Multi-Fuzzy linear spaces over multi-fuzzy field with respect to a bridge function

Introduction

The concept of fuzzy sets was introduced by Zadeh in 1965. Rosenfeld extended fuzzy set theory by defining fuzzy groupoids and fuzzy groups, which paved the way for further advancements such as fuzzy rings and fuzzy fields. The foundational concepts of fuzzy vector spaces were later formalized by Katsara and Liu, inspiring significant research and new theoretical developments in this area. Nanda advanced the field by introducing fuzzy vector spaces over fuzzy field, generalizing the notion of fuzzy vector spaces over conventional field. In 2010, Sabu Sebastian and T.V. Ramakrishnan introduced the concept of multi-fuzzy sets and investigated the properties of multi-fuzzy groups, broadening the scope of fuzzy mathematics. Inspired by these developments, this paper explores the properties of multi-fuzzy vector spaces μ of V over the multi-fuzzy field λ of the field F, assuming that the dimensions of μ and λ are equal. In addition, we introduce a new class of multi-fuzzy vector spaces using bridge functions, say h. When the dimensions of the multi-fuzzy set μ and the multi-fuzzy field λ differ, the bridge function h plays a crucial role in defining these spaces. The primary focus of this paper is to study into the algebraic structure of these spaces and analyze the influence of the bridge function on their properties, offering new insights into multi-fuzzy linear algebra.

Preliminaries Definition 2.1.

Let {〈𝐿 𝑖 ,∨ 𝑖 ,∧ 𝑖 〉, 1 ≤ 𝑖 ≤ 𝑘} be a family of complete distributive lattices. Then the product 𝐿 = ∏ 𝐿 𝑖 𝑘 𝑖=1 form a lattice with 𝑠𝑢𝑝𝑟𝑒𝑚𝑢𝑚 ∨ 𝑎𝑛𝑑 𝑖𝑛𝑓𝑖𝑚𝑢𝑚 ∧ are defined by

Remark: In all cases we consider each lattice to be a complete distributive lattice. The supremum and infimum elements of the lattice L are denoted by 1 𝐿 and 0 𝐿 respectively. Definition2.2 [3]. 𝐿𝑒𝑡 𝐿 = ∏ 𝐿 𝑗 𝑘 𝑗=1 be a product complete distributive lattices Let 𝑡 𝑗: 𝐿 𝑗 × 𝐿 𝑗 → 𝐿 𝑗 𝑓𝑜𝑟 𝑗 = 1,2, … , 𝑘 be t-norms. The T-operator (or T-norm) T:𝐿 × 𝐿 → 𝐿 is defined by T(𝑥, 𝑦)=(𝑡 1 (𝑥

𝑏) 𝜆(-𝑝) ≥ 𝜆(𝑝), for all 𝑝 𝑖𝑛 𝐹.

𝑐) 𝜆(𝑝. 𝑞) ≥ T(𝜆(𝑝), 𝜆(𝑞)) =(𝑡 1 (𝜆 1 (𝑝), 𝜆 1 (𝑞)), 𝑡 2 (𝜆 2 (𝑝), 𝜆 2 (𝑞)), … , 𝑡 𝑘 , (𝜆 𝑘 (𝑝), 𝜆 𝑘 (𝑞))).

𝑑) 𝜆(𝑝 -1 ) ≥ 𝜆(𝑝), for all non zero element 𝑝𝜖𝐹.

Where 0 𝐹 is the zero element and 1 𝐹 is the unity in F.1 𝐿 𝑗 is the supremum of 𝐿 𝑗 for each 1 ≤ 𝑗 ≤ 𝑘 so that 1 𝐿 is supremum of L. The set all multi fuzzy field of F under T-norm T=(𝑡 1 , 𝑡 2 , … , 𝑡 𝑘 ) is denoted by 𝑀𝐹𝐹 𝑘 (𝐹, 𝐿, 𝑇)

We say that 𝜆 =(𝜆 1, , 𝜆 2 , … , 𝜆 𝑘 ) is a multi-fuzzy field of F and write 𝜆𝜖𝑀𝐹𝐹 𝑘 (𝐹, 𝐿, 𝑇).

