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Vague Strong Implicative Filters of Lattice Wajsberg Algebras 

 

1.Praveen Vardhan Kuppili, 2.V.B.V.N.Prasad, 3.Rama Devi Burri 
1Research Scholar, Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation , Vaddeswaram, 

Guntur, A.P, India. 

pvkuppili@yahoo.co.in 

2.V.B.V.N.Prasad, Professor, Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, 

Vaddeswaram, Guntur, A.P, India. 

vbvnprasad@kluniversity.in 

3Professor, department of Information technology, Institute of Aeronautical Engineering, Dundigal, Hyderabad, 

Telangana, India 

Ramaburri5@gmail.com 

 

Article History: 

Received: 10-04-2024 

Revised: 24-05-2024 

Accepted: 12-06-2024 

Abstract:  

In this paper, we introduce the notation of a vague strong implicative filter of lattice 

wajsberg algebra. Also, we investigate some of its properties with illustrations. Further, 

we obtain the relation between vague implicative filter and anti vague strong implicative 

filter In lattice wajssberg algebra. Finally, we establish the equivalent condition of a vague 

strong implicative filter.  

Keywords: wajsberg algebra; Lattice wajsberg algebra; Implicative filter; strong 

Implicative filter, vague Implicative strong filter; vague strong implicative filter; vague 

implicative filter, vague strong implicative filter. 

1. Introduction:  

The concept of Lattice was first defined by Dedekind in 1897 and then developed by Birkhoft.G, 

imposed an operation an open problem "Is there a common abstraction which includes Boolean 

algebra, Boolean rings and lattice ordered group or L-group is an algebraic structure connecting lattice 

and group. To answer this problem many common abstractions, namely dually residuated lattice 

ordered semigroups, commutative lattice ordered groups. lattice ordered rings,latice ordered near rings 

and lattice ordered semirings are presented.Amoung them the algebraic structure lattice ordered 

semirings or L-semiring was introduced by RangaRao.P.,[9].Also the concept proposed by 

Zadeh.L.A.[13] defining a fuzzy subset A of a given universe X characterizing the membership of an 

element x of X belonging to A by means of a membership function  µA(x) defined from X in to [0 1] 

has revolutionized the theory of Mathematical modeling. Decision making etc.,in handling the 

imprecise real life situations mathematically. Now several branches of fuzzy mathematics like fuzzy 

algebra,fuzzy topology ,fuzzy control theory ,fuzzy measure theory etc.,haveemerged.But in the 

decision making, the fuzzy theory takes care of membership of an element x only, that is the evidence 

against x belonging to A . 

Gau and Buehrer.D.J and some other areas of Mathematical modeling.Since then the theory of fuzzy 

sets developed extensively and embraced almost all subjects like engineering science and technology. 

But the membership function µA (x) gives only a approximation belong to A .To avid this and obtain a 



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better estimation and analysis of data decision making. Gau.W.L and Bueher D.J. [3] have initiated 

the study of vague sets with the hope that they form a better tool to understand, interpret and solve real 

life problems which are in general vague, than the theory of vague sets do. Ranjit Biswas[6] initiated 

the study of vague groups by Ramakrishna.N [4 ],[5 ],[7]  are grate extended the study of vague algebra. 

The objective of this paper is to contribute further to the study of vague algebra by In this section, we 

introduce vague strong implicative filter of lattice of  wajsberg  algebra and vague strong implicative 

filter of lattice of wajsberg algebra A with illustrations and investigate some properties with suitable 

examples. 

2. Preliminaries 

In this section, we recall some basic definitions and properties which are useful to develop the main 

results. 

Definition 2.1 [2] Let ( A, →,  *,1 ) be an  algebra with a binary operation  “→”and a quasi 

complement  “ * ” is called a wajsberg algebra if and only if it satisfies the following axioms for all 

x,y,z ∈ A, 

 1. 1→x=x         

 2. (x→ 𝑦)  → ((𝑦 → 𝑧) → (𝑥 → 𝑧))=1   

3. (x→ 𝑦) → 𝑦 = (y→ 𝑥) → 𝑥 

4. (x* → y∗) → (y→ 𝑥)=1. 

