Ratio Mathematica                                                                                  Volume 42, 2022 

 
 

 

Vague Positive Implicative and Associative     

W- Implicative Ideals of Lattice Wajsberg 

Algebras 

 

A. Ibrahim * 

M. Mohamed Rajik ** 
 

Abstract 

In the present paper, we introduce the notions of vague positive implicative, and vague 

associative W-implicative ideals of lattice Wajsberg algebra. Further, we investigate some 

relevant properties.  Moreover, we obtain some relationship between the vague 

associative W-implicative ideal, and the vague  W-implicative ideal.  

 

Keywords: Wajsberg algebra; Lattice Wajsberg algebra; W-implicative ideal; Vague set; 
Vague W-implicative ideal; Vague positive implicative W-implicative ideal; Vague 
associative W-implicative ideal. 

AMS Mathematical Subject Classification 2020: 06B10, 06D35, 06D72.* 

 

 

 

 
 

* P.G. and Research Department of Mathematics,  H. H. The Rajah’s College, Pudukkottai, Affiliated to    
  Bharathidasan University, Tamilnadu, India.  Email: dribra@hhrc.ac.in; dribrahimaadhil@gmail.com 
** Department of Mathematics, Jansons Institute of Technology, Coimbatore, Research Scholar, P.G. and 
Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan 

University, Tamilnadu, India.Email: rajik4u@gmail.com. 
* Received on April 2nd, 2022. Accepted on June 12nd, 2022. Published on June 30th, 2021.  

doi: 10.23755/rm.v39i0.747. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY licence agreement. 

 
 

 

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A. Ibrahim and M. Mohamed Rajik 
 

1.  Introduction 

 The concept of Wajsberg algebra was first proposed by Mordchaj Wajsberg[10] in 

1935, and analysed by Font, Rodriguez and Torrens[4] in 1984. Zadeh introduced the 

notion of fuzzy set in 1965. Fuzzy logic has been applied to many fields, from control 

theory to artificial intelligence.  In 1993, the idea of vague set was introduced by Gau and 

Buehrer[5]. Vague set as an extension of fuzzy sets, the idea of vague set is that the 

membership of every element can be divided into two aspects including supporting and 

opposing. It is the new extension not only provides a significant addition to existing 

theories for handling uncertainties, but it leads to potential areas of further field research 

and pertinent applications. The authors [9], introduced the notions of vague W-implicative 
ideals, vague implicative W-implicative ideals of lattice Wajsberg algebra, and 
investigated some properties. 

 In this paper, we introduce the definitions of the vague positive implicative              

W-implicative ideal, and the vague associative W-implicative ideal of lattice Wajsberg 
algebra. Also, we discuss the relationship between the vague positive implicative                 

W-implicative ideal, and the vague W-implicative ideal. 
 

2.  Preliminaries 
 In this section, we recall some basic definitions and results that are helpful in 

developing our main results. 

Definition 2.1[4] Let (𝕎, →, ∗ ,1) be an algebra with a binary operation “→” and a quasi 
complement “∗ ”. Then it is called Wajsberg algebra, if the following axioms are satisfied 
for all 𝑥, 𝑦, 𝑧 ∈ 𝕎,  
i 1 → 𝑥 = 𝑥 
ii (𝑥 → 𝑦) → ((𝑦 → 𝑧) → (𝑥 → 𝑧)) = 1 

iii (𝑥 → 𝑦) → 𝑦 = (𝑦 → 𝑥) → 𝑥 
iv (𝑥∗ → 𝑦∗) → (𝑦 → 𝑥) = 1. 
 

Proposition 2.2[4] A Wajsberg algebra (𝕎, →, ∗ ,1)is satisfied the following axioms for 
all 𝑥, 𝑦, 𝑧 ∈ 𝕎, 
i 𝑥 → 𝑥 = 1 

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 (𝑥 → 𝑦) → ((𝑧 → 𝑥) → (𝑧 → 𝑦)) = 1 

vii 𝑥 → (𝑦 → 𝑧) = 𝑦 → (𝑥 → 𝑧) 

viii 𝑥 → 0 = 𝑥 → 1∗ =  𝑥 ∗ 

ix (𝑥∗)∗ = 𝑥 

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x (𝑥∗ → 𝑦∗) = 𝑦 → 𝑥. 

