Generalized Bifuzzy Lie Subalgebras

Abstract

We introduce the concept of (𝛾,𝛿)-bifuzzy Lie subalgebra, where 𝛾, 𝛿 are any two of {∈,𝑞,∈∨𝑞,∈∧𝑞} with 𝛾≠∈∧𝑞, by using belongs to relation (∈) and quasi-coincidence with relation (𝑞) between bifuzzy points and bifuzzy sets and discuss some of its properties. Then we introduce bifuzzy soft Lie subalgebras and investigate some of their properties.

1. Introduction

The concept of Lie groups was first introduced by Sophus Lie in nineteenth century through his studies in geometry and integration methods for differential equations. Lie algebras were also discovered by him when he attempted to classify certain smooth subgroups of a general linear group. The importance of Lie algebras in mathematics and physics has become increasingly evident in recent years. In applied mathematics, Lie theory remains a powerful tool for studying differential equations, special functions, and perturbation theory. It is noted that Lie theory has applications not only in mathematics and physics but also in diverse fields such as continuum mechanics, cosmology, and life sciences. A Lie algebra has nowadays even been applied by electrical engineers in solving problems in mobile robot control [1].

After introducing the concept of fuzzy sets by Zadeh [2] in 1965, there are many generalizations of this fundamental concept. Among these new concepts, the concept of intuitionistic fuzzy sets given by Atanassov [3] in 1983 is the most important and interesting one because it is simply an extension of fuzzy sets. In 1995, Gerstenkorn and Mańko [4] renamed the intuitionistic fuzzy sets as bifuzzy sets. The elements of the bifuzzy sets are featured by an additional degree which is called the degree of uncertainty. This kind of fuzzy sets have now gained a wide recognition as a useful tool in the modeling of some uncertain phenomena. Bifuzzy sets have drawn the attention of many researchers in the last decades. This is mainly due to the fact that bifuzzy sets are consistent with human behavior, by reflecting and modeling the hesitancy present in real-life situations. In fact, the fuzzy sets give the degree of membership of an element in a given set, while bifuzzy sets give both a degree of membership and a degree of nonmembership. As for fuzzy sets, the degree of membership is a real number between 0 and 1. This is also the case for the degree of nonmembership, and furthermore the sum of these two degrees is not greater than 1.

In 1999, Molodtsov [5] initiated the novel concept of soft set theory to deal with uncertainties which can not be handled by traditional mathematical tools. He successfully applied the soft set theory several disciplines, such as game theory, Riemann integration, Perron integration, and measure theory. Applications of soft set theory in real life problems are now catching momentum due to the general nature parametrization expressed by a soft set. Maji et al. [6] gave first practical application of soft sets in decision making problems. They also presented the definition of intuitionistic fuzzy soft set [7]. Yehia introduced the notion of fuzzy Lie subalgebras of Lie algebras in [8] and studied some results. Akram and Feng introduced the notion of soft Lie subalgebras of Lie algebras in [9] and studied some of their results. Akram et al. introduced the notions of fuzzy soft Lie subalgebras in [10]. In this paper, we introduce a new kind of generalized bifuzzy Lie algebra, an (𝛾,𝛿)-bifuzzy Lie algebra, and discuss some of its properties. Then we introduce bifuzzy soft Lie algebras and investigate some of their properties. We use standard definitions and terminologies in this paper.

2. Preliminaries

In this section, we review some known basic concepts that are necessary for this paper.

A Lie algebra is a vector space 𝐿 over a field 𝐹 (equal to 𝐑 or 𝐂) on which 𝐿×𝐿→𝐿 denoted by (𝑥,𝑦)→[𝑥,𝑦] is defined satisfying the following axioms:(L1)[𝑥,𝑦] is bilinear,(L2)[𝑥,𝑥]=0 for all 𝑥∈𝐿,(L3)[[𝑥,𝑦],𝑧]+[[𝑦,𝑧],𝑥]+[[𝑧,𝑥],𝑦]=0 for all 𝑥,𝑦,𝑧∈𝐿 (Jacobi identity).

Throughout this paper, 𝐿 is a Lie algebra and 𝐹 is a field. We note that the multiplication in a Lie algebra is not associative; that is, it is not true in general that [[𝑥,𝑦],𝑧]=[𝑥,[𝑦,𝑧]]. But it is anticommutative; that is, [𝑥,𝑦]=−[𝑦,𝑥]. A subspace 𝐻 of 𝐿 closed under [⋅,⋅] will be called a Lie subalgebra.

