Communications on Applied Nonlinear Analysis 

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Fuzzy Soft Set Applications Explored in W-Algebras 
 

M. Indhumathi 1, K. Jeya Lekshmi2, A. Ibrahim3 

1 Department of Mathematics, RVS College of Arts and Science, Sulur, Affiliated to Bharathiar University, Coimbatore, 

Tamil Nadu, India. Email: indumathi@rvsgroup.com 

2 Department of Mathematics RVS College of Arts and Science, Sulur, Affiliated to Bharathiar University, Coimbatore, 

Tamil Nadu, India. Email: jeyalekshmi@rvsgroup.com 

3 P.G. and Research Department of Mathematics, H. H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan 

University, Tiruchirappalli, Tamil Nadu, India. 

Email: ibrahimaadhil@yahoo.com, dribrahimaadhil@gmail.com 

 

Article History: 

Received: 12-04-2024 

Revised: 24-05-2024 

Accepted: 06-06-2024 

Abstract 

Fuzzy soft set theory in Wajsberg algebras (W-algebras) is presented in this paper and its 

application to a problem of decision making is illustrated. We first introduce the notion of 

fuzzy soft ideals (fs-ideals) and then analyse some of the associated characteristics. 

Keywords: W- algebras, soft set, fs-set, fs-ideals, SOFT ENGINE. 

2010 MSC: 03B20, 03B52, 18B20 

 

1. INTRODUCTION 

Research on the soft set theory is now progressing at a rapid pace. The application of soft set theory to 

a decision-making issue was covered by Maji et al. [12]. Maji and [13] additionally investigated some 

techniques related to soft set theory. A new definition of soft set parametrization reduction was 

developed out by Chen et al. [4], who also compared it to the same attribute reduction idea in rough 

set theory. The algebraic structure of uncertainty-aware set theories has been studied by a few authors. 

The best theory for dealing with uncertainty is Zadeh's notion of fuzzy sets. M. Wajsberg introduced 

the ideas of W-algebra [15]. LPW-Algebras were first presented by Ceterchi Rodica [2]. A 

generalisation of regular soft sets, known as fs-sets, is presented in [11]. The use of fs-sets to a 

decision-making scenario is then demonstrated. In this work, we use the notion of                           fs-

ideals and fs-W-algebras and then derive their basic characteristics. In the future, this paper's approach 

will be expanded upon to analyse many kinds of ideals in W-algebras. 

2. BASIC RESULTS ON SOFT SETS 

The following is a description of Molodtsov's [14] definition of the soft set. Suppose that ℳ is an 

initial universe set and that 𝒦 is a set of parameters. Let 𝐼⊂ 𝒦 and 𝒯(ℳ) represent the power set of 

ℳ and 𝐼 ⊂ 𝒦. 

Definition 2.1. [14] [17] A pair (𝒮, 𝐼) is called a soft set over ℳ, where 𝒮 is a mapping given by              

 𝒮: 𝐼 ⟶ 𝒯(ℳ).  

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A soft set over ℳ, stated differently, defines a parameterized family of universe ℳ subsets. It is 

possible to consider elements of the soft set (𝒮, 𝐼) that are 𝛼 –approximate if 𝛼 ∈ 𝐼. It is evident that a 

soft set is not a set. 

Definition 2.2. [11] Let 𝒦 be a set of parameters and let ℳ be the initial universe set. 𝒯(ℳ) denote 

the set of fuzzy sets within ℳ. Where 𝐼 ⊆ 𝒦 and �̌� ∶  𝐼 ⟶ 𝒯(ℳ). 

�̌� [a] is often referred to as the fuzzy value set of parameter a. It is a fuzzy set in ℳ for every         a ∈

𝐼. (�̌�, 𝐼) degenerates to the usual soft set. If �̌� [a] is a crisp subset of ℳ for every a in 𝐼,                fs- 

sets are therefore a generalization of standard soft sets due to the above concept. 

3. MAIN RESULT (FUZZY SOFT WAJSBERG-ALGEBRAS) 

Unless otherwise noted, let 𝒦 be a collection of the following text’s parameters. The expression “Soft 

engine” will be used, referring to the fact that it generates a W-algebras. 