Definition 2.3[4] A Wajsberg algebra (𝕎, →, ∗ ,1) is called a lattice Wajsberg algebra, if 
the following conditions are satisfied for all 𝑥, 𝑦 ∈ 𝕎,  

i The partial ordering " ≤ "on a Wajsberg algebra 𝕎, such that 𝑥 ≤ 𝑦 if and only 
if  𝑥 → 𝑦 = 1 

ii  (𝑥 ∨ 𝑦) = (𝑥 → 𝑦) → 𝑦 

iii (𝑥 ∧ 𝑦) = ((𝑥∗ → 𝑦∗) → 𝑦∗)∗ .  

Thus, (𝕎, ∨, ∧, ∗, 0, 1) is a lattice Wajsberg algebra with lower bound 0 and upper 
bound 1. 

Proposition 2.4[4] Let (𝕎, →, ∗ ,1) be lattice Wajsberg algebra. Then the following 
axioms hold for all 𝑥, 𝑦, 𝑧 ∈ 𝕎, 

i If 𝑥 ≤ 𝑦 then 𝑥 → 𝑧 ≥ 𝑦 → 𝑧 and 𝑧 → 𝑥 ≤ 𝑧 → 𝑦 

ii 𝑥 ≤ 𝑦 → 𝑧 if and only if 𝑦 ≤ 𝑥 → 𝑧 

iii (𝑥 ∨ 𝑦)∗ = (𝑥 ∗ ∧ 𝑦∗) 

iv (𝑥 ∧ 𝑦)∗ = (𝑥∗ ∨ 𝑦∗) 

v (𝑥 ∨ 𝑦) → 𝑧 = (𝑥 → 𝑧) ∧ (𝑦 → 𝑧) 

vi 𝑥 → (𝑦 ∧ 𝑧) = (𝑥 → 𝑦) ∧ (𝑥 → 𝑧) 

vii (𝑥 → 𝑦) ∨ (𝑦 → 𝑥) = 1 

viii 𝑥 → (𝑦 ∨ 𝑧) = (𝑥 → 𝑦) ∨ (𝑥 → 𝑧) 

ix (𝑥 ∧ 𝑦) → 𝑧 = (𝑥 → 𝑧) ∨ (𝑦 → 𝑧) 

x (𝑥 ∧ 𝑦) ∨ 𝑧 = (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧) 

xi (𝑥 ∧ 𝑦) → 𝑧 = (𝑥 → 𝑦) ∧ (𝑥 → 𝑧). 

Definition 2.5[6] Let(𝕎, →, ∗ ,1)be a lattice Wajsberg algebra. Then it is called lattice   
H-Wajsberg algebra, if it satisfied (𝑥 ∨ 𝑦) ∨ (((𝑥 ∧ 𝑦) → 𝑧) = 1 for all 𝑥, 𝑦, 𝑧 ∈ 𝕎.  

Note. In a lattice H-Wajsberg algebra 𝕎, the following hold: 

i 𝑥 → (𝑥 → 𝑦) = (𝑥 → 𝑦) 

ii 𝑥 → (𝑦 → 𝑧) = (𝑥 → 𝑦) → (𝑥 → 𝑧)for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. 

Definition 2.6[6] Let 𝐿 be a lattice. An ideal 𝐼 of 𝐿 is a non-empty subset of 𝐿 is called a 
lattice ideal, if the following axioms are satisfied for all 𝑥, 𝑦 ∈ 𝐼, 

i 𝑥 ∈ 𝐼and𝑦 ≤ 𝑥 imply 𝑦 ∈ 𝐼 

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A. Ibrahim and M. Mohamed Rajik 
 

ii 𝑥, 𝑦 ∈ 𝐼implies 𝑥 ∨ 𝑦 ∈ 𝐼. 

Definition 2.7[6] Let (𝕎, →, ∗ ,1) be a lattice Wajsberg algebra. Let 𝐼 be a non-empty 
subset of 𝕎. Then 𝐼 is called a W-implicative ideal of 𝕎, if the following axioms are 
satisfied for all 𝑥, 𝑦 𝜖𝕎, 

i 0 ∈  𝐼 

ii (𝑥 → 𝑦)∗  ∈ 𝐼 and 𝑦 ∈  𝐼 imply 𝑥 ∈  𝐼. 