Let 𝜇 be a fuzzy set on 𝑋; that is, a map 𝜇∶𝑋→[0,1]. As an important generalization of the notion of fuzzy sets in 𝑋, Atanassov introduced the concept of a bifuzzy set defined on a nonempty set 𝑋 as objects having the form 𝐴=𝜇_𝐴,𝜈_𝐴=𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)∣𝑥∈𝑋, (1) where the functions 𝜇∶𝑋→[0,1] and 𝜈∶𝑋→[0,1] denote the degree of membership (viz., 𝜇_𝐴(𝑥)) and the degree of nonmembership (viz., 𝜈_𝐴(𝑥)) of each element 𝑥∈𝑋 to the set 𝐴, respectively, and 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1 for all 𝑥∈𝑋.

Definition 1 (see [3, 11]). Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be a bifuzzy set on 𝑋 and let 𝑠,𝑡∈[0,1] be such that 𝑠+𝑡≤1. Then the set 𝐴_(𝑠,𝑡)=𝑥∣𝜇_𝐴(𝑥)≥𝑠,𝜈_𝐴(𝑥)≤𝑡 (2) is called an (𝑠,𝑡)-level subset of 𝐴. 𝐴_(𝑠,𝑡) is a crisp set.

Definition 2 (see [12]). A bifuzzy set 𝐴=(𝜇_𝐴,𝜈_𝐴) on 𝐿 is called a bifuzzy Lie subalgebra if the following conditions are satisfied: (i)𝜇_𝐴(𝑥+𝑦)≥min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)},(ii)𝜈_𝐴(𝑥+𝑦)≤max{𝜈_𝐴(𝑥),𝜈_𝐴(𝑦)},(iii)𝜇_𝐴(𝛼𝑥)≥𝜇_𝐴(𝑥), 𝜈_𝐴(𝛼𝑥)≤𝜈_𝐴(𝑥),(iv)𝜇([𝑥,𝑦])≥max{𝜇(𝑥),𝜇(𝑦)},(v)𝜈([𝑥,𝑦])≤min{𝜈(𝑥),𝜈(𝑦)}
for all 𝑥, 𝑦∈𝐿 and 𝛼∈𝐹.

Definition 3 (see [13]). Let 𝑐 be a point in a nonempty set 𝑋. If 𝛾∈(0,1] and 𝛿∈[0,1) are two real numbers such that 0≤𝛾+𝛿≤1, then the bifuzzy set 𝑐(𝛾,𝛿)=⟨𝑥,𝑐_𝛾,1−𝑐_1−𝛿⟩ is called a bifuzzy point (BP for short) in 𝑋, where 𝛾 (resp., 𝛿) is the degree of membership (resp., nonmembership) of 𝑐(𝛾,𝛿) and 𝑐∈𝑋 is the support of 𝑐(𝛾,𝛿). Let 𝑐(𝛾,𝛿) be a BP in 𝑋 and let 𝐴=⟨𝑥,𝜇_𝐴,𝜈_𝐴⟩ be a bifuzzy set in 𝑋. Then 𝑐(𝛾,𝛿) is said to belong to 𝐴, written 𝑐(𝛾,𝛿)∈𝐴, if 𝜇_𝐴(𝑐)≥𝛾 and 𝜈_𝐴(𝑐)≤𝛿. We say that 𝑐(𝛾,𝛿) is quasicoincident with 𝐴, written 𝑐(𝛾,𝛿)𝑞𝐴, if 𝜇_𝐴(𝑐)+𝛾>1 and 𝜈_𝐴(𝑐)+𝛿<1. To say that 𝑐(𝛾,𝛿)∈∨𝑞𝐴 (resp., 𝑐(𝛾,𝛿)∈∧𝑞𝐴) means that 𝑐(𝛾,𝛿)∈𝐴 or 𝑐(𝛾,𝛿)𝑞𝐴 (resp., 𝑐(𝛾,𝛿)∈𝐴 and 𝑐(𝛾,𝛿)𝑞𝐴) and 𝑐(𝛾,𝛿)∈∨𝑞𝐴 means that 𝑐(𝛾,𝛿)∈∨𝑞𝐴 does not hold.