Definition 3.1. Let (�̌�, 𝐼) be a fs-set over a W-algebra 𝒳, because 𝐼 is a subset of 𝒦. We say that (�̌�, 𝐼) 

is a fs-set set on ℳ over 𝒳, if there exist 𝓂 ∈ 𝐼  such that �̌�(𝓂) is a fuzzy W-algebra in 𝒳. It may 

be stated that (�̌�, 𝐼) is a fs-W-algebra over 𝒳. If (�̌�, 𝐼) is based on parameter 𝓂 over 𝒳 for all 𝓂 ∈ 𝐼. 

Example 3.2. Assuming the universe ℳ has five colours, that is  

ℳ = {𝑝𝑖𝑛𝑘, 𝑔𝑟𝑎𝑦, 𝑟𝑒𝑑, 𝑜𝑙𝑖𝑣𝑒, 𝑡𝑎𝑛}. 

Assume the role of ⊛ a Soft engine that combines two colours in the prescribed order to get the 

desired results. 

𝑝𝑖𝑛𝑘 ⊛ 𝜚 = 𝑝𝑖𝑛𝑘 for all 𝜚 ∈ ℳ 

 

𝑔𝑟𝑎𝑦 ⊛ 𝜍 = {
𝑝𝑖𝑛𝑘 ∶  𝜍 ∈  {𝑔𝑟𝑎𝑦, 𝑜𝑙𝑖𝑣𝑒, 𝑡𝑎𝑛}

𝑔𝑟𝑎𝑦 ∶  𝜍 ∈  {𝑝𝑖𝑛𝑘, 𝑟𝑒𝑑 }         
} 

 

𝑟𝑒𝑑 ⊛ 𝑧 = {
𝑝𝑖𝑛𝑘 ∶  𝑧 ∈ {𝑟𝑒𝑑, 𝑡𝑎𝑛}               

𝑔𝑟𝑎𝑦 ∶  𝑧 ∈ { 𝑝𝑖𝑛𝑘, 𝑔𝑟𝑎𝑦, 𝑜𝑙𝑖𝑣𝑒}
} 

 

𝑜𝑙𝑖𝑣𝑒 ⊛ 𝑢 = {
𝑝𝑖𝑛𝑘 ∶  𝑢 ∈ {𝑜𝑙𝑖𝑣𝑒, 𝑡𝑎𝑛}            

𝑜𝑙𝑖𝑣𝑒 ∶  𝑢 ∈ { 𝑝𝑖𝑛𝑘, 𝑔𝑟𝑎𝑦, 𝑟𝑒𝑑}
} 

 

𝑡𝑎𝑛 ⊛ 𝑣 = {

𝑝𝑖𝑛𝑘 ∶  𝑣 = 𝑡𝑎 𝑛  
𝑟𝑒𝑑 ∶  𝑣 = 𝑜𝑙𝑖𝑣𝑒
𝑜𝑙𝑖𝑣𝑒 ∶  𝑣 = 𝑟𝑒𝑑

 𝑡𝑎𝑛 ∶  𝑣 = {𝑝𝑖𝑛𝑘, 𝑔𝑟𝑎𝑦}

  } 

Then (ℳ,⊛,𝑝𝑖𝑛𝑘) is a W-algebra.  

 



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(i) Let (�̌�,𝒦) be a fs-set over ℳ, then �̌�[𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠], �̌�[𝑒𝑙𝑒𝑔𝑎𝑛𝑡] & �̌�[𝑚𝑜𝑑𝑒𝑠𝑡] are fuzzy sets in 

ℳ. Here is how we define them: 

�̌� 𝑝𝑖𝑛𝑘 𝑔𝑟𝑎𝑦 𝑟𝑒𝑑 𝑜𝑙𝑖𝑣𝑒  𝑡𝑎𝑛 

𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠 0.7 0.7 0.7 0.4 0.4 

𝑒𝑙𝑒𝑔𝑎𝑛𝑡 0.8 0.7 0.4 0.6 0.4 

𝑚𝑜𝑑𝑒𝑠𝑡 0.6 0.2 0.5 0.2 0.2 

Then, fs-sets W-algebras �̌�[𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠], �̌�[𝑒𝑙𝑒𝑔𝑎𝑛𝑡] and �̌�[𝑚𝑜𝑑𝑒𝑠𝑡] are based on parameters 

“𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠”, “𝑒𝑙𝑒𝑔𝑎𝑛𝑡” and “𝑚𝑜𝑑𝑒𝑠𝑡” over ℳ, respectively. Thus, (�̌�,𝒦) is a fs-W-algebras over 

ℳ. 