If T is the minimum t-norm, then 𝑀𝐹𝐹 𝑘 (𝐹, 𝐿, 𝑇) is denoted by 𝑀𝐹𝐹 𝑘 (𝐹, 𝐿)

be a product of lattices, and let T=(𝑡 1, , 𝑡 2 , … , 𝑡 𝑘 ) be a T-norm on L. Let V be a vector space over the field F, and let 𝜆 = (𝜆 1 , 𝜆 2 , … , 𝜆 𝑘 )𝜖𝑀𝐹𝐹 𝑘 (𝐹, 𝐿, 𝑇).A multi-fuzzy set A={(𝑥, 𝜇 1 (𝑥), 𝜇 2 (𝑥), … , 𝜇 𝑘 (𝑥)): 𝑥 ∈ 𝑉}, where 𝜇 𝑗 𝜖𝐿 𝑗 𝑉 for 1 ≤ 𝑗 ≤ 𝑘,with membership function 𝜇 =(𝜇 1 , 𝜇 2 , … , 𝜇 𝑘 ), is called a multi-fuzzy vector space over the multi-fuzzy field 𝜆 of F under T-norm T if it satisfies the following conditions for all 𝑥, 𝑦 𝜖𝑉 and 𝑝𝜖𝐹:

,where 0 𝑉 is the zero vector in V and 1 𝐿 is the largest element in L The set all multi-fuzzy vector space of V over the multi-fuzzy field 𝜆 of F under the T-norm T is denoted by 𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿, 𝑇).If the T-norm is the minimum, then 𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿, 𝑇) is denoted by 𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿).

We say that 𝜇 =(𝜇 1 , 𝜇 2 , … , 𝜇 𝑘 ) is a multi-fuzzy vector space of V over the multi-fuzzy Field 𝜆 of F and denote it as 𝜇𝜖𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿, 𝑇). All other conditions remain the same. The set of all multi-fuzzy vector space of V over the ordinary field F under T-norm T=(𝑡 1 , 𝑡 2 , … , 𝑡 𝑘 ) is denoted by 𝑀𝐹𝑉 𝑘 (𝑉, 𝐿, 𝑇). Theorem 3.1. Let V be a vector space over the field F, and let 𝜇 be a multi-fuzzy subset of V with 𝜇(0 𝑉 ) = 1 𝐿 = (1 𝐿 1 , 1 𝐿 2 , … , 1 𝐿 𝑘 ). Then: The set of all multi-fuzzy vector space of V over the multi-fuzzy field 𝜆 =(𝜆 1, , 𝜆 2 , … , 𝜆 𝑘 ) of F, based on lattice operations of 𝐿 = ∏ 𝐿 𝑗 𝑘 𝑗=1

is denoted by 𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿).

We say that 𝜇 = (𝜇 1, , 𝜇 2 , … , 𝜇 𝑘 ), is a multi-fuzzy vector space of V over the multi fuzzy field 𝜆 of F and denote it as 𝜇𝜖𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿).

Theorem 3.2. Let V be a vector space over field F, and let W be a subset of V. Let 𝜇: 𝑉 → 𝐿 = ∏ 𝐿 𝑗 𝑘 𝑗=1

be defined by: 𝜇(𝑥) = 1 𝐿 = (1 𝐿 1 , 1 𝐿 2 , … , 1 𝐿 𝑘 ), 𝑖𝑓 𝑥 ∈ 𝑊 𝑎𝑛𝑑 𝜇(𝑥) = 0 𝐿 𝑖𝑓 𝑥 ∉ 𝑊 Then, 𝜇𝜖𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿) if and only if W is a subspace of V. Theorem 3.3. Let V be a vector space over field F, and W be a subset of V. Let 𝜇𝜖𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿), Then W={𝑥 ∈ 𝑉: 𝜇(𝑥) = 1 𝐿 } is a subspace of V.

