose that 𝑥, 𝑦 ∈𝐿 is arbitrary. For every 𝜀>0, 𝐴 𝐵 𝐿 𝑉 (𝑥) ≤ Let and be vague sets of a residuated lattice .If 𝐴 we can take 𝑎1,...,𝑎𝑛,𝑏1,...,𝑏𝑚 ∈𝐿such that 𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛, 𝑉 (𝑥) 𝑥∈𝐿 𝐵 𝐴 𝐵 ,forall ,thenwesay containing and denote it 𝑦≥𝑏1 ∗⋅⋅⋅∗𝑏𝑚,and by 𝐴⪯𝐵. The unite vague set of a residuated lattice 𝐿 is a vague 𝑡⟨𝐴⟩ (𝑥) −𝜀
= min {𝑡⟨𝐴⟩ (𝑥) ,𝑡⟨𝐴⟩ (𝑦)}− 𝜀. Example 26. Consider the vague lattice filter 𝐴 in Example 20 𝐿 which is not a vague filter of . Then the vague filter generated Since 𝜀>0is arbitrary, we obtain that 𝑡⟨𝐴⟩(𝑥 ∗ 𝑦) ≥ 𝐴 by is min{𝑡⟨𝐴⟩(𝑥),⟨𝐴⟩ 𝑡 (𝑦)}. Similarly, we can show that 1−𝑓⟨𝐴⟩(𝑥 ∗ 𝑦) ≥ min{1 − ⟨𝐴⟩ = {⟨0, [0.3, 0.4]⟩ , ⟨𝑎, [0.3, 0.4]⟩ , 𝑓⟨𝐴⟩(𝑥), 1 −⟨𝐴⟩ 𝑓 (𝑦)}.Hence𝑉𝐴(𝑥 ∗ 𝑦) ≥ imin{𝑉𝐴(𝑥),𝐴 𝑉 (𝑦)}, (14) ⟨𝑏, [0.3, 0.4]⟩ , ⟨𝑐, [0.3, 0.8]⟩ , ⟨0, [0.5, 0.9]⟩} . for all 𝑥, 𝑦. ∈𝐿 (VF2) Suppose that 𝑥≤𝑦where 𝑥, 𝑦.Let ∈𝐿 𝑎1,...,𝑎𝑛 ∈ 𝐿 𝑎 ∗⋅⋅⋅∗𝑎 ≤𝑥 𝑎 ∗⋅⋅⋅∗𝑎 ≤𝑦 Theorem 27. Let 𝐴 be a vague set of a residuated lattice 𝐿. such that 1 𝑛 .Then 1 𝑛 .Andwe Then the vague value of 𝑥 in ⟨𝐴⟩ is get that 𝑉 (𝑦) ≥ {𝑉 (𝑎 ),𝑉 (𝑎 ),...,𝑉 (𝑎 )}. 𝑉 (𝑥) = { { ( ), ( ),..., ( )}: 𝐴 imin 𝐴 1 𝐴 2 𝐴 𝑛 (19) ⟨𝐴⟩ isup imin VA a1 VA a2 VA an Since 𝑎1,...,𝑎𝑛 ∈𝐿,where𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛 is arbitrary, then 𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛}, (15) 𝑉⟨𝐴⟩ (𝑦) ≥ isup {imin {𝑉𝐴 (𝑎1),𝑉𝐴 (𝑎2),...,𝑉𝐴 (𝑎𝑛)}: 𝑥∈𝐿 for all . 𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛} ⟨𝐴⟩ 𝐿 Proof. First, we will prove that is a vague set of : =𝑉⟨𝐴⟩ (𝑥) . (20) 𝑡⟨𝐴⟩ (𝑥) ⟨𝐴⟩ 𝐿 = { {𝑡 (𝑎 ),𝑡 (𝑎 ),...,𝑡 (𝑎 )}: Therefore is a vague filter of by Theorem 16.Itisclear sup min 𝐴 1 𝐴 2 𝐴 𝑛 that 𝐴⪯⟨𝐴⟩.Now,let𝐶 be a vague filter of 𝐿 such that 𝐴⪯𝐶. Thenwehave 𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛} 𝑉⟨𝐴⟩ (𝑥) ≤ sup {min {1 − 𝑓𝐴 (𝑎1),1−𝑓𝐴 (𝑎2),...,1−𝑓𝐴 (𝑎𝑛)}: = isup {imin {𝑉𝐴 (𝑎1),...,𝑉𝐴 (𝑎𝑛)}: 𝑥 ≥ 𝑎 1 ∗⋅⋅⋅∗𝑎𝑛} 𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛}
≤ isup {imin {𝑉𝐶 (𝑎1),...,𝑉𝐶 (𝑎𝑛)}:𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛} = sup {1 − max {𝑓𝐴 (𝑎1),𝑓𝐴 (𝑎2),...,𝑓𝐴 (𝑎𝑛)}: =𝑉 (𝑥) , 𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛} 𝐶 (21) =1−inf {max {𝑓𝐴 (𝑎1),𝑓𝐴 (𝑎2),...,𝑓𝐴 (𝑎𝑛)}: for all 𝑥∈𝐿.Hence⟨𝐴⟩ ⪯ 𝐶 and then ⟨𝐴⟩ is a vague filter 𝑥≥𝑎1 ∗⋅⋅⋅∗𝑎𝑛}, generated by 𝐴.