(ii) �̌�[𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠], �̌�[𝑒𝑙𝑒𝑔𝑎𝑛𝑡] and �̌�[𝑚𝑜𝑑𝑒𝑠𝑡] are fuzzy sets in ℳ. Let (�̌�,𝒦) be fs-set over ℳ. We 

define them as listed below, 

�̌� 𝑝𝑖𝑛𝑘 𝑔𝑟𝑎𝑦 𝑟𝑒𝑑 𝑜𝑙𝑖𝑣𝑒  𝑡𝑎𝑛 

𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠 0.8 0.8 0.8 0.8 0.8 

e𝑙𝑒𝑔𝑎𝑛𝑡 0.4 0.6 0.5 0.5 0.3 

𝑚𝑜𝑑𝑒𝑠𝑡 0.9 0.1 0.7 0.1 0.1 

 

(�̌�,𝒦) is not a fs-W-algebras over ℳ. As (�̌�,𝒦) is not a fs-W-algebras. It is based on a parameter 

“elegent” over ℳ.  

Infect �̌�[elegent] (gray ⊛  olive) = �̌�[elegent] (pink) =  0.4 ≱ 0.5 

                                                     = min {�̌�[elegent](gray), �̌�[elegent] (olive)}  

A fs-W-algebras based on both a parameter “glorious” and a parameter “modest” over ℳ, we are 

able to confirm that(�̌�,𝒦). 

Example 3.3: Think about the universe ℳ = {pink, gray, red, olive, tan} and examine the soft engine 

◬ that generates the products listed below 

𝑝𝑖𝑛𝑘 ◬ 𝜚 = {
𝑝𝑖𝑛𝑘 ∶  𝜚 ∈  {𝑝𝑖𝑛𝑘, 𝑔𝑟𝑎𝑦, 𝑟𝑒𝑑}

𝑜𝑙𝑖𝑣𝑒 ∶ 𝜚 ∈  {𝑜𝑙𝑖𝑣𝑒, 𝑡𝑎𝑛 }         
} 

 

𝑔𝑟𝑎𝑦 ◬ 𝜍 = { 

𝑝𝑖𝑛𝑘 ∶  𝜍 = 𝑔𝑟𝑎𝑦                 

𝑟𝑒𝑑 ∶  𝜍 = {𝑝𝑖𝑛𝑘, 𝑟𝑒𝑑 }     
  𝑜𝑙𝑖𝑣𝑒 ∶ 𝜍 = 𝑡𝑎𝑛                   
 𝑡𝑎𝑛 ∶  𝜍 = 𝑜𝑙𝑖𝑣𝑒                  

} 

 

𝑟𝑒𝑑 ◬ 𝑧 = {

𝑝𝑖𝑛𝑘 ∶  𝑧 = 𝑟𝑒𝑑                 
𝑟𝑒𝑑 ∶   𝑧 ∈ {𝑝𝑖𝑛𝑘, 𝑔𝑟𝑎𝑦} 
𝑜𝑙𝑖𝑣𝑒 ∶  𝑧 ∈ {𝑜𝑙𝑖𝑣𝑒, 𝑡𝑎𝑛}

} 

 



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𝑜𝑙𝑖𝑣𝑒 ◬ 𝑢 = {
𝑝𝑖𝑛𝑘 ∶  𝑢 ∈ {𝑜𝑙𝑖𝑣𝑒, 𝑡𝑎𝑛}            

