Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions


Ratio Mathematica                                           Volume 40, 2021, pp. 179-190 

 

 
 

179 

 

 

 

Intuitionistic FWI-ideals of residuated 

lattice Wajsberg algebras 
 

 

R. Shanmugapriya* 

A. Ibrahim† 

 
 

Abstract 

The notions of intuitionistic Fuzzy Wajsberg Implicative ideal (𝐹𝑊𝐼 –ideal) 
and intuitionistic fuzzy lattice ideal of residuated Wajsberg algebras are 

introduced. Also, we show that every intuitionistic 𝐹𝑊𝐼- ideal of residuated 
lattice Wajsberg algebra is an intuitionistic fuzzy lattice ideal of residuated 

lattice Wajsberg algebra. Further, we discussed its converse part. 

 

Keywords: Wajsberg algebra; Lattice Wajsberg algebra; Residuated lattice 

Wajsberg algebra; 𝑊𝐼–ideal; 𝐹𝑊𝐼 –ideal; Intuitionistic 𝐹𝑊𝐼 –ideal; 
Intuitionistic fuzzy lattice ideal.   

2010 AMS subject classification: 06B10, 03E72, 03G10. 

 

 

 

 

 

 

 
*Research Scholar, P. G. and Research Department of Mathematics, H. H. The Rajah’s 

College, Pudukkotai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India; 

priyasanmu@gmail.com 
†Assistant Professor, P.G. and Research Department of Mathematics, H. H. The Rajah’s 

College, Pudukkotai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India;   

ibrahimaadhil@yahoo.com; dribra@hhrc.ac.in 
†Received on January 12th, 2021. Accepted on May 12th, 2021. Published on June 30th, 2021. 

doi: 10.23755/rm.v40i1.587. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper 

is published under the CC-BY licence agreement. 

 

 

 



Shanmugapriya and Ibrahim 

180 

 

 

1. Introduction 
The concept of fuzzy set was introduced by Zadeh [13] in 1935.        

The concept of intuitionistic fuzzy set was introduced by Atanassov [1, 2].   

The idea of Wajsberg algebra was introduced by Mordchaj Wajsberg [10].  

The author [8] introduced the notions of FWI-ideals and investigated their 

properties with illustrations.  

In the present paper, we introduce the notions of intuitionistic          

𝐹𝑊𝐼 –ideal and intuitionistic fuzzy lattice ideal of residuated lattice Wajsberg 
algebras. Also, we show that every intuitionistic 𝐹𝑊𝐼–ideal of residuated 
lattice Wajsberg algebra is an intuitionistic fuzzy lattice ideal of residuated 

lattice Wajsberg algebra. Further, we verify its converse part. 

 

2. Preliminaries 

In this section, we recall some basic definitions and properties which 

are helpful to develop our main results. 

Definition 2.1[3]. Let (𝐴, →,∗ ,1) bean algebra with a binary operation          
 “ → " and a quasi-complement  “ ∗  ”.  Then it is called a Wajsberg algebra, if 
the following axioms are satisfied for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, 
(i) 1 → 𝑥 = 𝑥 
(ii) (𝑥 → 𝑦) → 𝑦 = ((𝑦 → 𝑧) → (𝑥 → 𝑧)) = 1 
(iii) (𝑥 → 𝑦) → 𝑦 = (𝑦 → 𝑥) → 𝑥 
(iv) (𝑥∗ → 𝑦∗) → (𝑦 → 𝑥) = 1.   

Definition 2.2[3].Let(𝐴, →,∗ ,1) be a Wajsberg algebra. Then the following 
axioms are satisfied for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, 
(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 
(x) (𝑥∗ → 𝑦∗) = 𝑦 → 𝑥. 

Definition 2.3[3]. Let (𝐴, →,∗ ,1) be a Wajsberg algebra. Then it is called a 
lattice Wajsberg algebra, if the following axioms are satisfied for all 𝑥, 𝑦 ∈ 𝐴, 



Intuitionistic FWI-ideals of residuated lattice Wajsberg algebras 

181 

 

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

(ii) 𝑥 ∨ 𝑦 = (𝑥 → 𝑦) → 𝑦  
(iii) 𝑥 ∧ 𝑦 = ((𝑥∗ → 𝑦∗) → 𝑦∗)∗.  

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

Proposition 2.4[3].Let(𝐴, →, ∗ ,1) be a lattice Wajsberg algebra. Then the 
following axioms are satisfied for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, 

(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[11]. Let(𝐴, ∨, ∧, ⊗, →, 0, 1) be an algebra of type (2, 2, 2, 2, 
0, 0). Then it is called a residuated lattice, if the following axioms are satisfied 

for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, 
(i) (𝐴, ∨, ∧, 0, 1) is a bounded lattice, 
(ii) (𝐴,⊗, 1) is commutative monoid, 
(iii)  𝑥 ⊗ y ≤ 𝑧 if and only if   𝑥 ≤ 𝑦 → 𝑧. 