=1−𝑓⟨𝐴⟩ (𝑥) . We denote the set of all vague filters of a residuated lattice (16) 𝐿 by VF(𝐿). 6 Journal of Discrete Mathematics
Corollary 28. (VF(𝐿), ∩, ∨) is a complete lattice. By Definition 13, 𝐴∗𝐵is a vague filter of 𝐿.
Proof. Clearly, if {𝐴𝑖}𝑖∈Γ is a family of vague filters of a Theorem 31. Let 𝐴 and 𝐵 be vague filters of a residuated lattice residuated lattice 𝐿, then the infimum of this family is ⋂𝑖∈Γ 𝐴𝑖 𝐿 such that 𝑉𝐴(1) = 𝑉𝐵(1).Then𝐴∗𝐵is the least upper bound and the supremum is ⋁𝑖∈Γ𝐴𝑖 =⟨⋃𝑖∈Γ 𝐴𝑖⟩.Itiseasytoprove of 𝐴 and 𝐵. that (IFF(𝐿), ∩, ∨) is a complete lattice. Proof. By Definition 13,wehave𝑉𝐴(1) = 𝑉𝐵(1) ≥ 𝑉𝐴(𝑦) for Let 𝐴 and 𝐵 be vague sets of a residuated lattice 𝐿.Then all 𝑦∈𝐿.Let𝑧∈𝐿.Since𝑧=1∗𝑧,wegetthat 𝐴∗𝐵is defined as follows:
𝑉(𝐴∗𝐵) (𝑧) = isup {imin {𝑉𝐴 (𝑥) ,𝑉𝐵 (𝑦)} :𝑧≥𝑥∗𝑦} , (22) 𝑉𝐴∗𝐵 (𝑧) = isup {imin {𝑉𝐴 (𝑥) ,𝑉𝐵 (𝑦)}: 𝑧 ≥ 𝑥 ∗𝑦} for all 𝑧∈𝐿. (26) ≥ imin {𝑉𝐴 (𝑧) ,𝑉𝐵 (1)}≥𝑉𝐴 (𝑧) . Theorem 29. Let 𝐴 and 𝐵 be vague sets of a residuated lattice 𝐿.Then𝐴∗𝐵is a vague set of 𝐿. Hence 𝐴⪯𝐴∗𝐵. Similarly, we can show that 𝐵⪯𝐴∗𝐵. Proof. Since 𝐴 and 𝐵 are vague sets of 𝐿,wehave𝑡𝐴(𝑥) ≤ Suppose that 𝐶 is the vague filter of 𝐿 containing 𝐴 and 𝐵. 1−𝑓𝐴(𝑥) and 𝑡𝐵(𝑦) ≤ 1 −𝐵 𝑓 (𝑦) for all 𝑥, 𝑦.Thus ∈𝐿 Then
𝑡𝐴∗𝐵 (𝑧) = sup {min {𝑡𝐴 (𝑥) ,𝑡𝐴 (𝑦)}: 𝑧 ≥ 𝑥 ∗𝑦} ≤ { {1 − 𝑓 (𝑥) ,1−𝑓 (𝑦)}:𝑧≥𝑥∗𝑦} sup min 𝐴 𝐴 𝑉𝐴∗𝐵 (𝑧) ≤1−𝑓 (𝑧) . 𝐴∗𝐵 = isup {imin {𝑉𝐴 (𝑥) ,𝑉𝐵 (𝑦)}: 𝑧 ≥ 𝑥 ∗𝑦} (23) (27) ≤ isup {imin {𝑉𝐶 (𝑥) ,𝑉𝐶 (𝑦)}: 𝑧 ≥ 𝑥 ∗𝑦} Hence 𝑡𝐴∗𝐵(𝑧) +𝐴∗𝐵 𝑓 (𝑧) ≤ 1 and then 𝐴∗𝐵is a vague set of 𝐿. ≤ isup {imin {𝑉𝐶 (𝑥∗𝑦)}:𝑧≥𝑥∗𝑦}=𝜇𝐶 (𝑧) .