Definition 2.2[2] The wajsberg algebra (A, → , *, 1 ) satisfies the following properties for all x,y,z ∈ 

A,     

(i).x →x=x       

(ii). If (𝑥 →y)= 𝑦 →x=1 then x=y  

(iii).x→ 1 = 1  

(iv).x→ (𝑦 →x)=1 

(v).If x→ 𝑦 =y→ 𝑧=1 then x→ 𝑧 = 1   

(vi). If ( 𝑥 → y) → ((𝑧 → 𝑥) → (𝑧 → 𝑦)) = 1 

(vii).x → (𝑦 → 𝑧) = 𝑦 → (𝑥 → 𝑧) 

(viii).x→ 0 = 𝑥 → 1* =x* 

(ix).(x*)* =x 

(x) .x* →y* = y→ 𝑥. 

Definition 2.3 [2] The wajsberg algebra (A, →,*,1) is called a lattice wajsberg algebra if it satisfies 

the following properties for all x,y ∈ A, 

(1) a partial ordering “≤” on a lattice wajsberg algebra A, such that x≤y if and only if x→ 𝑦 = 1 

(2) (xꓦy)=(x→ 𝑦)  → 𝑦 



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(3) (xꓥy)= ( (x*→ 𝑦*)→ 𝑦* )* Thus ,we have (A,ꓦ,ꓥ,*0,1) is a lattice wajsberg algebra with lower 

bound 0 and upper bound 1. 

Theorem 2.4 [2] The wajsberg algebra (A, →,*,1) satisfies the following properties for all x,y, 𝑧 ∈ 

A, 

1. If x≤y then x →z ≥ y→ 𝑧 

2. If x≤y then z →x ≤ z→ 𝑦 

3. If x≤y→z if and only if y ≤ x→ 𝑧 

4.(xꓦy)* =(x* ꓦy*) 

5.(xꓥy)* =(x* ꓥ y*) 

6. (xꓦy)→z=(x→ 𝑧)ꓥ(y→ 𝑧 

7.x→(yꓥz)=(x→ 𝑦)ꓥ(x→ 𝑧) 

8.(x→ 𝑦)ꓦ(y→x)=1 

9.x→ (𝑦 ꓦz)=(x→ 𝑦)ꓦ(x→ 𝑧) 

10.(xꓥy) → 𝑧 =(x→ 𝑦)ꓦ(x→ 𝑧) 

11.(xꓥy) ꓦz =(xꓦ𝑧)ꓥ(yꓦz) 

12.(xꓥy) →z =(x→ 𝑦) →(x→ z) for all x,y,z in A. 

Definition 2.5[2] A lattice  wajsberg algebra  (A, → , *, 1 ) is called a lattice H-Wajsberg algebra,if it 

satisfies  xꓦyꓦ((xꓥy)→z)=1 for all x,y.z ∈ A. In a lattice H-wajsberg algebra A ,the following hold. 

1.x→(x→ 𝑦)=(x→ 𝑦) 

2.x→ (𝑦 → 𝑧) =(x→ 𝑦) →(x→ 𝑧) foa all x,y,z in A. 

Definition 2.6[2] Let  (A1, → , *, 1 ) and  (A2, → , *, 1 ) be lattice wajsberg algebras, a maping  

f: A1→ A2 is called implication homomorphism if  f(x → y) =f(x) →f(y) holds, 

Definition 2.7[2] Let  (A1, → , *, 1 ) and  (A2, → , *, 1 ) be lattice wajsberg algebras, f: A1→ A2 is 

implication homomorphism from A1   to A2  satisfies the  following properties. 

 1. f(x ꓥ y) =f(x)ꓥf(y)    2. f(x ꓦ y) =f(x)ꓦf(y)  3.f(x* )=[f(x)]*  

Definition 2.8[2] Let (A1, → , *, 1 ) be a lattice wajsberg algebra. A subset F of A is called an 

implicative filter of A if it satisfies the following properties for all x,y ∈ 𝐴. 