Definition 2.6[3]. Let (𝐴, ∨, ∧, ∗, →, 1) be a lattice Wajsberg algebra. If a 
binary operation “ ⊗ " on 𝐴 satisfies 𝑥 ⊗ 𝑦 = (𝑥 → 𝑦∗)∗ for all 𝑥, 𝑦 ∈ 𝐴. 
Then (𝐴, ∨, ∧, ⊗, →, ∗, 0, 1) is called a residuated lattice Wajsberg 
algebra. 
Definition 2.7[4].Let 𝐴 be a lattice Wajsberg algebra. Let 𝐼 be a non-empty 
subset of 𝐴, then 𝐼 is called aWI-ideal of lattice Wajsberg algebra 𝐴, if the 
following axioms are satisfied for all𝑥, 𝑦 ∈ 𝐴, 
(i) 0 ∈ 𝐼 
(ii) (𝑥 → 𝑦)∗ ∈ 𝐼and 𝑦 ∈ 𝐼 imply 𝑥 ∈ 𝐼 . 

Definition 2.8[4].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 ∈ 𝐴, 
(i) 𝑥 ∈ 𝐼, 𝑦 ∈ 𝐿 and 𝑦 ≤ 𝑥 imply 𝑦 ∈ 𝐼 



Shanmugapriya and Ibrahim 

182 

 

(ii) 𝑥, 𝑦 ∈ 𝐼 implies 𝑥 ∨ 𝑦 ∈ 𝐼. 

Definition 2.9[7]. Let 𝐴 be a residuated lattice Wajsberg algebra and  𝐼 be a 
non-empty subset of 𝐴.Then 𝐼 is called a 𝑊𝐼-ideal of residuated lattice 
Wajsberg algebra 𝐴, if the following axioms are satisfiedfor all 𝑥, 𝑦 ∈ 𝐴, 
(i) 0 ∈ 𝐼 
(ii) 𝑥 ⊗ 𝑦 ∈ 𝐼 and 𝑦 ∈ 𝐼 imply 𝑥 ∈ 𝐼  
(iii) (𝑥 → 𝑦)∗ ∈ 𝐼 and  𝑦 ∈ 𝐼 imply 𝑥 ∈ 𝐼. 

Definition 2.10[13].Let 𝐴 be a set. A function 𝜇: 𝐴 → [0, 1] is called a fuzzy 
subset on 𝐴 for each 𝑥 ∈ 𝐴, the value of 𝜇(𝑥) describes a degree of 
membership of 𝑥 in 𝜇. 

Definition 2.11[5].Let 𝐴 be a lattice Wajsberg algebra. Then the fuzzy subset 
𝜇 of 𝐴 is called a fuzzy 𝑊𝐼-ideal of  𝐴, if the following axioms are satisfied 
for all 𝑥, 𝑦 ∈ 𝐴, 

(i) 𝜇(0) ≥ 𝜇(𝑥) 

(ii) 𝜇(𝑥) ≥ min{ 𝜇((𝑥 → 𝑦)∗), 𝜇(𝑦)}. 

Definition 2.12[5].A fuzzy subset𝜇 of a lattice Wajsberg algebra 𝐴 is called a 
fuzzy lattice ideal if for all 𝑥, 𝑦 ∈ 𝐴, 
(i) If  𝑦 ≤ 𝑥 then 𝜇(𝑦) ≥ 𝜇(𝑥) 
(ii) 𝜇(𝑥 ∨ 𝑦) ≥ min{𝜇(𝑥), 𝜇(𝑦)}. 

Definition 2.13[8]. Let 𝐴 be a residuated lattice Wajsberg algebra. Then the 
fuzzy subset 𝜇 of 𝐴 is called a 𝐹𝑊𝐼-ideal of residuated lattice Wajsberg 
algebra 𝐴, if the following axioms are satisfied for all 𝑥, 𝑦 ∈ 𝐴, 
(i) 𝜇(0) ≥ 𝜇(𝑥) 
(ii) 𝜇(𝑥) ≥ min{ 𝜇(𝑥 ⊗ 𝑦), 𝜇(𝑦)} 

(iii) 𝜇(𝑥) ≥ min{ 𝜇((𝑥 → 𝑦)∗), 𝜇(𝑦)}. 

Definition 2.14[2]. An intuitionistic fuzzy subset 𝑆 is a non-empty set 𝑋 is an 
object having the form 𝑆 = {(𝑥, 𝜇𝑠(𝑥), 𝛾𝑠(𝑥))|𝑥 ∈ 𝑋} = (𝜇𝑠, 𝛾𝑠)where the 
functions 𝜇𝑠(𝑥): 𝑋 → [0, 1]denote the degree of membership and the degree of 
non-membership respectively and 0 ≤ 𝜇𝑠(𝑥) + 𝛾𝑠(𝑥) ≤ 1 for any 𝑥 ∈ 𝑋.  

Definition 2.15[13]. If 𝜇 and ѵ are fuzzy sets in 𝐴, define 𝜇 ≤ ѵ if and only if 
𝜇(𝑥) ≤ ѵ(𝑥) for all 𝑥 ∈ 𝐴.  

Definition 2.16[13]. The level set 𝜇𝑡 defined by 𝜇𝑡 = {𝑥 ∈ 𝐴/𝜇(𝑥) ≥ 𝑡}, 
where𝑡 ∈ [0, 1], then 𝜇𝑡 is also denoted by 𝑈(𝜇; 𝑡). 