Theorem 30. Let 𝐴 and 𝐵 be vague filters of a residuated lattice 𝐿.Then𝐴∗𝐵is a vague filter of 𝐿. Hence 𝐴∗𝐵⪯𝐶;thatis,𝐴∗𝐵is the least upper bound of 𝐴 𝐵 Proof. By Theorem 29, 𝐴∗𝐵is a vague set of 𝐿.Wewillprove and . that 𝐴∗𝐵is a vague filter of 𝐿. Lemma 32. Let 𝐶 be a vague filter of a residuated lattice 𝐿 and (VF2) Let 𝑧1 ≤𝑧2 and 𝑥∗𝑦1.Wegetthat ≤𝑧 𝑧∈𝐿.Then imin{𝑉𝐴(𝑥),𝐵 𝑉 (𝑦)} ≤𝐴∗𝐵 𝑉 (𝑧2).Since𝑥∗𝑦≤𝑧1 is arbitrary, we have 𝑉 (𝑧 )= { {𝑉 (𝑥) ,𝑉 (𝑦)}: 𝑧 ≥𝑥∗𝑦} 𝐴∗𝐵 1 isup imin 𝐴 𝐵 1 (1) 𝑉𝐶(𝑧) = isup{VC(x ∗ y):z ≥ x ∗ y},forall𝑥, 𝑦,; 𝑧∈𝐿 (24) ≤𝑉𝐴∗𝐵 (𝑧2).
(2) 𝑉𝐶(𝑥 ∗ 𝑦) = imin{VC(x), VC(y)},for𝑥, 𝑦. ∈𝐿 (VF1) Let 𝑥, 𝑦.Then ∈𝐿
Proof. (1) Since 𝑥∗𝑦≤𝑧,then𝑉𝐶(𝑥 ∗ 𝑦)𝐶 ≤𝑉 (𝑧) by (VF2). imin {𝑉𝐴∗𝐵 (𝑥) ,𝑉𝐴∗𝐵 (𝑦)} Hence 𝑉𝐶(𝑧) ≥ isup{𝑉𝐶(𝑥∗𝑦):𝑧≥𝑥∗𝑦}. Also, we have 1∗𝑧=𝑧.Weobtainthat = imin {isup {imin {𝑉𝐴 (𝑎) ,𝑉𝐵 (𝑏)}:𝑥≥𝑎∗𝑏},
isup {imin {𝑉𝐴 (𝑐) ,𝑉𝐵 (𝑑)}:𝑦≥𝑐∗𝑑}} 𝑉𝐶 (𝑧) =𝑉𝐶 (1∗𝑧) ≤ isup {𝑉𝐶 (𝑥∗𝑦) :𝑧≥𝑥∗𝑦} . (28) = isup {isup {imin {𝑉𝐴 (𝑎∗𝑐) ,𝑉𝐵 (𝑐∗𝑑)}:
𝑥≥𝑎∗𝑏}:𝑦≥𝑐∗𝑑} (25) (2) It follows from Proposition 2 part (2) and Definition 13. ≤ isup {imin {𝑉𝐴 (𝑎∗𝑐) ,𝑉𝐵 (𝑐∗𝑑)}:
𝑥∗𝑦≥(𝑎∗𝑏) ∗ (𝑐∗𝑑)} Let VFD(𝐿)bethesetofallvaguefilters𝐴 and 𝐵 of 𝐿 such ≤ isup {imin {𝑉𝐴 (𝑒) ,𝑉𝐵 (𝑓)}:𝑥∗𝑦≥𝑒∗𝑓} that 𝑉𝐴(1) = 𝑉𝐵(1).