Definition 4. Let 𝑓 and 𝑔 be any two bifuzzy subsets of 𝐿. Then the sum 𝑓+𝑔 is a bifuzzy subset of 𝐿 defined by 𝜇_𝑓+𝜇_𝑔(𝑧)=_{𝑧=𝑥,𝑦}𝜇_𝑓(𝑎)∧𝜇_𝑓(𝑏)∧𝜇_𝑔(𝑎)∧𝜇_𝑔(𝑏),𝜈for𝑧=𝑥,𝑦0otherwise,_𝑓+𝜈_𝑔(𝑧)=_{𝑧=𝑥,𝑦}𝜈_𝑓(𝑎)∨𝜈_𝑓(𝑏)∨𝜈_𝑔(𝑎)∨𝜈_𝑔(𝑏)for𝑧=𝑥,𝑦,1otherwise. (3)

Let IF(𝑈) denote the family of all bifuzzy sets in 𝑈.

Definition 5 (see [7]). Let 𝑈 be an initial universe and 𝐴⊆𝐸 a set of parameters. A pair (𝑓,𝐴) is called a bifuzzy soft set over 𝑈, where 𝑓 is a mapping given by 𝑓∶𝐴→𝐼𝐹(𝑈). A bifuzzy soft set is a parameterized family of bifuzzy subsets of 𝑈. For any 𝜀∈𝐴, 𝑓_𝜀 is referred to as the set of 𝜀-approximate elements of the bifuzzy soft set (𝑓,𝐴), which is actually a bifuzzy set on 𝑈 and can be 𝑓_𝜀=𝜇_𝑓𝜀(𝑥),𝜈_𝑓𝜀(𝑥)∣𝑥∈𝑈, (4) where 𝜇_𝑓𝜀(𝑥) and 𝜈_𝑓𝜀(𝑥) are the membership degree and nonmembership degree that object 𝑥 holds on parameter 𝜀, respectively.

Definition 6 (see [7]). Let (𝑓,𝐴) and (𝑔,𝐵) be two bifuzzy soft sets over 𝑈. We say that (𝑓,𝐴) is a bifuzzy soft subset of (𝑔,𝐵) and write (𝑓,𝐴)⋐(𝑔,𝐵) if (i)𝐴⊆𝐵, (ii)for any 𝜀∈𝐴, 𝑓(𝜀)⊆𝑔(𝜀).
(𝑓,𝐴) and (𝑔,𝐵) are said to be bifuzzy soft equal and write (𝑓,𝐴)=(𝑔,𝐵) if (𝑓,𝐴)⋐(𝑔,𝐵) and (𝑔,𝐵)⋐(𝑓,𝐴).

Definition 7 (see [7, 14]). Let (𝑓,𝐴) and (𝑔,𝐵) be two bifuzzy soft sets over 𝑈. Then their extended intersection is a bifuzzy soft set denoted by (ℎ,𝐶), where 𝐶=𝐴∪𝐵 and ℎ(𝜀)=𝑓_𝜀𝑔if𝜀∈𝐴−𝐵,_𝜀𝑓if𝜀∈𝐵−𝐴,_𝜀∩𝑔_𝜀if𝜀∈𝐴∩𝐵, (5) for all 𝜀∈𝐶. This is denoted by (ℎ,𝐶)=(𝑓,𝐴)∩(𝑔,𝐵).

Definition 8 (see [7, 14]). If (𝑓,𝐴) and (𝑔,𝐵) are two bifuzzy soft sets over the same universe 𝑈 then “(𝑓,𝐴) AND (𝑔,𝐵)” is a bifuzzy soft set denoted by (𝑓,𝐴)∧(𝑔,𝐵) and is defined by (𝑓,𝐴)∧(𝑔,𝐵)=(ℎ,𝐴×𝐵), where, ℎ(𝑎,𝑏)=ℎ(𝑎)∩𝑔(𝑏) for all (𝑎,𝑏)∈𝐴×𝐵. Here ∩ is the operation of a bifuzzy intersection.

Definition 9 (see [7, 14]). Let (𝑓,𝐴) and (𝑔,𝐵) be two bifuzzy soft sets over 𝑈. Then their extended union is denoted by (ℎ,𝐶), where 𝐶=𝐴∪𝐵 and ℎ(𝜀)=𝑓_𝜀𝑔if𝜀∈𝐴−𝐵,_𝜀𝑓if𝜀∈𝐵−𝐴,_𝜀∪𝑔_𝜀if𝜀∈𝐴∩𝐵, (6) for all 𝜀∈𝐶. This is denoted by (ℎ,𝐶)=(𝑓,𝐴)∪(𝑔,𝐵).