Definition 2.8[5] A vague set 𝐴 in the universal of discourse 𝕎 is characterized by two 
membership functions given by: 

i A truth  membership function 𝑡𝐴: 𝑋 → [0,1] and 

ii A false  membership function 𝑓𝐴: 𝑋 → [0,1]. 

Where 𝑡𝐴(𝑥) is a lower bound of the grade of membership of 𝑥 derived from the 
“evidence for x”, and 𝑓𝐴 (𝑥) is a lower bound on the negation of 𝑥 derived from the 
“evidence against x” and 𝑡𝐴(𝑥) + 𝑓𝐴 (𝑥) ≤ 1. Thus the grade of membership of x in the 
vague set 𝐴 is bounded by subinterval [𝑡𝐴(𝑥), 1 − 𝑓𝐴 (𝑥)] of [0, 1]. The vague set 𝐴 is 
written as 𝐴 = {〈𝑥, [𝑡𝐴(𝑥), 𝑓𝐴(𝑥)]〉/ 𝑥 ∈ 𝕎}. 

Where the interval [𝑡𝐴(𝑥), 1 − 𝑓𝐴(𝑥)] is called the value of 𝑥 in the vague set 𝐴 and 
denoted by 𝑉𝐴(𝑥). 

Definition 2.9[5] A vague set 𝐴 of a universe 𝑋 with 𝑡𝐴(𝑥) = 0 and 𝑓𝐴(𝑥) = 1 for all 𝑥 ∈
𝕎, is called the zero vague set of 𝕎. 

Definition 2.10[5] A vague set 𝐴 of a universe 𝑋 with 𝑡𝐴(𝑥) = 1 and 𝑓𝐴(𝑥) = 0 for all 
𝑥 ∈ 𝕎 is called the zero vague set of 𝕎. 

Definition 2.11[5] Let 𝐴 be a vague set of a universe 𝑋 with the truth membership 
function 𝑡𝐴 and the false membership function𝑓𝐴. For any 𝛼, 𝛽 ∈ [0,1]with 𝛼 ≤ 𝛽, the 
(𝛼, 𝛽) − cut of a vague set A is a crisp subset 𝐴(𝛼,𝛽)of the set 𝑋 given by  

𝐴(𝛼,𝛽) = {𝑥 ∈ 𝒲/ 𝑉𝐴(𝑥) ≥  [𝛼, 𝛽]}. 

Definition 2.12[5] The 𝛼-cut, 𝐴𝛼  of the vague set is the (𝛼, 𝛼)-cut of 𝐴 and hence given 
by 𝐴𝛼 = {𝑥 ∈ 𝕎/ 𝑡𝐴(𝑥) ≥ 𝛼}. 
 

Definition 2.13[4] Let 𝐼 = [0,1] denote the family of all closed subintervals of [0,1]. 
If 𝐼1 = [𝑎1, 𝑏1], 𝐼2 = [𝑎2, 𝑏2] are two elements of 𝐼[0,1], we call 𝐼1 ≥ 𝐼2if 𝑎1 ≥ 𝑎2 and 
𝑏1 ≥ 𝑏2. We define the term rmax to mean the maximum of two intervals as  
𝑟𝑚𝑎𝑥[𝐼1,  𝐼2] = [max{𝑎1, 𝑎2} , max{𝑏1, 𝑏2}].   

Similarly, we can define the term rmin of any two intervals. 

Definition 2.14[5] The intersection of two vague sets 𝐴 and 𝐵 with respective truth 
membership functions and the false  membership functions 𝑡𝐴, 𝑡𝐵 , 𝑓𝐴 𝑎𝑛𝑑 𝑓𝐵 is a vague set 
𝐶 = 𝐴 ∩ 𝐵, whose truth membership function and false  membership functions are related 
to those of 𝐴 and 𝐵 by 
𝑡𝐶 = min{𝑡𝐴, 𝑡𝐵 } , 1 − 𝑓𝐶 = min{1 − 𝑓𝐴, 1 − 𝑓𝐵 } =  1 − max {𝑓𝐴,, 𝑓𝐵,}. 

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Definition 2.15[9] Let 𝐴 be a vague set of lattice Wajsberg algebra 𝕎. Then 𝐴 is called a 
vague WI-ideal of  𝕎, if the following axioms are satisfied for all 𝑥, 𝑦 𝜖 𝕎,  

i 𝑉𝐴(0) ≥ 𝑉𝐴(𝑥), 

ii        𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(𝑥 → 𝑦)
∗, 𝑉𝐴(𝑦)}. 