Intuitionistic FWI-ideals of residuated lattice Wajsberg algebras 

183 

 

3. Properties of Intuitionistic FWI-ideal of a 

residuated lattice Wajsberg algebra 

In this section, we introduce the concept of an intuitionistic FWI-ideal 

and intuitionistic fuzzy lattice ideals. Also, we obtain some properties of an 

intuitionistic 𝐹𝑊𝐼-ideal. 

Definition 3.1. Let 𝐴 be a residuated lattice Wajsberg algebra.                        
An intuitionistic fuzzy set  𝑆 = (𝜇𝑠, 𝛾𝑠) of 𝐴 is called an intuitionistic         
FWI-ideal of residuated lattice Wajsberg algebra 𝐴 if it satisfies the following 
inequalities for all 𝑥, 𝑦 ∈ 𝐴, 

(i) 𝜇𝑠(0) ≥ 𝜇𝑠(𝑥) and 𝛾𝑠(0) ≤ 𝛾𝑠(𝑥) 
(ii) 𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)}  
(iii) 𝛾𝑠(𝑥) ≤ max {𝛾𝑠(𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)}  
(iv) 𝜇𝑠(𝑥) ≥ min {𝜇𝑠((𝑥 → 𝑦)

∗, 𝜇𝑠(𝑦) 
(v) 𝛾𝑠(𝑥) ≤ max {𝛾𝑠((𝑥 → 𝑦)

∗, 𝛾𝑠(𝑦)} . 

Example 3.2. Consider a set 𝐴={0, 𝑎, 𝑏, 𝑐, 𝑑,𝑟, 𝑠, 𝑡, 1}. Define a partial 
ordering  “≤” on 𝐴, such that 0 ≤ 𝑎 ≤ 𝑏 ≤ 𝑐 ≤ 𝑑 ≤ 𝑟 ≤ 𝑠 ≤ 𝑡 ≤ 1  with  a 
binary operations“ ⊗ ”and" → ”and a quasi-complement " ∗ "on 𝐴  as in 
following tables 3.1 and 3.2. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.1: Complement                             Table 3.2: Implication 

Define ∨ and ∧ operations on 𝐴 as follows:                    
 (𝑥 ∨  𝑦) = (𝑥 → 𝑦) → 𝑦, 
 (𝑥 ∧  𝑦) = (𝑥∗ → 𝑦∗) → 𝑦∗)∗,  

𝑥 𝑥 ∗ 

0 1 

𝑎 𝑡 

𝑏 𝑏 

𝑐 𝑟 

𝑑 𝑑 

𝑟 𝑐 

𝑠 𝑏 

𝑡 𝑎 

1 0 

→ 0 𝑎 𝑏 𝑐   𝑑 𝑟 𝑠 𝑡 1 

0 1 1 1 1 1 1 1 1 1 

𝑎 𝑡 1 1 𝑡 1 1 𝑡 1 1 

𝑏 𝑏 𝑡 1 𝑠 𝑡 1 𝑠 𝑡 1 

𝑐 𝑟 𝑟 𝑟 1 1 1 1 1 1 

𝑑 𝑑 𝑟 𝑟 𝑡 1 1 𝑡 1 1 

𝑟 𝑐 𝑑 𝑟 𝑠 𝑡 1 𝑠 𝑡 1 

𝑠 𝑏 𝑏 𝑏 𝑟 𝑟 𝑟 1 1 1 

𝑡 𝑎 𝑏 𝑏 𝑑 𝑟 𝑟 𝑡 1 1 

1 0 𝑎 𝑏 𝑐   𝑑 𝑟 𝑠 𝑡 1 



Shanmugapriya and Ibrahim 

184 

 

 𝑥 ⊗ 𝑦 = (𝑥 → 𝑦∗)∗ for all 𝑥, 𝑦 ∈ 𝐴. 
Then, 𝐴 is a residuated lattice Wajsberg algebra. 
Consider an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) on 𝐴 as,  
 

 𝜇𝑠(𝑥) = {
1      if  𝑥 ∈ (0, 𝑞)    for all 𝑥 ∈ 𝐴
0.54     otherwise   for all 𝑥 ∈ 𝐴

; 

 

  𝛾𝑠(𝑥) = {
0  if  𝑥 ∈ (0, 𝑞)        for all 𝑥 ∈ 𝐴
0.36    otherwise    for all 𝑥 ∈ 𝐴

 

Then, 𝑆 is an intuitionistic FWI-ideal of 𝐴. 
In the same Example 3.2, we consider an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) 
on 𝐴 as,       

 𝜇𝑠(𝑥) = {
1      if  𝑥 ∈ {0, 𝑎, 𝑏}     for all 𝑥 ∈ 𝐴
0.55     otherwise        for all 𝑥 ∈ 𝐴

;  

 

  𝛾𝑠(𝑥) = {
0  if  𝑥 ∈ {0, 𝑎, 𝑏}        for all 𝑥 ∈ 𝐴
0.42    otherwise       for all 𝑥 ∈ 𝐴

 

Then, 𝑆 is not an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴.  
Since 𝜇𝑠(𝑥) ≱ min {𝜇𝑠(𝑠 ⊗ 𝑏), 𝜇𝑠(𝑏)} and 𝛾𝑠(𝑥) ≰ max{𝛾𝑠(𝑠 ⊗ 𝑏), 𝛾𝑠(𝑏)}. 