=𝑉𝐴∗𝐵 (𝑥 ∗ 𝑦) . Theorem 33. (𝑉𝐹𝐷(𝐿), ∩,∗) is a distributive lattice. Journal of Discrete Mathematics 7
Proof. Let 𝐴, 𝐵, 𝐶∈ VFD(𝐿) be such that 𝐴⪯𝐶.Wewill for all 𝛼, 𝛽 ∈ [0,,theset 1] 𝐴(𝛼,𝛽) is either empty or a subpositive show that (𝐴 ∗ 𝐵) ∩ 𝐶 = (𝐴 ∗ 𝐶) ∩(𝐵∗𝐶).ByLemma 32, implicative filter of 𝐿,where𝛼≤𝛽. 𝐴 𝑉(𝐴∗𝐵)∩𝐶 (𝑧) Proof. Suppose that is a subpositive implicative filter vague set of 𝐿 and 𝛼, 𝛽 ∈ [0,.Let 1] 𝐴(𝛼,𝛽) =0̸ .Since𝐴 is a vague = {(𝑉 ) 𝑧 ,𝑉 𝑧 } imin 𝐴∗𝐵 ( ) 𝐶 ( ) filter, then 𝐴(𝛼,𝛽) is a filter of 𝐿 by Theorem 21.Now,let(𝑥 → 𝑦) ((𝑦 𝑥)→∈𝐴(𝛼,𝛽).Then𝑉𝐴((𝑥 → 𝑦) = { { {𝑉 (𝑥) ,𝑉 (𝑦)}:𝑧≥𝑥∗𝑦},𝑉 (𝑧)} imin isup imin 𝐴 𝐵 𝐶 ((𝑦 𝑥) → 𝑥)) ≥[𝛼,𝛽].By(SVF3),𝑉𝐴((𝑥 → 𝑦) 𝑦) ≥ [𝛼,;thatis, 𝛽] (𝑥→ 𝑦)𝑦∈𝐴(𝛼,𝛽).Similarly,wecan = isup {imin {𝑉𝐴 (𝑥) ,𝑉𝐵 (𝑦) ,𝐶 𝑉 (𝑧)}:𝑧≥𝑥∗𝑦} show that 𝐴(𝛼,𝛽) satisfies (SVF4). Hence 𝐴(𝛼,𝛽) is a subpositive 𝐿 = isup {imin {𝑉𝐴 (𝑥) ,𝑉𝐵 (𝑦) ,𝐶 𝑉 (𝑥∗𝑦)}:𝑧≥𝑥∗𝑦} implicative filter of . Conversely, let 𝑡𝐴((𝑥 → 𝑦) ((𝑦 𝑥)→𝑥))= 𝛼 1−𝑓((𝑥 → 𝑦) ((𝑦 𝑥)→𝑥))=𝛽 = isup {imin {𝑉𝐴 (𝑥) ,𝑉𝐵 (𝑦) ,𝐶 𝑉 (𝑥) ,𝑉𝐶 (𝑦)}: 𝑧 ≥ 𝑥 ∗𝑦} and 𝐴 .Thus 𝑉𝐴((𝑥 → 𝑦) ((𝑦 𝑥) → 𝑥))≥[𝛼,𝛽].Wegetthat = { {𝑉 (𝑥) ,𝑉 (𝑥)}, isup imin 𝐴 𝐶 (𝑥 → 𝑦) ((𝑦 𝑥)→∈𝐴(𝛼,𝛽).So𝐴(𝛼,𝛽) =0̸ . Thus (𝑥 → 𝑦) 𝑦∈𝐴(𝛼,𝛽);thatis,𝑉𝐴((𝑥→ 𝑦)𝑦)≥ {𝑉𝐵 (𝑦) ,𝐶 𝑉 (𝑦)}: 𝑧 ≥ 𝑥 ∗𝑦} imin [𝛼, 𝛽]𝐴 =𝑉 ((𝑥 → 𝑦) ((𝑦 . 𝑥)→𝑥)) Similarly, we can prove that 𝐴 satisfies (SVF4). = isup {imin {𝑉𝐴∩𝐶 (𝑥) ,𝑉𝐵∩𝐶 (𝑦)}: 𝑧 ≥ 𝑥 ∗𝑦} In the following theorems, we obtain some characteriza- =𝑉(𝐴∩𝐶)∗(𝐵∩𝐶) (𝑧) . tions of subpositive implicative vague filter. (29) Theorem 38. Let 𝐴 be a vague filter of a