3.1.  Vague positive implicative W-implicative ideal  
 In this section, we introduce the definition of vague positive implicative              

W-implicative ideal of lattice Wajsberg algebra, and investigate some related properties. 

Definition 3.1.1 A vague set 𝐴  of lattice Wajsberg algebra 𝕎 is called a vague positive 
implicative W-implicative ideal of  𝕎,  if for all 𝑥, 𝑦, 𝑧 ∈ 𝕎, 

i  𝑉𝐴(0) ≥ 𝑉𝐴(𝑥)  

ii 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)}.  

Example 3.1.2 Consider a set 𝕎 = {0, 𝑎, 𝑏, 𝑐, 1} with partial ordering as in Figure 3.1.1. 
Defining a binary operation ′ → ′  and a quasi complement ‘∗’ on 𝕎 as given in tables 
3.1.1 and 3.1.2. 

 

 

 

 

 

 

 

 

                     Figure: 3.1.1                          Table: 3.1.1                     Table: 3.1.2 

                  Lattice diagram                       Complement                     Implication 

 

Define " ∨ " and " ∧ " operations on 𝕎 as follows: 
 (𝑥 ∨ 𝑦) = (𝑥 → 𝑦) → 𝑦 
 (𝑥 ∧ 𝑦) = ((𝑥∗ → 𝑦∗) → 𝑦∗)∗ for all 𝑥, 𝑦 ∈ 𝕎. 

Then, (𝕎, ∨, ∧, ∗, 0, 1) is a lattice Wajsberg algebra. 

Let 𝐴 be a vague set of 𝕎 defined by 

𝐴 = {〈0, [0.6,0.3]〉 , 〈𝑎, [0.5,0.2]〉 , 〈𝑏, [0.6,0.2]〉 , 〈𝑐, [0.7,0.2]〉 , 〈1, [0.7,0.3]〉}  

Then,  𝐴 is is vague positive implicative W-implicative ideal of  𝕎. 

Theorem 3.1.3 Every vague positive implicative W-implicative ideal of lattice Wajsberg 

algebra 𝕎 is a vague W-implicative ideal of 𝕎. 

Proof: Let A be a vague positive implicative W-implicative ideal of 𝕎. 

𝑥 𝑥 ∗ 
0 1 

a b 

b a 

c c 

1 0 

→ 0 a b c 1 
0 1 1 1 1 1 

a a a c 1 1 

b c b 1 1 1 

c b 1 1 1 1 

1 0 a b c 1 

1 

b 

a 

c 

0 

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A. Ibrahim and M. Mohamed Rajik 
 

Then from (ii) of definition 3.1.1,  

 we have 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)} for all 𝑥, 𝑦, 𝑧 ∈ 𝕎.                                   

                  (3.1.1) 

Taking  𝑥 = 𝑦, 𝑦 = 𝑥, and  𝑧 = 𝑥 in (3.1.1), we get 

                                𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛 {𝑉𝐴(((𝑥 → (𝑥 → 𝑥)
∗)∗ → 𝑦)∗), 𝑉𝐴(𝑦)}  

                                          = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑥 → 1)
∗ →)∗)∗, 𝑉𝐴(𝑦)}  

                                          = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑥 → 0)
∗ →)∗)∗, 𝑉𝐴(𝑦)} 

                                          = 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 𝑦))
∗, 𝑉𝐴(𝑦)}   

Thus, 𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 𝑦))
∗, 𝑉𝐴(𝑦)}, and 𝑉𝐴(0) ≥ 𝑉𝐴(𝑥). ∎ 

Note. The converse of the above proposition may not be true. 

Proposition 3.1.4 Let 𝑉𝐴 be a vague implicative W-implicative ideal of lattice Wajsberg 
algebra 𝕎. 𝑉𝐴 is a vague positive implicative W-implicative ideal of  𝕎 if and only if  
𝑉𝐴(𝑥) ≥  𝑉𝐴(((𝑥 → (𝑦 → 𝑥)

∗)∗ for all 𝑥, 𝑦 ∈ 𝕎. 