Proof. Let 𝑥, 𝑦 ∈ 𝑊 and p∈ 𝐹, Then 𝜇(𝑝𝑥 + 𝑦) ≥ inf{ 𝜇(𝑥), 𝜇(𝑦)}= inf{ 1 𝐿 , 1 𝐿 } = 1 𝐿 . Since 𝜇(𝑝𝑥 + 𝑦) ≤ 1 𝐿 , this implies 𝜇(𝑝𝑥 + 𝑦) = 1 𝐿 . Therefore,𝑝𝑥 + 𝑦 ∈ 𝑊, showing that W={𝑥 ∈ 𝑉: 𝜇(𝑥) = 1 𝐿 } is a subspace of V. Theorem 3.4.Let {𝜇 𝑗 𝜖𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿): 𝑗 ∈ 𝐽} be an indexed family of multi-fuzzy vector space of V over the multi-fuzzy field 𝜆 of F.Then their intersection 𝜇 = ⋂ 𝜇 𝑗 𝑗∈𝐽 𝜖𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿) Theorem 3.5.Let V be a vector space over the field F. Let 𝜇 𝑎𝑛𝑑 𝜏 𝑏𝑒 𝑖𝑛 𝑀𝐹𝑉 𝑘 (𝑉, 𝜆, 𝐿). Then the multi-𝑓𝑢𝑧𝑧𝑦 𝑠𝑒𝑡 𝜇 × 𝜏 ∶ 𝑉 × 𝑉 → 𝐿 is defined by (𝜇 × 𝜏)(𝑧 1 , 𝑧 2 ) = inf {𝜇(𝑧 1 ), 𝜏(𝑧 2 )} for all (𝑧 1 , 𝑧 2 ) ∈ 𝑉 × 𝑉, and is a multi-fuzzy vector space of 𝑉 × 𝑉 over the multi-fuzzy field 𝜆 of F. That is, 𝜇 × 𝜏 ∈ 𝑀𝐹𝑉 𝑘 (𝑉 × 𝑉, 𝜆, 𝐿). is defined by: = T(𝜆 ℎ (𝑝), T( 𝜇(𝑥), 𝜇(𝑥)))

Multi-fuzzy vector space

= T( 𝜆 ℎ (𝑝), 𝜇(𝑥)), since T is idempotent.

Finally,

𝜇 (-𝑥) ≥ T( 𝜆 ℎ (-1), 𝜇(𝑥)) = T(𝜆 ℎ (1), 𝜇(𝑥)), since 𝜆 ℎ (-1) = ℎ(𝜆(-1)) = ℎ(𝜆(1)) = 𝜆 ℎ (1) = T(1 𝐿 , 𝜇(𝑥)) = 𝜇(𝑥) for all x Hence, 𝜇𝜖𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇) Theorem 4.3.Let V be a vector space over the field F. Let 𝜇, 𝜏𝜖𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇), and T is idempotent. Then the intersection 𝜇 ∩ 𝜏 𝜖𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇) Proof Let V be a vector space over the field F and let 𝑥, 𝑦 𝜖 𝑉 and 𝑝𝜖𝐹. = 𝑇(T( 𝜆 ℎ (𝑝), 𝜆 ℎ (𝑝)), T( 𝜇(𝑥), 𝜏(𝑥) )), by the commutativity and associativity of the operator T, which simplifies to = T( 𝜆 ℎ (𝑝), (𝜇 ∩ 𝜏)(𝑥)) Also we have (𝜇 ∩ 𝜏)(0 𝑉 ) = T(𝜇(0 𝑉 ), 𝜏(0 𝑉 )) =T(1 𝐿 , 1 𝐿 )= 1 𝐿 Therefore, 𝜇 ∩ 𝜏 𝜖𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇) Theorem 4.4 Let V be a vector space over the field F. Let {𝜇 𝑗 𝜖 𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇): 𝑗 ∈ 𝐽} be an indexed family of multi-fuzzy vector spaces of V over the multi-fuzzy field 𝜆 of F and the T-norm is the minimum operator in L. Then their intersection: 𝜇 = ⋂ 𝜇 𝑗 𝑗∈𝐽 𝜖𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇) Proof Let V be a vector space over the field F. let 𝑥, 𝑦 𝜖 𝑉 and 𝑝𝜖𝐹.Let ⋀ 𝑗∈𝐽 denotes the infimum taken over all 𝑗 ∈ 𝐽 Let V be a vector space over field F and W be a subset of V and let 𝜇: 𝑉 → 𝐿 = ∏ 𝐿 𝑗 𝑘 𝑗=1 be defined by 𝜇(𝑥) = { 1 𝐿 , 𝑖𝑓 𝑥 ∈ 𝑊 0 𝐿, 𝑖𝑓 𝑥 ∉ 𝑊 .