residuated lattice 𝐿. Then 𝐴 is a subpositive implicative vague filter of 𝐿 if and only if it satisfies 5. Subpositive Implicative Vague Filters (SVF5) imin{VA((y → z)(x → y)), VA(x)} ≤ Definition 34. Let 𝐴 be a vague subset of a residuated lattice VA(y) for any 𝑥, 𝑦,, 𝑧∈𝐿 𝐿 𝐴 𝐿 . is called a subpositive implicative vague filter of if (SVF6) imin{VA((y z)→(x y)), VA(x)} ≤ VA(y) for any 𝑥, 𝑦,. 𝑧∈𝐿 (VF6) 𝑉𝐴(𝑥) ≤𝐴 𝑉 (1),forall𝑥∈𝐿,
(SVF1) imin{𝑉𝐴(((𝑥 → 𝑦) ∗ 𝑧) ((𝑦 𝑥)→ Proof. Let 𝐴 be a subpositive implicative vague filter of 𝐿.By 𝑥)),𝐴 𝑉 (𝑧)} ≤𝐴 𝑉 ((𝑥→ 𝑦)𝑦)for any Proposition 2 part (3), we have imin{𝑉𝐴((𝑦→ 𝑧)(𝑥→ 𝑥, 𝑦,, 𝑧∈𝐿 𝑦)),𝐴 𝑉 (𝑥)} = imin{𝑉𝐴(𝑥 → ((𝑦 → 𝑧) 𝐴 𝑦)),𝑉 (𝑥)} ≤ 𝑉𝐴((𝑦 → 𝑧) .By 𝑦) Proposition 2 part (7), (𝑦 → 𝑧) (SVF2) imin{𝑉𝐴((𝑧 ∗ (𝑥 𝑦)) → ((𝑦 →𝑥) 𝑦≤𝑧𝑦≤(𝑦→ 𝑧)((𝑧𝑦)→.Wegetthat 𝑦) 𝑥)),𝐴 𝑉 (𝑧)} ≤𝐴 𝑉 ((𝑥 𝑦) →𝑦) for any 𝑉 ((𝑦→ 𝑧)𝑦)≤𝑉(𝑧 𝑦)≤𝑉 ((𝑦 → 𝑧) ((𝑧 𝑥, 𝑦,. 𝑧∈𝐿 𝐴 𝐴 𝐴 𝑦)→𝑦))≤𝑉𝐴((𝑧 𝑦). →𝑦) Hence imin{𝑉𝐴((𝑦 → Proposition 35. Let 𝐴 be a vague filter of a residuated lattice 𝑧) 𝐴 𝑦),𝑉 (𝑧 𝑦)}≤ imin{𝑉𝐴((𝑧 𝑦)𝐴 →𝑦),𝑉 (𝑧 𝐿.Then𝐴 isasubpositiveimplicativevaguefilterofLifand 𝑦)} ≤𝐴 𝑉 (𝑦). Thus it satisfies (SVF5). Similarly, we can only if it satisfies prove (SVF6). Conversely, let 𝐴 satisfy (SVF5) and (SVF6); 𝑉𝐴((𝑥 → 𝑦) ((𝑦 𝑥)→𝑥))= imin{𝑉𝐴([((𝑥 → 𝑦) (SVF3) 𝑉𝐴((𝑥 → 𝑦) ((𝑦 𝑥)→𝑥))≤ 𝑦) → 1] [(𝑥 → 𝑦) ((𝑦 𝑥)→((𝑥 𝑉 ((𝑥→ 𝑦)𝑦) 𝑥, 𝑦 ∈𝐿 𝐴 for any , 𝑦) 𝑦)]),𝐴 𝑉 (1)} ≤ 𝑉𝐴((𝑥→ 𝑦)𝑦).Similarly,we 𝐴 (SVF4) 𝑉𝐴((𝑥 𝑦) → ((𝑦 → 𝑥)𝑥))≤ can prove that (SVF5) holds. Hence is a vague subpositive 𝑉𝐴((𝑥 𝑦) →𝑦) for any 𝑥, 𝑦. ∈𝐿 implicative filter.