Definition 10 (see [7, 14]). Let (𝑓,𝐴) and (𝑔,𝐵) be two fuzzy soft sets over a common universe 𝑈 with 𝐴∩𝐵≠∅. Then their restricted intersection is a bifuzzy soft set (ℎ,𝐴∩𝐵) denoted by (𝑓,𝐴)⋓(𝑔,𝐵)=(ℎ,𝐴∩𝐵), where ℎ(𝜀)=𝑓(𝜀)∩𝑔(𝜀) for all 𝜀∈𝐴∩𝐵.

Definition 11 (see [7, 14]). Let (𝑓,𝐴) and (𝑔,𝐵) be two bifuzzy soft sets over a common universe 𝑈 with 𝐴∩𝐵≠∅. Then their restricted union is denoted by (𝑓,𝐴)⋒(𝑔,𝐵) and is defined as (𝑓,𝐴)⋒(𝑔,𝐵)=(ℎ,𝐶), where 𝐶=𝐴∩𝐵 and for all 𝜀∈𝐶, ℎ(𝜀)=𝑓(𝜀)∪𝑔(𝜀).

Definition 12 (see [7, 14]). The extended product of two bifuzzy soft sets (𝑓,𝐴) and (𝑔,𝐵) over 𝑈 is a fuzzy soft set, denoted by (𝑓∘𝑔,𝐶), where 𝐶=𝐴∪𝐵, and defined by 𝜀)=𝑓(𝜀)𝑔(𝜀)𝑓(𝜀)(𝜀)𝑓∘𝑔(if𝜀∈𝐴−𝐵,if𝜀∈𝐵−𝐴,∘𝑔if𝜀∈𝐴∩𝐵, (7) for all 𝜀∈𝐶. This is denoted by (𝑓∘𝑔,𝐶)=(𝑓,𝐴)∘̃(𝑔,𝐵).

3. Bifuzzy Lie Algebras

In this section, we introduce a new kind of generalized bifuzzy Lie algebra, an (𝛾,𝛿)-bifuzzy Lie subalgebra, and discuss some of its properties.

Definition 13. A bifuzzy set 𝐴=(𝜇_𝐴,𝜈_𝐴) in 𝐿 is called an (𝛾,𝛿)-bifuzzy Lie subalgebra of 𝐿 if it satisfies the following conditions: (a)𝑥(𝑠₁,𝑡₁)𝛾𝐴,𝑦(𝑠₂,𝑡₂)𝛾𝐴⇒(𝑥+𝑦)(min(𝑠₁,𝑠₂),max(𝑡₁,𝑡₂))𝛿𝐴,(b)𝑥(𝑠,𝑡)𝛾𝐴⇒(𝑚𝑥)(𝑠,𝑡)𝛿𝐴,(c)𝑥(𝑠₁,𝑡₁)𝛾𝐴,𝑦(𝑠₂,𝑡₂)𝛾𝐴⇒[𝑥,𝑦](max(𝑠₁,𝑠₂),min(𝑡₁,𝑡₂))𝛿𝐴
for all 𝑥,𝑦∈𝐿,  𝑚∈𝐹,  𝑠,𝑠₁,𝑠₂∈(0,1],  𝑡,𝑡₁,𝑡₂∈[0,1).

Remark 14. Consider (i) 𝑥(𝑠,𝑡)𝛾𝐴⇒(−𝑥)(𝑠,𝑡)𝛿𝐴,
(ii) 𝑥(𝑠,𝑡)𝛾𝐴⇒(0)(𝑠,𝑡)𝛿𝐴.