Proposition 3.3. Every intuitionistic 𝐹𝑊𝐼-ideal 𝑆 = (𝜇𝑠, 𝛾𝑠) of residuated 
lattice Wajsberg algebra 𝐴 is an intuitionistic monotonic. That is, if  𝑥 ≤ 𝑦, 
then 𝜇𝑠(𝑥) ≥ 𝜇𝑠(𝑦) and   𝛾𝑠(𝑥) ≤ 𝛾𝑠(𝑦). 

Proof. Let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. 
Let 𝑥, 𝑦 ∈ 𝐴, 𝑥 ≤ 𝑦. 
Then  𝑥 ⊗ 𝑦 = (𝑥 → 𝑦∗)∗                                         [From the definition 2.6]               

          =  (𝑥 → 𝑥)∗ = 1∗ = 0                         [From (i) of definition 2.2]      
𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)}                             [From (ii) of definition 3.1] 
We have  𝜇𝑠(𝑥) ≥ 𝜇𝑠(𝑦) 
Now,𝛾𝑠(𝑥) ≤ max{𝛾𝑠(𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)}                     [From (iii) of definition 3.1] 

       = max{𝛾𝑠(0), 𝛾𝑠(𝑦)} = 𝛾𝑠 (𝑦)               [From the definition 2.6]                                  
Hence  𝛾𝑠(𝑥) ≤ 𝛾𝑠(𝑦) 
And      𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑥 → 𝑦)

∗, 𝜇𝑠(𝑦)}              [From (iv) of definition 3.1] 
           = min{𝜇𝑠(0), 𝜇𝑠(𝑦)} = 𝜇𝑠(𝑦)          [From (ii) of definition 2.7] 

We have 𝜇𝑠(𝑥) ≥ 𝜇𝑠(𝑦) 
Now, 𝛾𝑠(𝑥) ≤ max{𝛾𝑠(𝑥 → 𝑦)

∗, 𝛾𝑠(𝑦)}                   [From (v) of definition 3.1] 
        = max{𝛾𝑠(0), 𝛾𝑠(𝑦)} = 𝛾𝑠 (𝑦)               [From (ii) of definition 2.7] 

Therefore,  𝛾𝑠(𝑥) ≤ 𝛾𝑠(𝑦). ∎ 

Example 3.4. Let 𝐴 be a residuated lattice Wajsberg algebra defined in 
example 3.2, define an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) of 𝐴 as follows, 



Intuitionistic FWI-ideals of residuated lattice Wajsberg algebras 

185 

 

(i) 𝜇𝑠(0) = 𝜇𝑠(𝑐) = 1 
(ii) 𝜇𝑠(𝑥) = 𝑚 for any 𝑥 ∈ {𝑎, 𝑏, 𝑐, 𝑑, 𝑟, 𝑠, 𝑡, 1} 
(iii) 𝛾𝑠(0) = 𝛾𝑠(𝑐) = 0 
(iv) 𝛾𝑠(𝑥) = 𝑛 for any 𝑥 ∈ {𝑎, 𝑏, 𝑐, 𝑑, 𝑟, 𝑠, 𝑡, 1}. 

Where 𝑚, 𝑛 ∈ [0, 1] and 𝑚 + 𝑛 ≤ 1. Then 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic 
𝐹𝑊𝐼-ideal of 𝐴. 

Example 3.5. Consider a set𝐴 = {𝑎, 𝑏, 𝑝, 𝑞, 𝑐, 𝑑, 1}. Define a partial 
ordering  “≤” on 𝐴, such that 0 ≤ 𝑎 ≤ 𝑏 ≤ 𝑝 ≤ 𝑞 ≤ 𝑐 ≤ 𝑑 ≤ 1 with a binary 
operations“ ⊗ ”and " → ”and a quasi-complement " ∗ "on 𝐴  as in following 
tables 3.3 and 3.4. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.3: Complement                       Table 3.4: Implication 

Define ∨ and ∧ operations on 𝐴 as follows:              
 (𝑥 ∨  𝑦) = (𝑥 → 𝑦) → 𝑦, 
 (𝑥 ∧  𝑦) = (𝑥∗ → 𝑦∗) → 𝑦∗)∗, 
 𝑥 ⊗ 𝑦 = (𝑥 → 𝑦∗)∗ for all 𝑥, 𝑦 ∈ 𝐴. 
Then, 𝐴 is a residuated lattice Wajsberg algebra. 
Consider an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) on 𝐴 as,  

𝜇𝑠(𝑥) = {
1      if  𝑥 ∈ (0, 𝑞)    for all 𝑥 ∈ 𝐴
0.54     otherwise   for all 𝑥 ∈ 𝐴

; 

 

 𝛾𝑠(𝑥) = {
0  if  𝑥 ∈ (0, 𝑞)        for all 𝑥 ∈ 𝐴
0.36    otherwise    for all 𝑥 ∈ 𝐴

 

Then, 𝑆 is an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. 
In the same Example 3.5, we consider an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) 
on 𝐴 as,       