1.1∈ F     

2. x∈F and x→y ∈ F implies x,y ∈ 𝐹. 

Definition 2.9[1] Let X be a set. A function μ: X→[0,1] is called  a fuzzy subset on X, for all x ∈ X 

the value of μ(x) describes a degree of membership of x in μ. 



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Definition 2.10[11] Let μ be a fuzzy subset in set A. Then for t ∈ [0 ,1],the st μt ={ x ∈A : μ(x) ≥ t } 

is called a level subset of μ. 

Definition 2.11 [11]Let (A1, → , *, 1 ) be a lattice wajsberg algebra. A fuzzy subset μ of A is called a 

fuzzy implicative filter of A if it satisfies the following properties for all x,y in A. 

1. μ(1)≥ μ(x) 

2. μ(𝑧)≥min{ μ(𝑦), μ((y→z) }. 

Definition 2.12 [12]Let μ be a fuzzy implicative filter of a lattice  wajsberg algebra  A, then A is 

called x≤y implies μ(x)≤ μ(y) for all x,y in A 

Definition 2.13 [11]Let (A1, → , *, 1 ) be a lattice wajsberg algebra. A fuzzy subset μ of A is called a  

strong implicative filter  if it satisfies  

1. 1∈ 𝐹 

2. x→ (𝑦 → z) ∈ 𝐹    and x → 𝑦 ∈ F    implies    x→ 𝑧 ∈ F. 

Definition 2.14[2] Let μ be a fuzzy implicative filter of a lattice wajsberg algebra.A fuzzy sub set μ 

of A is called fuzzy strong implicative filter of A if it satisfies the properties. 

 1.μ(1)≥ μ(x) 

2.μ(𝑥 → 𝑧)≥min{ μ(𝑥 → 𝑦), μ((x→y→z) }. 

Definition 2.15 [3]: A vague  set  A  in  the  universe  of  discourse  X is  a  pair   ( tA ,  fA  )   where tA 

:X→[0 ,1] , fA: X→ [0,1] with tA(x)+fA(x) ≤ 1 for all x in X. Here tA is called the membership function 

and fA is called non-membership function and also called true membership function, false membership 

function respectively. 

3. Conclusions:  

In this paper, we have introduced the definitions of vague  WI-ideal lattice ideal of lattice Wajsberg 

algebra. We have discussed some of their properties with illustrations. Also, we have shown that every  

vague  WI-ideal of lattice Wajsberg algebra is an Finally, we have shown that collection of WI-ideals 

of lattice Wajsberg algebras is an  lattice ideal of lattice Wajsberg algebra. But, the converse part is 

true only in the lattice H-Wajsberg algebras. Finally, we have shown that collection of WI-ideals of 

lattice Wajsberg algebras is an 

Results : 3. Vague strong Implicative Filters 

In this section, we introduce vague implicative filter of lattice of   wajsberg algebra and vague strong 

implicative filter of lattice of wajsberg algebra A with illustrations and investigate some properties. 

Definition 3.1 Let  w =(A, → , *, 1 ) be a lattice wajsberg algebra. An vague set A= ( tA ,  fA  ) of w is 

called vague implicative filter of A if it satisfies the properties. 

1. mA 
(1) ≥  mA 

(x)     and nA 
(1)  ≤  nA 

(x) 

2. mA
(𝑦) ≥ min{ m(𝑥), m(𝑥→𝑦) } 

n(𝑦) ≤ max {nA
(𝑥), nA

(𝑥→𝑦) } for all x, y ,z ∈ A. 



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Definition 3.2 Let w= (A, → , *, 1)be a lattice wajsberg algebra. An vague set A= ( mA ,  nA  ) of w is 

called vague strong  implicative filter of A if it satisfies the properties. 