Example 15. Let 𝑉 be a vector space over a field 𝐹 such that dim(𝑉)=5. Let 𝑉={𝑒₁,𝑒₂,…,𝑒₅} be a basis of a vector space over a field 𝐹 with Lie brackets as follows: 𝑒₁,𝑒₂=𝑒₃,𝑒₁,𝑒₃=𝑒₅,𝑒₁,𝑒₄=𝑒₅,𝑒₁,𝑒₅𝑒=0,₂,𝑒₃=𝑒₅,𝑒₂,𝑒₄𝑒=0,₂,𝑒₅=0,𝑒₃,𝑒₄𝑒=0,₃,𝑒₅=0,𝑒₄,𝑒₅=0,𝑒_𝑖,𝑒_𝑗=−𝑒_𝑗,𝑒_𝑖 (8) and [𝑒_𝑖,𝑒_𝑗]=0 for all 𝑖=𝑗. Then 𝑉 is a Lie algebra over 𝐹.
We define a bifuzzy set 𝐴=(𝜇_𝐴,𝜈_𝐴)∶𝑉→[0,1]×[0,1] by 𝜇_𝐴(𝑥)𝜈∶=1if𝑥=0,0.5otherwise,_𝐴(𝑥)∶=0if𝑥=0,0.3otherwise. (9) Take 𝑠=0.4∈(0,1] and 𝑡=0.5∈[1,0). By routine computations, it is easy to see that 𝐴 is not an (𝛾,𝛿)-bifuzzy Lie subalgebra of 𝐿.

For a bifuzzy set 𝐴 in 𝐿, we denote 𝐿(0,1)={𝑥∈𝐿∶𝜇(𝑥)>0 and 𝜈(𝑥)<1}.

Theorem 16. Let 𝐴=(𝜇_𝐴,𝜈) be an (𝛾,𝛿)-bifuzzy Lie subalgebra of 𝐿; then the nonzero set 𝐿(0,1) is a Lie subalgebra of 𝐿.

Proof. Let 𝑥, 𝑦∈𝐿(0,1). Then 𝜇_𝐴(𝑥)>0 and 𝜈_𝐴(𝑥)<1, 𝜇_𝐴(𝑦)>0 and 𝜈_𝐴(𝑦)<1. Assume that 𝜇_𝐴(𝑥+𝑦)=0 and 𝜈_𝐴(𝑥+𝑦)=1. If 𝛾∈{∈,∈∨𝑞}, then we can see that 𝑥(𝜇_𝐴(𝑥),𝜈_𝐴(𝑥))𝛾𝐴 and 𝑦(𝜇_𝐴(𝑦),𝜈_𝐴(𝑦))𝛾𝐴, but (𝑥+𝑦)(min{𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)},max{𝜇_𝐴(𝑦),𝜈_𝐴(𝑦)})𝛿𝐴 for all 𝛿∈{∈,∈∨𝑞,∈∧𝑞}, a contradiction. Also, 𝑥(1,0)𝑞𝐴 and 𝑦(1,0)𝑞𝐴, but (𝑥+𝑦)(1,0)𝛿𝐴 for all 𝛿∈{∈,∈∨𝑞,∈∧𝑞}, a contradiction. Thus 𝜇_𝐴(𝑥+𝑦)>0 and 𝜈_𝐴(𝑥+𝑦)<1. Thus 𝑥+𝑦∈𝐿(0,1). For other conditions the verification is analogous. Consequently 𝐿(0,1) is a Lie subalgebra of 𝐿.

Definition 17. A bifuzzy set 𝐴=(𝜇_𝐴,𝜈_𝐴) in 𝐿 is called an (∈,∈∨𝑞)-bifuzzy Lie algebra of 𝐿 if it satisfies the following conditions: (f)𝑥(𝑠₁,𝑡₁)∈𝐴, 𝑦(𝑠₂,𝑡₂)∈𝐴⇒(𝑥+𝑦)(min(𝑠₁,𝑠₂), max(𝑡₁,𝑡₂))∈∨𝑞𝐴,(g)𝑥(𝑠,𝑡)∈𝐴⇒(𝑚𝑥)(𝑠,𝑡)∈∨𝑞𝐴,(h)𝑥(𝑠₁,𝑡₁)∈𝐴, 𝑦(𝑠₂,𝑡₂)∈𝐴⇒[𝑥,𝑦](max(𝑠₁,𝑠₂), min(𝑡₁,𝑡₂))∈∨𝑞𝐴
for all 𝑥, 𝑦∈𝐿, 𝑚∈𝐹, 𝑠, 𝑠₁, 𝑠₂∈(0,1], 𝑡, 𝑡₁, 𝑡₂∈[0,1).