𝑥 𝑥 ∗ 

0 1 

𝑎 𝑏 

𝑏 𝑎 

𝑝 0 

𝑞 0 

𝑐 0 

𝑑 0 

1 0 

→ 0 𝑎 𝑏 𝑝 𝑞 𝑐 𝑑 1 

0 1 1 1 1 1 1 1 1 

𝑎 𝑏 1 𝑏 1 1 1 1 1 

𝑏 𝑎 𝑎 1 1 1 1 1 1 

𝑝 0 𝑎 𝑏 1 1 1 1 1 

𝑞 0 𝑎 𝑏 𝑝 1 1 1 1 

𝑐 0 𝑎 𝑏 𝑝 𝑑 1 𝑑 1 

𝑑 0 𝑎 𝑏 𝑝 𝑐 𝑐 1 1 

1 0 𝑎 𝑏 𝑝 𝑞 𝑐 𝑑 1 



Shanmugapriya and Ibrahim 

186 

 

 𝜇𝑠(𝑥) = {
1      if  𝑥 ∈ {0, 𝑎, 𝑏}     for all 𝑥 ∈ 𝐴
0.55     otherwise        for all 𝑥 ∈ 𝐴

;     

 

 𝛾𝑠(𝑥) = {
0  if  𝑥 ∈ {0, 𝑎, 𝑏}        for all 𝑥 ∈ 𝐴
0.42    otherwise       for all 𝑥 ∈ 𝐴

 

Then, 𝑆 is not an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴.  
Since 𝜇𝑠(𝑥) ≱ min {𝜇𝑠(𝑐 ⊗ 𝑎), 𝜇𝑠(𝑎)} and 𝛾𝑠(𝑥) ≰ max{𝛾𝑠(𝑐 ⊗ 𝑎), 𝛾𝑠(𝑎)}. 

Proposition 3.6. Let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic 𝐹𝑊𝐼-ideal of residuated 
lattice Wajsberg algebra 𝐴. For any 𝑥, 𝑦, 𝑧 ∈ 𝐴 which satisfies 𝑥 ≤ 𝑦∗ → 𝑧 then 
𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑦), 𝜇𝑠(𝑧)} and 𝛾𝑠(𝑥) ≤ max{𝛾𝑠(𝑦), 𝛾𝑠(𝑧)}. 

Proof. Let𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. If  𝑥 ≤ 𝑦
∗ → 𝑧 

Then, we have 1 = 𝑥 → (𝑦∗ → 𝑧) = 𝑧∗ → (𝑥 → 𝑦) 
= (𝑥 → 𝑦)∗ → 𝑧 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴                      [From (x) of definition 2.2] 
And   ((𝑥 → 𝑦)∗ → 𝑧)∗) = 0.  
It follows that, 

𝜇𝑠(𝑥) ≥ min{𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)}                        [From (ii) of definition 3.1] 
 ≥ min {min {𝜇𝑠((𝑥 ⊗ 𝑦) → 𝑧), 𝜇𝑠(𝑧)}, 𝜇𝑠(𝑦)} 

= min{min{𝜇𝑠((0) → 𝑧), 𝜇𝑠(𝑧)} , 𝜇𝑠(𝑦)}          [From the definition 2.6] 

 = min{min{𝜇𝑠(0), 𝜇𝑠(𝑧)} , 𝜇𝑠(𝑦)} = min{𝜇𝑠(𝑦), 𝜇𝑠(𝑧)} 
                                                                           [From (ii) of definition 3.1] 

We have  𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑦), 𝜇𝑠(𝑧)}  for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 
Now,𝛾𝑠(𝑥) ≤ max {max{𝛾𝑠((𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦))} 

 ≤ max {max{ 𝛾𝑠 (((𝑥 ⊗ 𝑦) → 𝑧), 𝛾𝑠(𝑧)} , 𝛾𝑠(𝑦)} 

 = max{max{𝛾𝑠((0) → 𝑧), 𝛾𝑠(𝑧)} , 𝛾𝑠(𝑦)} [From the definition 2.6] 

 = max {max{𝛾𝑠(0), 𝛾𝑠(𝑧)} , 𝛾𝑠 (𝑦)} 
 = max {𝛾𝑠(𝑦), 𝛾𝑠(𝑧)}                                [From (iii) of definition 3.1] 

Hence 𝛾𝑠(𝑥) ≤ max {𝛾𝑠(𝑦), 𝛾𝑠(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 
Now, 𝜇𝑠(𝑥) ≥ min{𝜇𝑠((𝑥 → 𝑦)

∗), 𝜇𝑠(𝑦)}           [From (iv) of definition 3.1] 
        ≥ min{min{𝜇𝑠(𝑥 → 𝑦)

∗ → 𝑧)∗) , 𝜇𝑠(𝑧)} , 𝜇𝑠(𝑦)} 
        = min {min{𝜇𝑠(0),  𝜇𝑠(𝑧)} , 𝜇𝑠(𝑦)} 
        = min {𝜇𝑠(𝑦), 𝜇𝑠(𝑧)}                        [From (ii) of definition 3.1] 

We have  𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑦), 𝜇𝑠(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 

And 𝛾𝑠(𝑥) ≤ max{𝛾𝑠((𝑥 → 𝑦
∗), 𝛾𝑠(𝑦))}             [From (v) of definition 3.1] 