1. mA 
(1) ≥  mA 

(x)     and nA 
(1)  ≤  nA 

(x) 

𝑚𝐴
(𝑥→𝑧) ≥ min{ mA

(𝑥→𝑦), mA
(𝑥→(𝑦→𝑧)) } 

3. nA
(𝑥→𝑧) ≤  max{ nA

(𝑥→𝑦), nA
(𝑥→(𝑦→𝑧)) }. 

Example 3.2 Let a set A={0,a,b,c,d,1} with the following figures 3.2.1,3.2.2 and 3.2.3 as a partial 

ordering .Define a quasi complement  “ * ” and a binary operation → on A as in the tables 3.2.1 and 

3.2.2.    

Table 3.2.1 (Implication) 

→ 0 A b C D 1 

0 1 1 1 1 1 1 

A D 1 a C C 1 
b c 1 1 c c 1 

C B A b 1 A 1 

D A 1 a 1 1 1 

1 0 A b C D 1 

 

Table:3.2 .2    (complement) 

X x* 

   0 1 

 A C 

  B D 

C A 

D B 

1 0 

  

Define ‘ꓦ’ and ‘ꓥ’ operations on A as follows:  

a) (xꓦy)=(x→y)→ 𝑦  

b) (xꓥy)=((  x* →y* )→y* ))* for all x,y in A, then A is a lattice wajsberg algebra. 

Let A={(0,0.4,0.1)(a,0.7,0.2)(b,0.4,0.2)(c,0.6,0.3)(d 0.5,0.1),(1,0.7,0.3} be a vague strong implicative 

filter of A but not an vague strong implicative filter of lattice wajsberg algebra w. 

  Sol:-      (i) mA 
(1) ≥  mA 

(x)     and nA 
(1)  ≤  nA 

(x)     

     (ii)  mA
(𝑦) ≥ min{ mA

(𝑦), mA
(𝑥→𝑦) } and (iii). nA

(𝑦) ≤ max {nA
(𝑥), nA

(𝑥→𝑦) }  

for all x, y ,z ∈ A. 

Clearly (1) mA 
(1) ≥  mA 

(x)     and nA 
(1)  ≤  nA 

(x)    for all x in A 

     (ii)  mA
(𝑦)= mA

(𝑏)=0.4    mA
(𝑎→𝑏)=mA

(𝑎)=0.7 and min{mA
(𝑏), mA

(𝑎)}=min{0.4,0.7}=0.4 

      (iii)nA
(𝑎→𝑏)=nA

(𝑎)=0.2 max{nA
(𝑥→𝑦), nA

(𝑥→(𝑦→𝑧) } 



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=max {nA
(𝑎), nA

(𝑎→𝑐)}=max{nA
(𝑎), nA

(𝑐)}=0.7 

Therefore nA
(𝑥→𝑧) ≤ max { nA

(𝑥→𝑦), nA
(𝑥→(𝑦→𝑧) } 

Hence A={(0,0.4,0.1)(a,0.7,0.2)(b,0.4,0.2)(c,0.6,0.4)(d 0.5,0.1),(1,0.7,0.3} be a vague  implicative 

filter of A but not an vague strong implicative filter of lattice wajsberg algebra W. 

Theorem 3.3 Let W be a lattice wajsberg algebra, and let A= ( mA ,  nA  ) be a vague strong implicative 

filter of W, then A= ( mA ,  nA  ) is an vague implicative filter of W. 

Proof:- Let  w =(A, → , *, 1 ) be a lattice wajsberg algebra ,by the definition of w 

1. mA 
(1) ≥  mA 

(x)     and nA 
(1)  ≤  nA 

(x) 

2. mA
(𝑥→𝑧) ≥ min{ mA

(𝑥→𝑦), mA
(𝑥→(𝑦→𝑧)) } 

3. nA
(𝑥→𝑧) ≤  max{ nA

(𝑥→𝑦), nA
(𝑥→(𝑦→𝑧)) } 

Put x=1 in above  (2) and (3) then we get 2. mA
(1→𝑧) ≥ min{ mA

(1→𝑦), m(1→(𝑦→𝑧)) } 

Implies  mA
(𝑧) ≥ min{ mA

(𝑦), mA
(𝑦→𝑧) } and  

nA
(1→𝑧) ≤  max{ nA

(1→𝑦), nA
(1→(𝑦→𝑧)) } implies that nA

(𝑧) ≤  max{ nA
(𝑦), nA

(𝑦→𝑧) } 

Hence A= ( mA ,  nA  ) is an vague implicative filter of W. 