Theorem 18. Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be a bifuzzy set in a Lie algebra 𝐿. Then 𝐴 is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 if and only if(i)𝜇_𝐴(𝑥+𝑦)⩾min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦),0.5), 𝜈_𝐴(𝑥+𝑦)⩽max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦),0.5),(j)𝜇_𝑎(𝑚𝑥)⩾min(𝜇_𝐴(𝑥),0.5), 𝜈_𝑎(𝑚𝑥)⩽max(𝜈_𝐴(𝑥),0.5),(k)𝜇_𝐴([𝑥,𝑦])⩾max(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦),0.5), 𝜈_𝐴([𝑥,𝑦])⩽min(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦),0.5)
hold for all 𝑥,𝑦∈𝐿, 𝑚∈𝐹.

Proof. (f)⇒(i): Let 𝑥,𝑦∈𝐿. We consider the following two cases:(1)min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦))<0.5, max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦))>0.5,(2)min(𝜇_𝐴(𝑥,𝜇_𝐴(𝑦))⩾0.5, max(𝜈_𝐴(𝑥,𝜈_𝐴(𝑦))⩽0.5.
Case 1. Assume that 𝜇_𝐴(𝑥+𝑦)<min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦),0.5), 𝜈_𝐴(𝑥+𝑦)>max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦),0.5). Then 𝜇_𝐴(𝑥+𝑦)<min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)), 𝜈_𝐴(𝑥+𝑦)>max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦)). Take 𝑠, 𝑡 such that 𝜇_𝐴(𝑥+𝑦)<𝑠<min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)), 𝜈_𝐴(𝑥+𝑦)>𝑡>max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦)). Then 𝑥_𝑠,𝑦_𝑠∈𝜇_𝐴 and 𝑥_𝑡,𝑦_𝑡∈𝜈_𝐴, but (𝑥+𝑦)(min(𝑠₁,𝑠₂),max(𝑡₁,𝑡₂))∈∨𝑞𝐴, which is contradiction with (𝑓).
Case 2. Assume that 𝜇_𝐴(𝑥+𝑦)<0.5, 𝜈_𝐴(𝑥+𝑦)>0.5. Then 𝑥(0.5,0.5), 𝑦(0.5,0.5)∈𝐴 but (𝑥+𝑦)(0.5,0.5)∈∨𝑞𝐴, a contradiction. Hence (i) holds. (i)⇒(f): Let 𝑥(𝑠₁,𝑡₁), 𝑦(𝑠₂,𝑡₂)∈𝐴; then 𝜇_𝐴(𝑥)⩾𝑠₁, 𝜇_𝐴(𝑦)⩾𝑠₂, 𝜈_𝐴(𝑥)⩽𝑡₁, 𝜈_𝐴(𝑦)⩽𝑡₂. Now, we have 𝜇_𝐴𝑥+𝑦⩾min𝜇_𝐴(𝑥),𝜇_𝐴𝑦,0.5⩾min𝑠₁,𝑠₂,𝜈,0.5_𝐴𝑥+𝑦⩽max𝜈_𝐴(𝑥),𝜈_𝐴𝑦,0.5⩽max𝑡₁,𝑡₂,0.5. (10) If min(𝑠₁,𝑠₂)>0.5, max(𝑡₁,𝑡₂<0.5, then 𝜇_𝐴(𝑥+𝑦)⩾0.5⇒𝜇_𝐴(𝑥+𝑦)+min(𝑠₁,𝑠₂)>1, 𝜈_𝐴(𝑥+𝑦)⩽0.5⇒𝜈_𝐴(𝑥+𝑦)+max(𝑡₁,𝑡₂)<1. On the other hand, if min(𝑠₁,𝑠₂)⩽0.5, max(𝑡₁,𝑡₂)⩾0.5, then 𝜇_𝐴(𝑥+𝑦)⩾min(𝑠₁,𝑠₂), 𝜈_𝐴(𝑥+𝑦)⩽max(𝑡₁,𝑡₂). Hence (𝑥+𝑦)(min(𝑠₁,𝑠₂),max(𝑡₁,𝑡₂))∈∨𝑞𝐴. The verification of (g)⇔(j) and (h)⇔(k) is analogous and we omit the details. This completes the proof.

Theorem 19. Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be a bifuzzy set of Lie algebra of 𝐿. Then 𝐴 is an (∈,∈∨𝑞)-bifuzzy Li subalgebra of 𝐿 if and only if each nonempty 𝐴_(𝑠,𝑡), 𝑠∈(0.5,1], 𝑡∈[0.5,1) is a Lie subalgebra of 𝐿.