Proof: Let 𝑉𝐴 be a vague positive implicative W-implicative ideal of 𝕎, then from (ii) of 
definition 3.1.1 we have  

𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)} for all 𝑥, 𝑦, 𝑧 ∈ 𝕎.              (3.1.2) 

Substituting 𝑥 = 0, 𝑦 = 𝑥 and 𝑧 = 𝑦 in (3.1.2) we get 

                                      𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑥 → (𝑦 → 𝑥)
∗)∗ → 0)∗)∗, 𝑉𝐴(0)} 

                                                 = 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → (𝑦 → 𝑥)
∗), 𝑉𝐴(0)} 

                                                 = 𝑉𝐴((𝑥 → (𝑦 → 𝑥)
∗)∗) 

Conversely, suppose 𝑉𝐴 is a vague W-implicative ideal and it satisfies the inequality,  

𝑉𝐴(𝑥) ≥ 𝑉𝐴((𝑥 → (𝑦 → 𝑥)
∗)∗) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎                 (3.1.3) 

Put 𝑥 = 𝑦 in (3.1.3) then, we have 

                     𝑉𝐴(𝑦) ≥ 𝑉𝐴((𝑦 → (𝑧 → 𝑦)
∗)∗)  

                                ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)}  

Thus, we have 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)}, and  

                        𝑉𝐴(0) ≥ 𝑉𝐴(𝑥)                                                  [From (i) of definition 3.1.1] 

Hence, 𝑉𝐴 is a vague positive implicative W-implicative ideal of  𝕎. ∎ 

Proposition 3.1.5 If 𝑉𝐴 is a vague positive implicative W-implicative ideal of lattice 
Wajsberg algebra 𝕎 then, 𝐼 = {𝑥 ∈ 𝐴/𝑉𝐴(𝑥) = 𝑉𝐴(0)} is a positive implicative              
W-implicative ideal of  𝕎. 

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Proof: Let 𝑉𝐴 be a vague positive implicative W-implicative ideal of 𝕎 and                   
𝐼 = {𝑥 ∈ 𝐴/𝑉𝐴(𝑥) = 𝑉𝐴(0)}. 

Obviously, 0 ∈ 𝐴.  

Let ((𝑦 → (𝑧 → 𝑦)∗)∗ → 𝑥)∗) ∈ 𝐼, 𝑥 ∈ 𝐼 for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 

Then, we have 𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗) = 𝑉𝐴(0) and 𝑉𝐴(𝑥) = 𝑉𝐴(0)  (3.1.4) 

Since 𝑉𝐴 is a vague positive implicative W-implicative ideal, we have  

𝑉𝐴(𝑦) ≥  𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)}           [From (ii) of definition 3.1.1] 

           =  𝑉𝐴(0)              [From 3.1.4] 

and  𝑉𝐴(0) ≥ 𝑉𝐴(𝑦)               [From (i) of definition 3.1.1] 

Then, we get 𝑉𝐴(𝑦) = 𝑉𝐴(0) 

Thus, 𝑦 ∈ 𝐼 it follows that 𝐼 is a positive implicative W-implicative ideal of  𝕎. ∎ 

Theorem 3.1.6 Let 𝑉𝐴 be a vague subset of lattice Wajsberg algebra 𝕎. 𝑉𝐴 is a vague 
positive implicative W-implicative ideal of  𝕎 if and only if 𝑉𝐴(𝛼, 𝛽) ≠ ∅;  𝛼, 𝛽 ∈ [0,1]. 

Proof: Let 𝑉𝐴 is a vague positive implicative W-implicative ideal of 𝕎 and 𝛼, 𝛽 ∈ [0,1] 
such that 𝑉𝐴(𝛼, 𝛽) ≠ ∅. Clearly 0 ∈ 𝑉𝐴(𝛼, 𝛽). 

Let ((𝑦 → (𝑧 → 𝑦)∗)∗ → 𝑥)∗) ∈  𝑉𝐴(𝛼, 𝛽) and  𝑥 ∈  𝑉𝐴(𝛼, 𝛽) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 

Then, we have  𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗ ≥ [𝛼, 𝛽], 𝑉𝐴(𝑥) ≥ [𝛼, 𝛽]. 

It follows that, 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)} ≥ [𝛼, 𝛽]. 

Thus, 𝑦 ∈ 𝑉𝐴[𝛼, 𝛽].  

Hence, [𝛼, 𝛽] is a positive implicative W-implicative ideal of 𝕎. 