Theorem 3.4 Let w be a lattice wajsberg algebra. Then w is a H-wajsberj algebra if and only if each 

a vague implicative filter of w is an vague strong implicative filter. 

Proof:- Let  w =(A, → , *, 1 ) be a lattice wajsberg algebra and let A= ( mA ,  nA  ) be a vague strong 

implicative filter of w, we have 1. mA 
(1) ≥  mA 

(x)     and nA 
(1)  ≤  nA 

(x) 

2. mA
(𝑦) ≥ min{ mA

(𝑦), mA
(𝑥→𝑦) } 

3. nA
(𝑥→𝑧) ≤  max{ nA

(𝑥→𝑦), nA
(𝑥→(𝑦→𝑧) } also we have by the definition of H-wajsberj algebra   x→

(𝑦 → 𝑧) =(x→ 𝑦) →(x→ 𝑧) foa all x,y,z in W. 

Implies min{ mA
(𝑥→𝑦), mA

(𝑥→(𝑦→𝑧)) }= min{mA
(𝑥→𝑦), mA

(𝑥→𝑦)→(𝑥→𝑧)}≤ mA
(𝑥→𝑧). 

Therefore  mA
(𝑥→𝑧).≥ min{mA

(𝑥→𝑦), mA
(𝑥→𝑦)→(𝑥→𝑧)}…….(3.4.1) 

And  max{ nA
(𝑥→𝑦), nA

(𝑥→(𝑦→𝑧)) }= max{nA
(𝑥→𝑦), nA

(𝑥→𝑦)→(𝑥→𝑧)}≥nA
(𝑥→𝑧). 

Therefore  nA
(𝑥→𝑧)≤ max{nA

(𝑥→𝑦), nA
(𝑥→𝑦)→(𝑥→𝑧)} for all x,y,z in W…….(3.4.2) 

From (3.4.1) and (3.4.2)” w” is an vague strong implicative filter. 

Conversely, we consider an vague strong  implicative filter A= 

{(0,0,0.7)(a,0,0.7)(b,0,0.7),(c,0,0.7)(d,0,0.7)(1,0.8,0)} of w is vague strong implicative filter. Then A 

is implicative filter, and then implies that 

               A is lattice H- wajsberg algebra. 

Theorem 3.5 Let w be a lattice wajsberg algebra and  A= ( mA ,nA  )be a vague set of A,if  A=(mA ,  nA  

) vague  strong implicative filter, Then the following are satisfied and equivalent. 



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(i) If A= ( mA ,nA  ) is an vague implicative filter and for all x,y ∈ 𝐴, 

                   mA
(𝑥→𝑦) ≥ min{ mA

(𝑥→(𝑥→𝑦) and nA
(𝑥→𝑦) ≤ min{ nA

(𝑥→(𝑥→𝑦) } 

           (ii)If A= ( mA ,nA  ) is an vague implicative filter and for all x,y,z ∈ 𝐴, 

 mA
(𝑥→𝑦)→(𝑥→𝑧)≥ mA

(𝑥→(𝑦→𝑧))  and  nA
(𝑥→𝑦)→(𝑥→𝑧) ≤  nA

(𝑥→(𝑦→𝑧)) . 

 (iii)    mA
(1)≥ mA

(𝑥)  and  nA
(1)≤ nA

(𝑥) for all x,y,z in A. 

(iv)  mA
(𝑥→𝑦) ≥min{ mA

(𝑧→(𝑥→(𝑥→𝑦))),   mA
(𝑧) }  and nA

(𝑥→𝑦) ≤max{ nA
(𝑧→(𝑥→(𝑥→𝑦))),    nA

(𝑧) }  , for 

all x,y,z in A. 