Proof. Assume that 𝐴=(𝜇_𝐴,𝜈_𝐴) is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 and let 𝑠∈(0.5,1], 𝑡∈[0.5,1). If 𝑥,𝑦∈𝐴_(𝑠,𝑡) and 𝑚∈𝐹, then 𝜇_𝐴(𝑥)≥𝑠 and 𝜇_𝐴(𝑦)≥𝑠, 𝜈_𝐴(𝑥)≤𝑡 and 𝜈_𝐴(𝑦)≤𝑡. Thus, 𝜇_𝐴𝑥+𝑦⩾min𝜇_𝐴(𝑥),𝜇_𝐴𝑦()𝜈,0.5⩾min𝑠,0.5=𝑠,_𝐴𝑥+𝑦⩽max𝜈_𝐴(𝑥),𝜈_𝐴𝑦()𝜇,0.5⩽max𝑡,0.5=𝑡,_𝐴(𝑚𝑥)⩾min𝜇_𝐴(𝑥)()𝜈,0.5⩾min𝑠,0.5=𝑠,_𝐴(𝑚𝑥)⩽max𝜈_𝐴(𝑥)()𝜇,0.5⩽max𝑡,0.5=𝑡,_𝐴𝑥,𝑦⩾min𝜇_𝐴(𝑥),𝜇_𝐴𝑦()𝜈,0.5⩾min𝑡,0.5=𝑡,_𝐴𝑥,𝑦⩽max𝜈_𝐴(𝑥),𝜈_𝐴𝑦,0.5⩽max(𝑡,0.5)=𝑡, (11) and so 𝑥+𝑦,𝑚𝑥,[𝑥,𝑦]∈𝐴_(𝑠,𝑡). This shows that 𝐿(𝜈;𝑡) are Lie subalgebras of 𝐿. The proof of converse part is obvious. This ends the proof.

Theorem 20. Let 𝐴 be a bifuzzy set in a Lie algebra 𝐿. Then 𝐴_(𝑠,𝑡) is a Lie subalgebra of 𝐿 if and only if(1)max(𝜇_𝐴(𝑥+𝑦),0.5)⩾min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)), min(𝜈_𝐴(𝑥+𝑦),0.5)⩽max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦)),(2)max(𝜇_𝐴(𝑚𝑥),0.5)⩾𝜇_𝐴(𝑥), min(𝜈_𝐴(𝑚𝑥),0.5)⩽𝜈_𝐴(𝑥),(3)max(𝜇_𝐴([𝑥,𝑦]),0.5))⩾min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)), min(𝜈_𝐴([𝑥,𝑦]),0.5))⩽max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦)) for all 𝑥,𝑦∈𝐿, 𝑚∈𝐹.

Proof. Suppose that 𝐴_(𝑠,𝑡) is a Lie subalgebra of 𝐿. Let max(𝜇_𝐴(𝑥+𝑦),0.5)<min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦))=𝑠, min(𝜈_𝐴(𝑥+𝑦),0.5)>max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦))=𝑡 for some 𝑥,𝑦∈𝐿; then 𝑠∈(0.5,1], 𝑡∈[0.5,1), 𝜇_𝐴(𝑥+𝑦)<𝑠, 𝜈_𝐴(𝑥+𝑦)>𝑡, 𝑥,𝑦∈𝐴_(𝑠,𝑡). Since 𝑥,𝑦∈𝐴_(𝑠,𝑡) and 𝐴_(𝑠,𝑡) is a Lie subalgebra of 𝐿, so 𝑥+𝑦∈𝐴_(𝑠,𝑡) or 𝜇_𝐴(𝑥+𝑦)⩾𝑠, 𝜈_𝐴(𝑥+𝑦)⩽𝑡, which is contradiction with 𝜇_𝐴(𝑥+𝑦)<𝑠, 𝜈_𝐴(𝑥+𝑦)>𝑡. Hence (1) holds. For (2), (3) the verification is analogous.
Conversely, suppose that (1)–(3) hold. Assume that 𝑠∈(0.5,1], 𝑡∈[0.5,1), 𝑥,𝑦∈𝐴_(𝑠,𝑡). Then 0.5<𝑠⩽min𝜇_𝐴(𝑥),𝜇_𝐴𝑦⩽max𝜇_𝐴𝑥+𝑦,0.5⟹𝜇_𝐴𝜈𝑥+𝑦⩾𝑠,0.5>𝑡⩾max_𝐴(𝑥),𝜈_𝐴𝑦⩾min𝜈_𝐴𝑥+𝑦,0.5⟹𝜈_𝐴𝑥+𝑦⩽𝑡,0.5<𝑠⩽𝜇_𝐴(𝑥)⩽max𝜇_𝐴(𝑚𝑥),0.5⟹𝜇_𝐴(𝑚𝑥)⩾𝑠,0.5>𝑡⩾𝜈_𝐴(𝑥)⩾min𝜈_𝐴(𝑚𝑥),0.5⟹𝜈_𝐴()𝜇𝑚𝑥⩽𝑡,0.5<𝑠⩽min_𝐴(𝑥),𝜇_𝐴𝑦⩽max𝜇_𝐴𝑥,𝑦,0.5⟹𝜇_𝐴𝜈𝑥,𝑦⩾𝑠,0.5>𝑡⩾max_𝐴(𝑥),𝜈_𝐴𝑦⩾min𝜈_𝐴𝑥,𝑦,0.5⟹𝜈_𝐴𝑥,𝑦⩽𝑡, (12) and so 𝑥+𝑦,𝑚𝑥,[𝑥,𝑦]∈𝐴_(𝑠,𝑡). This shows that 𝐴_(𝑠,𝑡) is a Lie subalgebra of 𝐿.