𝑜𝑙𝑖𝑣𝑒 ∶  𝑢 ∈ { 𝑝𝑖𝑛𝑘, 𝑔𝑟𝑎𝑦, 𝑟𝑒𝑑}
} 

𝑡𝑎𝑛 ◬ 𝑣 = { 

𝑝𝑖𝑛𝑘 ∶  𝑣 = 𝑡𝑎𝑛            
𝑔𝑟𝑎𝑦 ∶  𝑣 = 𝑜𝑙𝑖𝑣𝑒             
  𝑜𝑙𝑖𝑣𝑒 ∶  𝑣 = 𝑔𝑟𝑎𝑦               

 𝑡𝑎𝑛 ∶  𝑣 = {𝑝𝑖𝑛𝑘, 𝑟𝑒𝑑}    

  } 

Then (ℳ,◬, 𝑝𝑖𝑛𝑘) is a W-algebra.  

 𝒦 = {𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠, 𝑒𝑙𝑒𝑔𝑎𝑛𝑡,𝑚𝑜𝑑𝑒𝑠𝑡}. Let (�̌�,𝒦) be a fs-set over ℳ, then �̌�[𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠], �̌�[𝑒𝑙𝑒𝑔𝑎𝑛𝑡] 

& �̌�[𝑚𝑜𝑑𝑒𝑠𝑡] are fuzzy sets in ℳ. We define, 

�̌� 𝑝𝑖𝑛𝑘 𝑔𝑟𝑎𝑦 𝑟𝑒𝑑 𝑜𝑙𝑖𝑣𝑒  𝑡𝑎𝑛 

𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠 0.9 0.6 0.2 0.1 0.1 

𝑒𝑙𝑒𝑔𝑎𝑛𝑡 0.6 0.4 0.6 0.5 0.4 

𝑚𝑜𝑑𝑒𝑠𝑡 0.7 0.1 0.5 0.4 0.1 

Then (�̌�,𝒦) is a fs-W-algebras over ℳ. 

Proposition 3.4: A fs-W-algebras (�̌�, 𝐼)over 𝒳 is defined as (�̌�(𝓂)(0) ≥ �̌�(𝓂)(ϱ) for every ϱ ∈

𝒳 where 𝓂 is any parameter in 𝐼.  

Proof: Let ϱ ∈ 𝒳 and 𝓂 ∈  𝐼, then �̌�(𝓂)(0) = �̌�(𝓂)(ϱ → ϱ) 

                                                                            ≥ min { �̌�(𝓂)(ϱ), �̌�(𝓂)(ϱ)} = �̌�(𝓂)(ϱ) 

Hence �̌�(𝓂)(0) ≥ �̌�(𝓂)(ϱ) for all ϱ ∈ 𝒳 and any parameter 𝓂 in 𝐼. 

Theorem 3.5: Let (�̌�, 𝐼) be a fs-W-algebras over 𝒳. If 𝒥 is a subset of 𝐼, then (�̌� 𝒥⁄ , 𝒥) is a               fs-

W-algebras over 𝒳. 

The example that follows demonstrates that a fs-set (�̌�, 𝐼) exists over a W-algebra 𝒳 in such a way 

that, 

 (i)  (�̌�, 𝐼) is not a fs-W-algebra over 𝒳. 

(ii) there exists a subset ℬ of 𝐼 such that (�̌� 𝒥⁄ , 𝒥) is a fs-W-algebra over 𝒳. 

Example 3.6: Let (ℳ,⊛,pink) is a W-algebra as in example-3.2 define parameters                               𝐼 =

{glorious, elegant,modest, clever, gentle}. 

If (�̌�, 𝐼) be a fs-set over ℳ, then �̌�[glorious], �̌�[elegant], �̌�[modest], �̌�[clever] & �̌�[gentle] 

defined as  

�̌� pink gray red olive  tan 

glorious 0.7 0.7 0.7 0.4 0.4 

elegant 0.8 0.7 0.4 0.6 0.4 

modest 0.6 0.2 0.6 0.2 0.2 

clever 0.2 0.3 0.4 0.5 0.5 

gentle 0.3 0.2 0.6 0.6 0.2 



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Then (�̌�, 𝐼) is not fs-W-algebra over ℳ. Since �̌�[clever] & �̌�[gentle] are not fuzzy W-algebra in ℳ. 