Conversely, if 𝑉𝐴(𝛼, 𝛽) ≠ ∅ is a positive implicative W-implicative ideal of  𝕎, where 
𝛼, 𝛽 ∈ [0,1]. For any 𝑥 ∈ 𝕎 and 𝑥 ∈ 𝑉𝐴(𝐴), it follows that 𝑉𝐴(𝐴)(𝑥) is a positive 
implicative W-implicative ideal of  𝕎. 

Thus, 0 ∈ 𝑉𝐴(𝐴)(𝑥). That is, 𝑉𝐴(0) ≥ 𝑉𝐴(𝑥) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. 

Let [𝛼, 𝛽] = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗)∗, 𝑉𝐴(𝑥)}, it follows that 𝑉𝐴(𝛼, 𝛽) is a 

positive implicative W-implicative ideal and ((𝑦 → (𝑧 → 𝑦)∗)∗ → 𝑥)∗)∗ ∈ 𝑉𝐴[𝛼, 𝛽] and                 
𝑥 ∈ 𝑉𝐴[𝛼, 𝛽]. 

This implies that 𝑦 ∈ 𝑉𝐴[𝛼, 𝛽].  

So, 𝑉𝐴(𝑦) ≥ [𝛼, 𝛽] = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦)
∗)∗ → 𝑥)∗, 𝑉𝐴(𝑥)}  

Thus, 𝑉𝐴 is a vague positive implicative W-implicative ideal of  𝕎. ∎ 

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A. Ibrahim and M. Mohamed Rajik 
 

Corollary 3.1.7 A vague subset 𝑉𝐴 of lattice Wajsberg algebra 𝕎 is a vague positive 
implicative W-implicative ideal of 𝕎 if and only if 𝑉𝛼 is a positive implicative                
W-implicative ideal of 𝕎, when 𝑉𝛼 ≠ ∅, 𝛼 ∈ [0,1]. 

Proposition 3.1.8 Let 𝑀 and 𝑁 be implicative W-implicative ideals of lattice Wajsberg 
algebra 𝕎, such that 𝑀 ⊆ 𝑁. If 𝑉𝐴 is a vague positive implicative W-implicative ideal 
of 𝑀. Then so on 𝑁. 

Proof: Let 𝑀 and 𝑁 be implicative W-implicative ideals of lattice Wajsberg algebra 𝕎. 
Let 𝑉𝐴 be a vague positive implicative W-implicative ideal of M.  

Since 𝑀 ⊆ 𝑁, 𝑉𝑀 (𝑥) ≤ 𝑉𝑁(𝑥) for all 𝑥 ∈ 𝕎.  

Then, clearly 𝑀𝛼 ≤ 𝑁𝛼 for every 𝛼 ∈ [0,1].  

If 𝑉𝑀 is a vague positive implicative W-implicative ideal of 𝕎.  

Hence, we get 𝑀𝛼  is a positive implicative W-implicative ideal of 𝕎.  

                                                                                                            [From corollary 3.1.7]  

Then, 𝑁𝛼 is a positive implicative W-implicative ideal of  𝕎.       [From Proposition 2.10] 

Thus, 𝑉𝑁 is a positive implicative W-implicative ideal.  

Hence 𝑉𝐴 is a vague positive implicative W-implicative ideal of N. ∎ 

 

3.2.  Vague Associative W-implicative ideal 

 In this section,we introduce an notation of vague associative W-implicative ideal 

of lattice Wajsberg algebra 𝕎 and examine its properties. 

Definition 3.2.1 A vague subset 𝑉𝐴 of lattice Wajsberg algebra 𝕎 is said to be a vague 
associative W-implicative ideal of 𝕎 with respect to 𝑥, where 𝑥 is a fixed element of 𝕎, 
if it satisfies, 

 i           𝑉𝐴(0) ≥ 𝑉𝐴(𝑦)  

 ii          𝑉𝐴(𝑧) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → 𝑦)
∗ → 𝑥)∗, 𝑉𝐴((𝑦 → 𝑥)

∗)} for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 

Note. A vague associative W-implicative ideal with respect to all 𝑥 ≠ 1 is called a vague 
associative W-implicative ideal. Vague associative W-implicative ideal with respect to 1 

is constant. 

Example 3.2.2 Consider a set  𝐴 = {0, 𝑓, 𝑔, 1} with partial ordering as in Figure 3.2.1. 
Define ‘*’ and ‘→’ on 𝕎 as given in tables 3.2.1 and 3.2.2.                                      

→ 0 f g 1 

0 1 1 1 1 

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      Figure 3.2.1                      Table 3.2.1                                       Table 3.2.2    

    Lattice diagram                Complement                                     Implication    

 

Here, 𝕎 is a lattice Wajsberg algebra.  