Let crisp function h be such that 𝜆 ℎ (𝑝) = 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑝 and let T be idempotent. Then, 𝜇𝜖𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇) if and only if W is a subspace of V.

Proof

Let W be a subset of V and 𝑥, 𝑦 ∈ 𝑊, P∈ 𝐹, and 𝜇𝜖𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇)

Then, 𝜇(𝑥) =1 𝐿 , 𝜇(𝑦) =1 𝐿 so that 𝜇(𝑝𝑥 + 𝑦) ≥ T (T(𝜆 ℎ (𝑝), 𝜇(𝑥)), 𝜇(𝑦))=T (T( 𝜆 ℎ (𝑝), 1 𝐿 ), 1 𝐿 ) = T( 𝜆 ℎ (𝑝), 1 𝐿 ) = h(𝜆(𝑝))

Thus, 𝑝𝑥 + 𝑦 ∈ 𝑊 if 𝜆 ℎ (𝑝) = 1 for all 𝑝.

Therefore, W is a subspace of V if 𝜆 ℎ (𝑝) = 1 for all 𝑝.

Conversely suppose that W is a subspace of V. Consider the multi-fuzzy set 𝜇(𝑥) = { 1 𝐿 , 𝑖𝑓 𝑥 ∈ 𝑊 0 𝐿, 𝑖𝑓 𝑥 ∉ 𝑊 .

For 𝑥, 𝑦 ∈ 𝑊 and P∈ 𝐹, we have 𝑝𝑥 + 𝑦 ∈ 𝑊 Then, 𝜇(𝑝𝑥 + 𝑦) = 1 𝐿 = 𝑇(1 𝐿 , 1 𝐿 ) = T(T( 𝜆 ℎ (𝑝), 1 𝐿 ), 1 𝐿 )= T(𝜆 ℎ (𝑝), 𝜇(𝑥)), 𝜇(𝑦)) Thus, 𝜇𝜖𝑀𝐹𝑉 (𝑘,𝑚) (𝑉, (𝜆, ℎ ∈ 𝐿 𝑀 ), 𝑇)

Conclusion

This paper analysed the structure and characteristics of a multi-fuzzy vector space 𝜇 = (𝜇 1, , 𝜇 2 , … , 𝜇 𝑘 ) of V over the multi-fuzzy field 𝜆 = (𝜆 1, , 𝜆 2 , … , 𝜆 𝑘 ) of F, when the dimensions of μ and λ are equal. After that we discussed a new type of multi-fuzzy vector space μ over the multi-fuzzy field λ using a bridge function when the dimensions of the multi-fuzzy sets μ and λ differ. Further research is required to investigate the role of different bridge functions in multi-fuzzy vector spaces. This study provides a foundation for future exploration in multi-fuzzy linear algebra and its potential applications.

References

1. Abdukhalikov KS. The dual of fuzzy subspace. Fuzzy Sets Syst. 1996;82:375-381.
2. Katsaras AK, Liu DB. Fuzzy vector spaces and fuzzy topological vector spaces. J Math Anal Appl. 1977;58:135-146.
3. Ramakrishnan TV, Sebastian S. A study on multi-fuzzy sets. Int J Appl Math. 2010;23(4):713-721.
4. Rasuli R. Fuzzy vector spaces under norms. Ann Math Comput Sci. 2022;9:7-24.
5. Sebastian S, Ramakrishnan TV. Multi-fuzzy sets: an extension of fuzzy sets. Fuzzy Inf Eng. 2010;4:389-397.
6. Nanda S. Fuzzy linear spaces over valued fields. Fuzzy Sets Syst. 1991;42(3):351-354.
7. Sebastian S, Ramakrishnan TV. Multi-fuzzy sets. Int Math Forum. 2010;5(50):2471-2476.