But 𝒥 = {glorious, elegant,modest}  then (�̌� 𝒥⁄ , 𝒥) is a fs-W-algebra over ℳ.   

Definition 3.7: [1] The soft set (�̌�, 𝒟) where 𝒟 = 𝐼 ∪ 𝒥 and for every t ∈ 𝒟 is the extended 

intersection (Ei) of two soft sets (�̌�, 𝐼) and (�̌�, 𝒥) over ℳ.  

�̌�[t] = {

�̌�[t] ∶  t ∈ 𝐼 ∖ 𝒥

�̌�[t] ∶  t ∈ 𝒥 ∖ 𝐼

�̌�[t] ∩ �̌�[t] ∶  t ∈ 𝐼 ∩ 𝒥

} we write (�̌�, 𝐼) ∩̌t (�̌�, 𝒥) =(�̌�, 𝒟) .   

Definition 3.8: Let (�̌�, 𝐼) & (�̌�, 𝒥) be two soft sets over ℳ such that  𝐼 ∩ 𝒥 ≠ ϕ the Ei of (�̌�, 𝐼)  and 

(�̌�, 𝒥)  is denoted by (�̌�, 𝐼) ∩̌𝐬 ((�̌�, 𝒥) and is defined as (�̌�, 𝐼) ∩̌𝐬 (�̌�, 𝒥)  = (�̌�, 𝒟)  where 𝒟 = 𝐼 ∩ 𝒥 

and for all u ∈ 𝒟, �̌�[u] = �̌�[u] ∩ �̌�[u].  

Theorem 3.9: The Ei of (�̌�, 𝐼) and (�̌�, 𝒥)  is a fs-W-algebra over 𝒳, If (�̌�, 𝐼) and (�̌�, 𝒥) are            fs-

W-algebras over 𝒳. 

Proof: The Ei of (�̌�, 𝐼) and (�̌�, 𝒥) is (�̌�, 𝐼)  ∩̌t (�̌�, 𝒥) = (�̌�,𝒟) then  𝒟 = 𝐼 ∪ 𝒥 for any u ∈ 𝒟. 

�̌�(u) = �̌�(u) (resp �̌�(u) = �̌�(u)) is a fuzzy W-algebra if u ∈ 𝐼 ∖ 𝒥 (resp ∈ 𝒥 ∖ 𝐼).                         �̌�[u] =

�̌�[u] ∩ �̌�[u] is a fuzzy W-algebra for all u ∈ 𝐼 ∩ 𝒥 if 𝐼 ∩ 𝒥 ≠ ϕ. Given that two fuzzy      W-algebra 

intersect to form a fuzzy W-algebra. Therefore, over W-algebra  𝒳, (�̌�, 𝒟) is a                    fs-W-

algebra. 

Corollary 3.10: If (�̌�, 𝐼) and (�̌�, 𝒥)  are two fs-W-algebras over 𝒳, then (�̌�, 𝐼) ∩̌ (�̌�, 𝒥) is a                    

fs-W-algebra over 𝒳 through its Ei. 

Corollary 3.11: A fs-W-algebra is the limited Ei of two fs-W-algebras. 

Theorem 3.12: A W-algebra 𝒳 has two fs-W-algebras: (�̌�, 𝐼) and (�̌�, 𝒥). The union                         

(�̌�, 𝐼) ∪̌ (�̌�, 𝒥) is a fuzzy W-algebra over 𝒳, If 𝐼 & 𝒥 are disjoint. 

Proof: We can write (�̌�, 𝐼) ∪̌ (�̌�, 𝒥)  = (�̌�, 𝒟) where 𝒟 = 𝐼 ∪ 𝒥 and for all t ∈ 𝒟  

�̌�[t] = {

�̌�[t] ∶  t ∈ 𝐼 ∖ 𝒥

�̌�[t] ∶  t ∈ 𝒥 ∖ 𝐼

�̌�[t] ∩ �̌�[t] ∶  t ∈ 𝐼 ∩ 𝒥

} 

Since 𝐼 ∩ 𝒥 = ϕ, for every u ∈ 𝒟, either u ∈ 𝐼 ∖ 𝒥 (or) u ∈ 𝒥 ∖ 𝐼. Because (�̌�, 𝐼) is a fuzzy                W-

algebra over 𝒳.  If u ∈ 𝐼 ∖ 𝒥, then �̌�(u) = �̌�(u) is a fuzzy W-algebra in a 𝒳. �̌�(u) = �̌�(u) is a fuzzy 

W-algebra in a 𝒳, if u ∈ 𝒥 ∖ 𝐼, for (�̌�, 𝒥).  