      ≤ max {max{𝛾𝑠((𝑥 → 𝑦
∗) → 𝑧)∗), 𝛾𝑠(𝑧)} , 𝛾𝑠(𝑦)}  

      = max {max{𝛾𝑠(0), 𝛾𝑠(𝑧)} , 𝛾𝑠(𝑦)} 
      = max {𝛾𝑠(𝑦), 𝛾𝑠(𝑧)}                          [From (iii) of definition 3.1] 

Hence, 𝛾𝑠(𝑥) ≤ max {𝛾𝑠(𝑦), 𝛾𝑠(𝑧)}  for all 𝑥, 𝑦, 𝑧 ∈ 𝐴.∎ 



Intuitionistic FWI-ideals of residuated lattice Wajsberg algebras 

187 

 

Definition 3.7. An intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) of residuated lattice 
Wajsberg algebra 𝐴 is called an intuitionistic fuzzy lattice ideal of 𝐴 if it 
satisfies the following axioms for all𝑥, 𝑦 ∈ 𝐴, 

(i) 𝑆 = (𝜇𝑠, 𝛾𝑠) is intuitionistic monotonic 
(ii) 𝜇𝑠(𝑥 ∨  𝑦) ≥ min{𝜇𝑠(𝑥), 𝜇𝑠(𝑦)} 
(iii) 𝛾𝑠(𝑥 ∨  𝑦) ≤ max{𝛾𝑠(𝑥), 𝛾𝑠(𝑦)}. 

Remark 3.8. In the Definition 3.7(ii) and (iii) can be equivalently replaced by   

𝜇𝑠(𝑥 ∨  𝑦) = min{𝜇𝑠(𝑥), 𝜇𝑠(𝑦)}  and 𝛾𝑠(𝑥 ∨  𝑦) = max {𝛾𝑠(𝑥), 𝛾𝑠(𝑦)} 
respectively by 𝛾. 

Example 3.9. Let 𝐴 be a residuated lattice Wajsberg algebra defined in the 
Example 3.2 and 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic fuzzy set of 𝐴 defined by 

                  𝜇𝑠(𝑥) = {
1      if  𝑥 ∈ (0, 𝑑)   for all 𝑥 ∈ 𝐴
𝑚       otherwise    for all 𝑥 ∈ 𝐴

; 

𝛾𝑠(𝑥) = {
0      if  𝑥 ∈ (0, 𝑑)   for all 𝑥 ∈ 𝐴
𝑛       otherwise    for all 𝑥 ∈ 𝐴

 

 

Where 𝑚, 𝑛 ∈ [0, 1] and 𝑚 + 𝑛 ≤ 1.                       [From the definition 3.11] 
Then, 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic fuzzy lattice ideal of residuated lattice 
Wajsberg   algebra 𝐴. 

Proposition 3.10. Let𝐴be a residuated lattice Wajsberg algebra. Every 
intuitionistic  𝐹𝑊𝐼-ideal of 𝐴 is an intuitionistic fuzzy lattice ideal of 𝐴. 

Proof. Let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic fuzzy lattice ideal of 𝐴. 
Then we have 𝑆 = (𝜇𝑠, 𝛾𝑠) is intuitionistic monotonic.   [From proposition 3.6] 
Now ((𝑥 ∨  𝑦) → 𝑦)∗ = (((𝑥 → 𝑦) → 𝑦)) → 𝑦)∗   From (ii) of definition 2.3] 

= (𝑥 → 𝑦)∗ ≤ (𝑥∗)∗ for all  𝑥, 𝑦 ∈ 𝐴       [From (ix) of proposition 2.2] 
It follows that 

𝜇𝑠(𝑥 ∨  𝑦) ≥ min{𝜇𝑠(𝑥 ∨  𝑦) ⊗ 𝑦, 𝜇𝑠(𝑦)} 
                                                             [From definition 3.1 and definition 3.7] 

                                    ≥ min{𝜇𝑠(𝑥 → 𝑦) → 𝑦) ⊗ 𝑦, 𝜇𝑠(𝑦)}  
                                                                    [From (ii) of definition 2.3] 

                                    ≥ min {𝜇𝑠(0), 𝜇𝑠(𝑦)} 
                                    ≥ min{𝜇𝑠(𝑥), 𝜇𝑠(𝑦)}for all 𝑥, 𝑦 ∈ 𝐴  
                                                                  [From (i) of proposition 2.10]                    

                         𝛾𝑠(𝑥) ≤ max {𝛾𝑠((𝑥 ∨  𝑦) ⊗ 𝑦), 𝛾𝑠(𝑦)} 

                                   ≤ max{𝛾𝑠((𝑥 → 𝑦) → 𝑦) ⊗ 𝑦) , 𝛾𝑠(𝑦)}  
                                                                    [From (ii) of definition 2.3] 

                                   ≤ max {𝛾𝑠(0), 𝛾𝑠(𝑦)} 



Shanmugapriya and Ibrahim 

188 

 

                                   ≤ max{𝛾𝑠(𝑥), 𝛾𝑠(𝑦)}for all 𝑥, 𝑦 ∈ 𝐴 
                                                                  [From (ii) of definition 2.10] 