Proposition 36. Let 𝐴 be a subpositive implicative vague filter Theorem 39. Let 𝐴 be a vague filter of a residuated lattice 𝐿. of a residuated lattice 𝐿.Then𝐴 is a vague filter of 𝐿. Then 𝐴 is a subpositive implicative vague filter of 𝐿 if and only if it satisfies Proof. We have imin{𝑉𝐴(𝑥 𝐴 𝑦),𝑉 (𝑥)} = imin{𝑉𝐴((𝑦 → 𝑦) ∗ 𝑥) ((𝑦 𝑦)→𝑦),𝑉 (𝑥)} ≤ 𝑉 ((𝑦 → 𝑦) (SVF7) 𝑉𝐴((𝑥 → 𝑦) ((𝑦 → 𝑥) 𝑥))≤ 𝐴 𝐴 𝑉 ((𝑦 𝑥) →𝑥) 𝑥, 𝑦 ∈𝐿 𝑦) =𝐴 𝑉 (𝑦).By(VS1)andTheorem 17, 𝐴 is a vague filter of a 𝐴 for any ; residuated lattice 𝐿. (SVF8) 𝑉𝐴((𝑥 𝑦) → ((𝑦 𝑥)→𝑥))≤ 𝑉𝐴((𝑦→ 𝑥)𝑥)for any 𝑥, 𝑦. ∈𝐿 The relation between subpositive implicative vague filter anditslevelsubsetisasfollows. Proof. Let (SVF7) and (SVF8) hold. We will show that 𝐴 satisfies (SVF5) and (SVF6). We have imin{𝑉𝐴((𝑦 → 𝑧) Theorem 37. Let 𝐴 be a vague set of a residuated lattice 𝐿. (𝑥 → 𝑦)),𝐴 𝑉 (𝑥)} = imin{𝑉𝐴(𝑥 → ((𝑦 → 𝑧) Then 𝐴 is a subpositive implicative vague filter of 𝐿 if and only if 𝑦)),𝐴 𝑉 (𝑥)} ≤𝐴 𝑉 ((𝑦→ 𝑧)𝑦)≤𝑉𝐴((𝑦 → 𝑧) 8 Journal of Discrete Mathematics
((𝑧 → 𝑦) 𝑦))𝐴 ≤𝑉 ((𝑧 𝑦). →𝑦)) Also, we have (𝑥→ 𝑦)𝑥≤¬𝑥𝑥by Proposition 2 part (6). 𝑉𝐴((𝑦 → 𝑧) 𝐴 𝑦)≤𝑉 (𝑧 𝑦).Since𝐴 is a vague Therefore 𝑉𝐴((𝑥→ 𝑦)𝑥)≤𝑉𝐴(¬𝑥 𝑥)𝐴 ≤𝑉 (𝑥). filter, 𝑉𝐴((𝑦→ 𝑧)𝑦)≤imin{𝑉𝐴(𝑧 𝐴 𝑦),𝑉 ((𝑧 Similarly, we can prove that (SVF10) holds. Hence 𝐴 is a 𝑦)→𝑦)}≤𝑉𝐴(𝑦).Hence𝐴 satisfies (SVF5). Analogously, subpositive implicative vague filter of 𝐿 by Theorem 40. we can prove that 𝐴 satisfies (SVF6). Hence 𝐴 is a subpositive implicative vague filter of 𝐿 by Theorem 38.Conversely,let𝐴 be a subpositive implicative vague filter of 𝐿.Wehave The extension theorem of subpositive implicative vague filters is obtained from the following theorem. [((𝑦 𝑥) → 𝑥)→1] Theorem 42. Let 𝐴 and 𝐵 be two vague filters of a residuated [((𝑥 → 𝑦) ((𝑦 →𝑥)𝑥)) lattice 𝐿 which satisfies 𝐴≤𝐵and 𝑉𝐴(1) = 𝑉𝐵(1).If𝐴 is a subpositive implicative vague filter of L, so 𝐵 is. → ( ( 𝑦 𝑥 ) →𝑥)] (30) Proof. Suppose that 𝑢=(𝑥→ 𝑦)𝑥.Then = [(𝑥 → 𝑦) ((𝑦 → 𝑥)𝑥)]
→ [((𝑦 → 𝑥) 𝑥) ((𝑦→𝑥)𝑥)] 1 = 𝑢 → 𝑢 = 𝑢 → ( ( 𝑥 → 𝑦)𝑥 (32) =(𝑥→𝑦)((𝑦→𝑥)𝑥). =(𝑥→𝑦)(𝑢→𝑥) =1.