Hence (�̌�, 𝒟) = (�̌�, 𝐼) ∪̌ (�̌�, 𝒥)  is a fuzzy W-algebra over 𝒳.  

If 𝐼 & 𝒥 are not disjoint, theorem 3.12 is not valid as the following example displays. 

Example 3.13: Let ℳ = { pink, gray, red, olive, tan} be a universe, and determine of a soft engine ⍟ 

that generates the items listed below. 



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pink⍟ϱ = {

pink ∶ ϱ ∈ {pink, gray}
red ∶  ϱ = red
olive ∶ ϱ = olive
 tan ∶  ϱ =  tan

  } 

gray⍟ς =

{
 
 

 
 
 gray ∶ ς = pink
  pink ∶  ς = gray
red ∶  ς = red
olive ∶ ς = olive
tan ∶  ς = tan }

 
 

 
 

 

red⍟z = {

pink ∶  z = red
red ∶  z ∈ { pink, gray} 

tan ∶  z = olive
olive ∶  z = tan

} 

olive⍟u = {

pink ∶  u = olive
olive ∶  u ∈ {pink, tan} 

tan ∶  u = red
red ∶  u = tan

} 

 tan⍟v = {

pink ∶  v = tan
red ∶ v = olive 
olive ∶  v = red

tan ∶  v ∈ {pink, tan}

} 

Then (ℳ,⍟, pink) is a W-algebra, define 𝐼 & 𝒥 as, 

𝐼 = {glorious, elegant,modest, clever}  

𝒥 = { modest, clever, gentle} 

𝐼 and 𝒥 are therefore not disjoint. If (�̌�, 𝐼) be a fs-set over ℳ, then fuzzy sets in ℳ include 

�̌�[glorious], �̌�[elegant] , �̌�[modest] & �̌�[clever]. Then, define as: 

�̌� 𝑝𝑖𝑛𝑘 𝑔𝑟𝑎𝑦 𝑟𝑒𝑑 𝑜𝑙𝑖𝑣𝑒 𝑡𝑎𝑛 

𝑔𝑙𝑜𝑟𝑖𝑜𝑢𝑠 0.8 0.7 0.4 0.4 0.4 

𝑒𝑙𝑒𝑔𝑎𝑛𝑡 0.7 0.6 0.5 0.3 0.3 

𝑚𝑜𝑑𝑒𝑠𝑡 0.9 0.6 0.2 0.4 0.2 

𝑐𝑙𝑒𝑣𝑒𝑟 0.6 0.6 0.3 0.3 0.5 

 

Then a fs-W-algebra over ℳ is (�̌�, 𝐼). If (�̌�, 𝒥) be a fs-W-algebra over ℳ, then the fuzzy sets in ℳ 

�̌�[modest], �̌�[clever] and �̌�[gentle]. Here is how we define them: 

�̌� 𝑝𝑖𝑛𝑘 𝑔𝑟𝑎𝑦 𝑟𝑒𝑑 𝑜𝑙𝑖𝑣𝑒 𝑡𝑎𝑛 

𝑚𝑜𝑑𝑒𝑠𝑡 0.7 0.6 0.4 0.2 0.2 

𝑐𝑙𝑒𝑣𝑒𝑟 0.6 0.6 0.4 0.4 0.6 

𝑔𝑒𝑛𝑡𝑙𝑒 0.8 0.5 0.3 0.5 0.3 

 



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Hence, a fs-W-algebra over ℳ is (�̌�, 𝒥). However, the union (�̌�, 𝐼) ∪ (�̌�, 𝒥) is not a fs-W-algebra 

over ℳ. 