And we have  

𝜇𝑠(𝑥 ∨  𝑦) ≥ min{𝜇𝑠(𝑥 ∨  𝑦) → 𝑦)
∗) , 𝜇𝑠(𝑦)} ≥ min  {𝜇𝑠(𝑥), 𝜇𝑠(𝑦)} 

𝛾𝑠(𝑥) ≤ max {𝛾𝑠((𝑥 ∨  𝑦) → 𝑦)
∗), 𝛾𝑠(𝑦)} ≤ max{𝛾𝑠(𝑥), 𝛾𝑠(𝑦)} 

for all 𝑥, 𝑦 ∈ 𝐴. 
Hence, we have 𝑆 = (𝜇𝑠, 𝛾𝑠)is an intuitionistic fuzzy lattice ideal of    
residuated   lattice Wajsberg algebra 𝐴. ∎ 

Proposition 3.11. Let 𝐴 be a residuated lattice Wajsberg algebra. An 
intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic FWI-ideal of 𝐴 if and 
only if the fuzzy subsets 𝜇𝑠 and 𝛾𝑠

𝑐  are 𝐹𝑊𝐼-ideal of 𝐴, where 𝛾𝑠
𝑐 (𝑥) = 1 −

𝛾𝑠(𝑥)for all 𝑥 ∈ 𝐴. 

Proof. Let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic FWI-ideal of 𝐴. 
Then 𝜇𝑠 is a FWI-ideal of 𝐴.  
Now, we have 𝛾𝑠

𝑐 = 1 − 𝛾𝑠(0) 
                  ≥ 1 − 𝛾𝑠(𝑥)                             [From (i) of proposition 2.10] 

𝛾𝑠
𝑐 (0) = 𝛾𝑠

𝑐 (𝑥)  for all 𝑥, 𝑦 ∈ 𝐴 
 And   𝛾𝑠

𝑐 (𝑥) = 1 − 𝛾𝑠(𝑥) 
          ≥ 1 − max {𝛾𝑠(𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)} 
          = min{ 1 − 𝛾𝑠(𝑥 ⊗ 𝑦), 1 −  𝛾𝑠(𝑦)} 
          = min{ 𝛾𝑠

𝑐 (𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)}   
 𝛾𝑠

𝑐 (𝑥) = 1 − 𝛾𝑠(𝑥) 
            ≥ 1 − max {𝛾𝑠((𝑥 → 𝑦)

∗), 𝛾𝑠(𝑦)} 
            = min{ 1 − 𝛾𝑠((𝑥 → 𝑦)

∗), 1 −  𝛾𝑠(𝑦)} 
𝛾𝑠

𝑐 (𝑥) = min{ 𝛾𝑠
𝑐 ((𝑥 → 𝑦)∗), 𝛾𝑠(𝑦)}for all 𝑥, 𝑦 ∈ 𝐴 

Hence, we have 𝛾𝑠
𝑐  is a FWI-ideal of 𝐴. 

Conversely, assume that 𝜇𝑠 and 𝛾𝑠
𝑐  are FWI-ideal of 𝐴. 

Then, we have 𝜇𝑠(0) ≥ 𝜇𝑠(𝑥)and 1 − 𝛾𝑠(0) = 𝛾𝑠
𝑐 (0) ≥ 𝛾𝑠

𝑐 (𝑥) = 1 − 𝛾𝑠(𝑥) 
𝛾𝑠(0) ≤ 𝛾𝑠(𝑥)   for all 𝑥, 𝑦 ∈ 𝐴 

Now, 𝜇𝑠(𝑥) ≥ min {𝜇𝑠
𝑐 (𝑥 ⊗ 𝑦), 𝜇𝑠

𝑐 (𝑦)} 
        = min {1 − 𝜇𝑠(𝑥 ⊗ 𝑦), 1 − 𝜇𝑠(𝑦)} 
        = 1 − max {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} 

𝛾𝑠(𝑥) ≤ max {𝛾𝑠(𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)}   for all 𝑥, 𝑦 ∈ 𝐴 
 𝜇𝑠(𝑥) ≥ min {𝜇𝑠

𝑐 (𝑥 → 𝑦)∗, 𝜇𝑠
𝑐 (𝑦)} 

           = min {1 − 𝜇𝑠((𝑥 → 𝑦)
∗), 1 − 𝜇𝑠(𝑦)} 

           = 1 − max {𝜇𝑠((𝑥 → 𝑦)
∗), 𝜇𝑠(𝑦)} 

𝛾𝑠(𝑥) ≤ max{𝛾((𝑥 → 𝑦)
∗), 𝛾𝑠(𝑦)}for all 𝑥, 𝑦 ∈ 𝐴 

Hence, we have 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴.∎ 



Intuitionistic FWI-ideals of residuated lattice Wajsberg algebras 

189 

 

Proposition 3.12. Let 𝐴 be a residuated lattice Wajsberg algebra and 𝑆 =
(𝜇𝑠, 𝛾𝑠) is an intuitionistic 𝐹𝑊𝐼-ideal of𝐴. Then 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic 
FWI-ideal of 𝐴 if and only if (𝜇𝑠, 𝜇𝑠

𝑐 ) and (𝛾𝑠
𝑐 , 𝛾𝑠) are intuitionistic 𝐹𝑊𝐼-ideal 

of 𝐴. 