Proof:- (i) implies (ii): 

             Let A= ( mA ,nA  ) is an vague strong implicative filter and for all x,y ∈ 𝐴, 

We have mA 
(1) ≥  mA 

(x)     and nA 
(1)  ≤  nA 

(x) 

     mA
(𝑥→𝑧) ≥ min{ mA

(𝑥→𝑦)  ,  mA
(𝑥→(𝑦→𝑧) }and  nA

(𝑥→𝑧) ≤ max{ nA
(𝑥→𝑦) ,  nA

(𝑥→(𝑦→𝑧) } 

     we have 1. mA 
(1) ≥  mA 

(x)     and nA 
(1)  ≤  nA 

(x) 

 mA
(𝑥→𝑧) ≥ min{ mA

(𝑥→𝑦), mA
(𝑥→(𝑦→𝑧)) } and nA

(𝑥→𝑧) ≤  max{ nA
(𝑥→𝑦), nA

(𝑥→(𝑦→𝑧)) } 

Put  x= 1 then  mA
(𝑧) ≥ min{ mA

(𝑦), mA
(𝑦→𝑧)) } and  

nA
(𝑧) ≤  max{ nA

(𝑦), nA
(𝑦→𝑧)) } for all x,y z ∈ 𝐴. 

Therefore A= ( mA ,nA  ) is an vague implicative filter and for all x,y,z ∈ 𝐴. 

From mA
(𝑥→𝑧) ≥ min{ mA

(𝑥→𝑦), mA
(𝑥→(𝑦→𝑧)) } and nA

(𝑥→𝑧) ≤  max{ nA
(𝑥→𝑦), nA

(𝑥→(𝑦→𝑧)) } 

Put z=y in the above condition then 

 mA
(𝑥→𝑦) ≥  mA

(𝑥→(𝑥→𝑦))    and   nA
(𝑥→𝑦) ≤   nA

(𝑥→(𝑦→𝑧))  

If for any x, y, z ∈ 𝐴 mA
(𝑥→𝑦) ≥  mA

(𝑥→(𝑥→𝑦), and nA
(𝑥→𝑦) ≤  nA

(𝑥→𝑦), nA
(𝑥→(𝑥→𝑦)) }. 

Implies  mA
(𝑥→𝑦)→(𝑥→𝑧) =mA

(𝑥→((𝑥→𝑦)→𝑧)) ≥ mA
((𝑥→𝑥→((𝑥→𝑦)→𝑧))). 

Therefore A= ( mA ,nA  ) is an vague implicative filter and for all x,y,z ∈ 𝐴, 

 mA
(𝑥→𝑦)→(𝑥→𝑧)≥ mA

(𝑥→(𝑦→𝑧))  and  nA
(𝑥→𝑦)→(𝑥→𝑧) ≤  nA

(𝑥→(𝑦→𝑧)) . 

 (ii) implies (iii):- Let (ii) be hold .Then, it is clear    mA
(1)≥ mA

(𝑥)  and  

 nA
(1)≤ nA

(𝑥) . If any x,y ∈ A, We have  mA
(𝑥→𝑦)→(𝑥→𝑧)≥ mA

(𝑥→(𝑦→𝑧) and  

nA
(𝑥→𝑦)→(𝑥→𝑧) ≤  nA

(𝑥→(𝑦→𝑧)).Put y=x, then, we get 𝑚 A
(𝑥→𝑥)→(𝑥→𝑧)≥ mA

(𝑥→(𝑥→𝑧) implies 

 mA
(𝑥→𝑧)≥ mA

(𝑥→(𝑥→𝑧)  and we have for any x,y in A  mA
(𝑥→𝑦)≥ mA

(𝑥→(𝑥→𝑦) since  mA
(𝑥)  is an 

implicative filter and t hen ,we have   mA
(𝑥→(𝑥→𝑦)≥ min{ mA

(𝑧→(𝑥→(𝑥→𝑦)), mA
(𝑧)}. 