 (�̌�[modest] ∪ �̌�[modest])(olive ⍟ red) = (�̌�[modest] ∪ �̌�[modest])(tan) 

 = max{�̌�modest](tan), �̌�[modest](tan)}=0.2 and  

min{(�̌�[modest] ∪ �̌�[modest])(olive), �̌�[modest] ∪ �̌�[modest])(red)} 

= min{max{�̌�[modest] (olive), �̌�[modest])(olive)}, max{�̌�[modest] (red),      

   �̌�[modest](red)}} 

= min {max{0.4, 0.2}, max{0.2, 0.4}} = 0.4. 

Definition 3.14. A fs-set (�̌�, 𝐼) over a W-algebra ℳ. (�̌�, 𝐼) is a fs-ideal of ℳ based on a parameter 

u, if there exists a parameter u ∈ 𝐼 such that �̌�[u] is a fuzzy ideal of ℳ. We say that (�̌�, 𝐼) is a         fs- 

ideal of ℳ if and only if it is based on all parameters. 

Example 3.15. Let ℳ = {guava,mango, beetroot, plum, turnip} be a universe, and look at a soft 

machine that produces the items listed below. 

guava ⋔ ϱ = guava for all ϱ ∈ ℳ, 

mango ⋔ ς = {
guava ∶  ς ∈ {mango, plum, turnip}       
banana ∶  ς ∈ {guava, beetroot}              

} 

carrot ⋔ z = {
beetroot ∶  z ∈ {guava,mango, turnip} 

guava ∶  z ∈ {beetroot, plum}                 
  } 

plum ⋔ u = {

plum ∶  u = guava,                     
mango ∶ u = beetroot,              
guava ∶  u = plum,                   
beetroot: u ∈ {mango, turnip}

} 

turnip ⋔ v = {

guava ∶  v = turnip,
turnip ∶   v ∈ {guava, beetroot} 
mango ∶  v ∈ {mango, plum}

} 

then (ℳ, ⋔, guava) is a W-algebra. Define 𝒦 = {puss, goat, yak, hen, sheep, rabbit}. 

Let (�̌�,𝒦) be a fs-set over ℳ. Then �̌�[puss], �̌�[goat], �̌�[yak], �̌�[hen], �̌�[sheep] & �̌�[rabbit] are 

fuzzy sets in ℳ.  

�̌� 𝑔𝑢𝑎𝑣𝑎 𝑚𝑎𝑛𝑔𝑜 𝑏𝑒𝑒𝑡𝑟𝑜𝑜𝑡  𝑝𝑙𝑢𝑚 𝑡𝑢𝑟𝑛𝑖𝑝 

𝑝𝑢𝑠𝑠 0.6 0.4 0.4 0.4 0.4 

𝑔𝑜𝑎𝑡 0.7 0.5 0.7 0.5 0.5 

𝑦𝑎𝑘 0.8 0.8 0.3 0.3 0.8 

ℎ𝑒𝑛 0.4 0.2 0.4 0.2 0.2 

𝑠ℎ𝑒𝑒𝑝 0.5 0.5 0.9 0.7 0.4 

𝑟𝑎𝑏𝑏𝑖𝑡 0.6 0.5 0.2 0.2 0.5 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

89 
https://internationalpubls.com 

A fs-ideal that is based on the parameters “puss”, “goat”, “yak”, “hen” and “rabbit”, then (�̌�,𝒦). 

However, based on the parameters “sheep”, (�̌�,𝒦) is not a fs-ideal of ℳ, as �̌�[sheep] (turnip) = 

0.4 < 0.5 = min {�̌�[sheep] ](turnip ⋔  plum), �̌�[sheep]( plum)}.   

We know for the most part, that sheeps prefer beetroot. Based on the parameter “sheep”, we may 

determine that (�̌�,𝒦) in the case above is not a fs-ideal of ℳ. This implies that (�̌�,𝒦) cannot be a 

fuzzy ideal of ℳ,  just as sheep prefers a beetroot over another.  

4. CONCLUSION 

In this paper, we combine the notion of fs-sets with the theory of W-algebras. Fs-ideals and                    fs-

W-algebras were introduced, and then some of their associated features were investigated. We shall 

investigate the applicability of fs-sets to different W-algebraic ideals on the basis of these results. 

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