We get that 𝑉𝐴((𝑥 → 𝑦) ((𝑦 → 𝑥) 𝑥))= We have (𝑥 → 𝑦) (𝑢 → 𝑥) ≤ ((𝑢 →𝑥)𝑦) imin{𝑉𝐴((𝑥 → 𝑦) ((𝑦 → 𝑥), 𝑥)) 𝑉𝐴([((𝑦 𝑥)→ (𝑢 → 𝑥). Therefore 𝑉𝐴(((𝑢 → 𝑥) → 𝑦) (𝑢 →𝑥))= 𝑥) → 1] [((𝑥 → 𝑦) ((𝑦 → 𝑥) 𝑥))→((𝑦 𝑉𝐴(1).By(SVF10),𝑉𝐴(1) = 𝑉𝐴(((𝑢 → 𝑥) → 𝑦) (𝑢→ 𝑥) → 𝑥)])}𝐴 ≤𝑉 ((𝑦 𝑥) →𝑥) by (SVF5). Similarly, we 𝑥)) ≤𝐴 𝑉 (𝑢 → 𝑥).Hence𝑉𝐴(𝑢 → 𝑥)𝐴 =𝑉 (1) = 𝑉𝐵(1). can prove that (SVF8) holds. Since 𝐴≤𝐵,then𝑉𝐵(𝑢 → 𝑥)𝐴 ≥𝑉 (𝑢 → 𝑥)𝐵 =𝑉 (1). We get that 𝑉𝐵((𝑥→ 𝑦)𝑥)=𝑉𝐵(𝑢) = imin{𝑉𝐵(𝑢 → Theorem 40. 𝐴 𝐿 Let be a vague filter of a residuated lattice . 𝑥),𝐵 𝑉 (𝑢)} ≤𝐵 𝑉 (𝑥).Similarly,wecanshowthat𝐺 satisfies Then 𝐴 is a subpositive implicative vague filter of 𝐿 if and only (SVF11). Hence 𝐺 is a vague subpositive implicative filter of if it satisfies 𝐿,byTheorem 40. 𝑉 ((𝑥→ 𝑦)𝑥)≤𝑉(𝑥) 𝑥, 𝑦 ∈𝐿 (SVF9) 𝐴 𝐴 for any , Definition 43. Let 𝐴 beavaguefilterofaresiduatedlattice𝐿. (SVF10) 𝑉𝐴((𝑥 𝑦)𝐴 →𝑥)≤𝑉 (𝑥) for any 𝑥, 𝑦. ∈𝐿 𝐴 is called a vague Boolean filter of 𝐿 if 𝑉 (𝑥 ∨ ¬𝑥) =𝑉 (1) 𝑥∈𝐿 Proof. Let 𝐴 be a subpositive implicative vague filter of 𝐿. (BVF1) 𝐴 𝐴 ,forall , 𝐴 Then satisfies (SVF5) and (SVF6): (BVF2) 𝑉𝐴(𝑥∨∼𝑥)=𝑉𝐴(1),forall𝑥∈𝐿.