Proof. Let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. 
Then, 𝜇𝑠 and 𝛾𝑠

𝑐  are 𝐹𝑊𝐼-ideal of 𝐴[From proposition 3.11] 
Hence, we have (𝜇𝑠, 𝜇𝑠

𝑐 ) and (𝛾𝑠
𝑐 , 𝛾𝑠) are intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. 

Conversely, if (𝜇𝑠, 𝜇𝑠
𝑐 ) and (𝛾𝑠

𝑐 , 𝛾𝑠) are intuitionistic 𝐹𝑊𝐼-idealof 𝐴 
                                                                                    [From proposition 3.11] 

Then, the fuzzy sets 𝜇𝑠 and 𝛾𝑠
𝑐  are 𝐹𝑊𝐼-ideal of 𝐴 

Hence, 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitonistic 𝐹𝑊𝐼-ideal of 𝐴. ∎ 

Proposition 3.13. Let 𝐴 be residuated lattice Wajsberg algebra, 𝑉 a non-empty 
subset of [0, 1] and {𝐼𝑡 / 𝑡 ∈ 𝑉} a collection of 𝐹𝑊𝐼 -ideal of 𝐴 such that  

(i) 𝐴 = 𝐼𝑡  𝑡∈𝑣
⋃  

(ii) 𝑟 > 𝑡 if and only if 𝐼𝑟 ⊆ 𝐼𝑡 for any 𝑟, 𝑡 ∈ 𝑉 then the intuitionistic fuzzy 
set 𝑆 = (𝜇𝑠, 𝛾𝑠) of 𝐴  defined by 𝜇𝑠 = sup{𝑡 ∈ 𝑉/𝑥 ∈ 𝐼𝑡 } and𝛾𝑠 =
inf{𝑡 ∈ 𝑉/𝑥 ∈ 𝐼𝑡 } for any 𝑥 ∈ 𝐴 is intuitionistic 𝐹𝑊𝐼 -ideal of 𝐴. 

Proof. According to proposition 3.10, it is sufficient to show that 𝜇𝑠 and 𝛾𝑠
𝑐    

are    𝐹𝑊𝐼–idealof 𝐴 for all𝑥 ∈ 𝐴. 
𝜇𝑠(0) = sup {𝑡 ∈ 𝑉/0 ∈ 𝐼𝑡 } = sup𝑉 ≥ 𝜇𝑠(𝑥)      [From (i) of definition 3.1] 
If there exists 𝑥, 𝑦 ∈ 𝐴 such that 𝜇𝑠(𝑥) < min {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} and      
𝜇𝑠(𝑥) < min {𝜇𝑠((𝑥 → 𝑦)

∗), 𝜇𝑠(𝑦)}. 
There exists 𝑡1 such that 𝜇𝑠(𝑥) < 𝑡1 < min {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} and    

𝜇𝑠(𝑥) < 𝑡1 < min {𝜇𝑠((𝑥 → 𝑦)
∗), 𝜇𝑠(𝑦)} 

It follows that 𝑡1 such that𝑡1 < 𝜇𝑠(𝑥 ⊗ 𝑦),𝑡1 < 𝜇𝑠((𝑥 → 𝑦)
∗),𝑡1 < 𝜇𝑠(𝑦) and 

Hence, there exist 𝑡2, 𝑡3 ∈ 𝑉, 𝑡2 > 𝑡1, 𝑡3 > 𝑡1, (𝑥 ⊗ 𝑦) ∈ 𝐼𝑡2 , (𝑥 → 𝑦)
∗) ∈ 𝐼𝑡2  

and 𝑦 ∈ 𝐼𝑡3  

It follows that (𝑥 ⊗ 𝑦) ∈ 𝐼𝑡2⋀𝑡3  , (𝑥 → 𝑦)
∗) ∈ 𝐼𝑡2⋀𝑡3  and 𝑦 ∈ 𝐼𝑡2⋀𝑡3  

Now, we have 𝑥 ∈ 𝐼𝑡2⋀𝑡3  

That is, 𝜇𝑠(𝑥) = sup {𝑡 ∈
𝑉

𝑥
∈ 𝐼𝑡 } ≥ 𝑡2⋀𝑡3 > 𝑡1            [From definition 2.16] 

Therefore, 𝜇𝑠(𝑥) > 𝑡1 
This is a contradiction. 

Hence, we have 𝜇𝑠 is a 𝐹𝑊𝐼 -ideal of 𝐴. 𝛾𝑠
𝑐 is a𝐹𝑊𝐼 -ideal, which can be 

proved by similar method. ∎ 
 

 



Shanmugapriya and Ibrahim 

190 

 

4.  Conclusions 

In this paper, we have introduced the notions of intuitionistic       

𝐹𝑊𝐼 –ideal and intuitionistic fuzzy lattice ideal of residuated Wajsberg 
algebras. Also, we have shown that every intuitionistic 𝐹𝑊𝐼- ideal of 
residuated lattice Wajsberg algebra is an intuitionistic fuzzy lattice ideal of 

residuated lattice Wajsberg algebra. Further, we have discussed its converse 

part. 

 

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