A vague subset 𝑉𝐴 of 𝕎 is defined by, 

𝐴 = {〈0, [0.7,0.2]〉 , 〈𝑓, [0.7,0.2]〉 , 〈𝑔, [0.5,0.3]〉 , 〈1, [0.5,0.3]〉}. 

Then, 𝑉𝐴 is a vague associative W-implicative ideal of  𝕎.  

Proposition 3.2.3 If 𝑉𝐴 is a vague associative W-implicative ideal of  𝕎 with respect to 𝑥 
then 𝑉𝐴(0) = 𝑉𝐴(𝑦). 

Proof: Let 𝑉𝐴 be a vague associative W-implicative ideal of 𝕎 with respect to 𝑥 if 𝑥 =
(0, 1). Then it is trivial. 

If 𝑥 is neither 0 nor 1.  Then,   

            𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 0)
∗ → 𝑥)∗, 𝑉𝐴((𝑦 → 𝑥)

∗)}  [From (ii) of definition 3.2.1] 

Thus, 𝑉𝐴(𝑥) = 𝑉𝐴(0). ∎ 

Proposition 3.2.4 Every vague associative W-implicative ideal of lattice Wajsberg 

algebra 𝕎 with respect to 0 is a vague W-implicative ideal of  𝕎. 

Proof: Let 𝑉𝐴 be a vague associative W-implicative ideal of  𝕎 with respect to 0.  

Then, we have   𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 𝑦)
∗ → 0)∗, 𝑉𝐴((𝑦 → 0)

∗)} for all 𝑥, 𝑦 ∈ 𝕎  

[From (ii) of definition 3.2.1] 

                                     =  𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 𝑦)
∗), 𝑉𝐴(𝑦)} 

Thus, 𝑉𝐴 is a vague W-implicative ideal of  𝕎. ∎ 

Theorem 3.2.5 Let 𝑉𝐴 be a vague W-implicative ideal of lattice Wajsberg algebra of 𝕎. 
𝑉𝐴 is a vague associative W-implicative ideal of 𝕎 if and only if it satisfies, 𝑉𝐴((𝑧 →
(𝑦 → 𝑥)∗)∗ ≥ 𝑉𝐴((𝑧 → 𝑦)

∗ → 𝑥)∗ for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. 

Proof: Let 𝑉𝐴 be a vague W-implicative ideal of  𝕎 satisfying 

f f 1 1 1 

g f g 1 1 

1 0 f g 1 

𝑥 𝑥 ∗ 

0 1 

f g 

g f 

1 0 0 

g f 

1 

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A. Ibrahim and M. Mohamed Rajik 
 

                                            𝑉𝐴((𝑧 → (𝑦 → 𝑥)
∗)∗ ≥ 𝑉𝐴((𝑧 → 𝑦)

∗ → 𝑥)∗ for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 

Then, 𝑉𝐴(𝑧) ≥  𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → (𝑦 → 𝑥)
∗)∗), 𝑉𝐴((𝑦 → 𝑥)

∗)} 

                     = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑧 → 𝑦)
∗ → 𝑥)∗), 𝑉𝐴((𝑦 → 𝑥)

∗)} 

Thus, 𝑉𝐴 is a vague associative W-implicative ideal of  𝕎. 

Conversely, if 𝑉𝐴 be a vague associative W-implicative ideal of 𝕎.  

Then, 𝑉𝐴((𝑧 → (𝑦 → 𝑥)
∗)∗) ≥  𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑧 → (𝑦 → 𝑥)

∗)∗ → (𝑧 → 𝑦)∗)∗ → 𝑥), 𝑉𝐴((𝑧 →
                                                                       (𝑦 → 𝑥)∗)∗)} 

Let us consider, (((𝑧 → (𝑦 → 𝑥)∗)∗ → (𝑧 → 𝑦)∗)∗ → 𝑥) 

= 𝑥 ∗ → ((𝑧 → (𝑦 → 𝑥)∗)∗ → (𝑧 → 𝑦)∗)  

        =  𝑥∗ → ((𝑧 → 𝑦) → (𝑧 → (𝑦 → 𝑥)∗)) 