𝑉𝐴 ((𝑥 → 𝑦) 𝑥) Similar to Theorem 40, we have the following theorem with respect to Boolean vague filters. = imin {𝑉𝐴 ( 1 → ( ( 𝑥 → 𝑦 ) 𝑥),𝑉𝐴 (1)} (31) Theorem 44. Let 𝐴 be a vague set of a residuated lattice 𝐿. = imin {𝑉𝐴 ((𝑥 → 𝑦) (1→𝑥)),𝑉𝐴 (1)} Then 𝐴 is a Boolean vague filter of 𝐿 if and only if for all 𝛼, 𝛽 ∈ [0, 1] 𝐴 ≤𝑉𝐴 (𝑥) . ;theset (𝛼,𝛽) is either empty or a Boolean filter of 𝐿,where𝛼≤𝛽. Similarly, we can prove that (SVF10) holds. Conversely, let Theorem 45. 𝐴 𝐿 𝐴 satisfy (SVF9) and (SVF10). We will show that 𝐴 satisfies Let be a vague filter of a residuated lattice . 𝐴 𝐿 (SVF5) and (SVF6); imin{𝑉𝐴((𝑦→ 𝑧)(𝑥→ Then is a subpositive implicative vague filter of if and if only 𝐴 𝑦)),𝐴 𝑉 (𝑥)} = imin{𝑉𝐴((𝑥 → (𝑦 → 𝑧) 𝐴 𝑦)),𝑉 (𝑥)} ≤ if is a Boolean vague filter. 𝑉 ((𝑦→ 𝑧)𝑦)≤𝑉(𝑦) 𝐴 𝐴 𝐴 .Hence satisfies (SVF5). 𝐴 Similarly, we can prove (SVF7) holds. Therefore 𝐴 is a Proof. Let be a subpositive implicative vague filter. We have subpositive implicative vague filter of L by Theorem 38. 𝑉𝐴 (1) =𝑉𝐴 (¬ (𝑥∨¬𝑥) (𝑥∨¬𝑥)) Theorem 41. 𝐴 𝐿 (33) Let be a vague filter of a residuated lattice . =𝑉 ((¬𝑥 ∧ ¬¬𝑥) (𝑥∨¬𝑥)) ≤𝑉 (𝑥∨¬𝑥) . Then 𝐴 is a subpositive implicative vague filter of 𝐿 if and only 𝐴 𝐴 if it satisfies Hence 𝑉𝐴(𝑥 ∨ ¬𝑥)𝐴 =𝑉 (1). Similarly, we can prove that 𝐴 𝑉 (¬𝑥𝑥)≤𝑉(𝑥) 𝑥∈𝐿 (SVF11) 𝐴 𝐴 for any , satisfies (BVF2). Hence 𝐴 is a Boolean vague filter. 𝑉 (𝑥∨¬𝑥) =𝑉 (1) 𝑉 (𝑥∨¬𝑥) = (SVF12) 𝑉𝐴(∼ 𝑥 → 𝑥)𝐴 ≤𝑉 (𝑥) for any 𝑥∈𝐿. Conversely, let 𝐴 𝐴 .Wehave 𝐴 𝑉𝐴((𝑥 𝑥) ∨ (¬𝑥𝐴 𝑥))𝑉 ((𝑥 ∨ ¬𝑥) 𝑥)= imin{𝑉𝐴((𝑥 ∨ Proof. Let 𝐴 be a subpositive implicative vague filter of 𝐿. ¬𝑥) 𝑥),𝐴 𝑉 (1)} = imin{𝑉𝐴((𝑥 ∨ ¬𝑥) 𝐴 𝑥),𝑉 (𝑥 ∨ ¬𝑥)} ≤ Then(SVF11)and(SVF12)followby(SVF9)and(SVF10). 𝑉𝐴(𝑥). Similarly, we can show that 𝐴 satisfies (SVF12). Hence Conversely, since 0≤𝑦,then𝑥→0≤𝑥→𝑦.Hence 𝐴 is a subpositive implicative vague filter by Theorem 41. Journal of Discrete Mathematics 9
6. 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.
Conflict of Interests There is no conflict of interests regarding the publication of this paper.
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