Hence  mA
(𝑥→𝑧)≥ min{ mA

(𝑧→(𝑥→(𝑥→𝑦)),  mA
(𝑧)}. 

Similarly nA
(𝑥→𝑧)≤ max{ nA

(𝑧→(𝑥→(𝑥→𝑦)),  nA
(𝑧)}. 



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(iii) Implies (i):  Let (iii) be hold. 

Put x=1then , we get  mA
(𝑦)≥min{ mA

(𝑧→𝑦) ,  mA
(𝑧) } and  nA

(𝑥→𝑦)≤ nA
(𝑥→(𝑥→𝑦) 

Implies    mA
(1)≥ mA

(𝑥)  and n(1)≤ nA
(𝑥) for all x, y, z in A. 

Theorem 3.5 A= ( mA ,nA  ) be a vague strong  implicative filter of lattice wajsberg algebra A if and 

only if and the fuzzy sets  mA ,𝑛𝐴
𝑐

 strong  implicative filter  A where 𝑛𝐴
𝑐(𝑥)=1- nA

(x) for all x ∈ 𝐴. 

Proof:- Let A= ( mA ,nA  ) be a vague strong  implicative filter of lattice wajsberg algebra A. From 

the definition we have the fuzzy subset mA (Membership function) is strong implicative filter of A. 

Now 𝑛𝐴
𝐶(1) =1-𝑛𝐴

(1)≥1-𝑛𝐴
(x) =𝑛𝐴

𝐶 (x)   and  

 𝑛𝐴
𝐶(x→ 𝑧) =1-𝑛𝐴(x→ 𝑧)≥1-𝑛𝐴(x→ 𝑧) =𝑛𝐴

𝐶 (x→ 𝑧)≥1-max{ 𝑛𝐴(x→ 𝑦), 𝑛𝐴(x→ (𝑦 → 𝑧)) } 

  =min{ 1 − 𝑛𝐴(x→ 𝑦), 1 − 𝑛𝐴(x→ (𝑦 → 𝑧)) }=min{ 𝑛𝐴
𝐶(x→ 𝑧), 𝑛𝐴

𝐶(x→ (𝑦 → 𝑧)) } 

𝑛𝐴
𝐶(x→ 𝑧)≥ min{ 𝑛𝐴

𝐶(x→ 𝑧), 𝑛𝐴
𝐶(x→ (𝑦 → 𝑧)) } 

Therefore 𝑛𝐴
𝑐

 strong implicative filter A. 

Conversely, if mA ,𝑛𝐴
𝑐

   fuzzy strong  implicative filter  A. Then we have mA
(1)≥  mA

(x)  and  

1-nA
(1)= 𝑛𝐴

𝑐(1)≥𝑛𝐴
𝑐(𝑥)=1- nA

(x) implies that nA
(x)≥ nA

(1)      also we have  

 mA
(𝑥→𝑧) ≥ min{ mA

(𝑥→𝑦), mA
(𝑥→(𝑦→𝑧)) } and 

  𝑛𝐴
𝐶(x→ 𝑧)≥ min{ 𝑛𝐴

𝐶(x→ 𝑧), 𝑛𝐴
𝐶(x→ (𝑦 → 𝑧)) } 

Implies that 1 − 𝑛𝐴(x→ 𝑧)≥min{1 − 𝑛𝐴(x→ 𝑦), 1 − 𝑛𝐴(x→ (𝑦 → 𝑧)) } 

                                       =1- max{𝑛𝐴(x→ 𝑦), 𝑛𝐴(x→ (𝑦 → 𝑧)) }. 

Therefore 𝑛𝐴(x→ 𝑧)≤ max{𝑛𝐴(x→ 𝑦), 𝑛𝐴(x→ (𝑦 → 𝑧)) }. 

Discussion (Acknowledgements): The authors are grateful to Prof.T.Eswarlal for his valuable 

suggestions and discussions on this work. 

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ISSN: 1074-133X 

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302 
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