        = (𝑥 → 𝑦) → (𝑥∗ → ((𝑦 → 𝑥) → 𝑧∗)  

        = (𝑧 → 𝑦) → ((𝑦 → 𝑥) → (𝑥∗ → 𝑧∗)) 

        =  (𝑧 → 𝑦) → ((𝑧 → 𝑦) → (𝑧 → 𝑦)) 

          = 1 

It follows that 𝑉𝐴((𝑧 → (𝑦 → 𝑥)
∗)∗) ≥  𝑟𝑚𝑖𝑛{𝑉𝐴(0), 𝑉𝐴(((𝑧 → 𝑦)

∗ → 𝑥)∗)}  

                          =  𝑉𝐴(((𝑧 → 𝑦)
∗ → 𝑥 ∗) 

Thus,  𝑉𝐴((𝑧 → (𝑦 → 𝑥)
∗)∗) ≥ 𝑉𝐴(((𝑧 → 𝑦)

∗ → 𝑥 ∗). ∎ 

Theorem 3.2.6 Let 𝑉𝐴 be a vague W-implicative ideal of lattice Wajsberg algebra 𝕎.     
𝑉𝐴 is a vague associative W-implicative ideal of 𝕎 if and only if 𝑉𝐴(𝑧) ≥ 𝑉𝐴(((𝑧 →
𝑥)∗ → 𝑥 ∗) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. 

Proof: Let 𝑉𝐴 be a vague associative W-implicative ideal of 𝕎.  

Then,  𝑉𝐴(𝑧) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → 𝑦)
∗ → 𝑥)∗, 𝑉𝐴((𝑦 → 𝑥)

∗)} for all 𝑥, 𝑦, 𝑧 ∈ 𝕎.  
                                                                                                [From (ii) of definition 3.2.1] 

Taking 𝑦 = 𝑥 we get, 

𝑉𝐴(𝑧) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → 𝑥)
∗ → 𝑥)∗, 𝑉𝐴((𝑥 → 𝑥)

∗)} 

           = 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → 𝑦)
∗ → 𝑥)∗, 𝑉𝐴(0)} 

                                = 𝑉𝐴(((𝑧 → 𝑦)
∗ → 𝑥∗)  

Conversely, if 𝑉𝐴 is a vague W-implicative ideal and satisfies 

𝑉𝐴(𝑧) ≥ 𝑉𝐴(((𝑧 → 𝑦)
∗ → 𝑥)∗) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 

Clearly, (((𝑧 → 𝑥)∗) → (𝑦 → 𝑥)∗)∗ → (𝑧 → 𝑦)∗)∗ = 0   

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Vague Positive Implicative and Associative W- Implicative Ideals of Lattice Wajsberg Algebra 

 

 
 

and ((𝑧 → 𝑦)∗ → (𝑧 → 𝑥)∗)∗ ≤ (𝑥 → 𝑦)∗ 

It follows that, (((𝑧 → (𝑦 → 𝑥)∗)∗ → 𝑥)∗ → ((𝑧 → 𝑦)∗ → 𝑥)∗)∗ = 0  

𝑉𝐴((𝑧 → (𝑦 → 𝑥)
∗)∗ ≥ 𝑉𝐴(((𝑧 → 𝑦 → 𝑥)

∗) → 𝑥)∗ → 𝑥)∗)  

            ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑧 → (𝑦 → 𝑥)
∗ → 𝑥)∗, 𝑉𝐴((𝑧 → 𝑦)

∗ → 𝑥)∗)∗} 

𝑉𝐴(((𝑧 → 𝑦)
∗ → 𝑥)∗) = 𝑟𝑚𝑖𝑛{𝑉𝐴(0), 𝑉𝐴(((𝑧 → 𝑦)

∗ → 𝑥)∗)  

   = 𝑉𝐴(((𝑧 → 𝑦)
∗ → 𝑥)∗) 

From the proposition 3.2.3, 𝑉𝐴 is a vague associative W-implicative ideal of 𝕎. ∎ 

4.  Conclusions 
In this paper, we have introduced the notions of vague positive implicative         

W-implicative ideal and vague associative W-implicative ideal of lattice Wajsberg 

algebras. Further, we have discussed the relationship between vague positive implicative 

W-implicative ideal, and vague W-implicative ideal. Moreover, we have given some of 

the characterization of the vague associative W-implicative ideal. 

 

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