Theorem 21. The intersection of any family of (∈,∈∨𝑞)-bifuzzy Lie subalgebras of 𝐿 is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra.

Proof. Let {𝐴_𝑖∶𝑖∈Λ} be a family of (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 and let 𝐴∶=⋂_𝑖∈Λ𝐴_𝑖=(sup_𝑖∈Λ𝜇_𝑖,inf_𝑖∈Λ𝜈_𝑖). Let 𝑥, 𝑦∈𝐿, then by Theorem 19, we have 𝜇_𝐴(𝑥+𝑦)⩾min(𝜇_𝐴(𝑥),𝜇_𝐴(𝑦),0.5), 𝜈_𝐴(𝑥+𝑦)⩽max(𝜈_𝐴(𝑥),𝜈_𝐴(𝑦),0.5), and hence 𝜇_𝐴𝑥+𝑦=sup_𝑖∈Λ𝜇_𝑖𝑥+𝑦⩾sup_𝑖∈Λmin𝜇_𝑖(𝑥),𝜇_𝑖𝑦,0.5=minsup_𝑖∈Λ𝜇_𝑖(𝑥),sup_𝑖∈Λ𝜇_𝑖𝑦,0.5=min_𝑖∈Λ𝜇_𝑖(𝑥),_𝑖∈Λ𝜇_𝑖𝑦𝜇,0.5=min_𝐴(𝑥),𝜇_𝐴𝑦,𝜈,0.5_𝐴𝑥+𝑦=inf_𝑖∈Λ𝜈_𝑖𝑥+𝑦⩽inf_𝑖∈Λmax𝜈_𝑖(𝑥),𝜈_𝑖𝑦,0.5=maxinf_𝑖∈Λ𝜈_𝑖(𝑥),inf_𝑖∈Λ𝜈_𝑖𝑦,0.5=max_𝑖∈Λ𝜈_𝑖(𝑥),_𝑖∈Λ𝜈_𝑖𝑦𝜈,0.5=max_𝐴(𝑥),𝜈_𝐴𝑦,0.5. (13) For other conditions the verification is analogous. By Theorem 19, it follows that 𝐴 is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿.

Theorem 22. Let 𝐿₀⊂𝐿₁⊂⋯⊂𝐿_𝑛=𝐿 be a strictly increasing chain of (∈,∈)-bifuzzy Lie subalgebras of a Lie algebra 𝐿; then there exists (∈,∈)-bifuzzy Lie subalgebra 𝐴=(𝜇_𝐴,𝜈_𝐴) of 𝐿 whose level subalgebras are precisely the members of the chain with 𝐴_0.5=(𝜇_0.5,𝜈_0.5)=𝐿(0,1).

For any 𝑟,𝑠∈[0,1] and fuzzy subset 𝜇 in 𝐿, denotes 𝜇𝑟̂={𝑥∈𝐿∣𝜇(𝑥)>𝑟}, ⟨𝜇⟩_𝑟={𝑥∈𝐿∣𝑥_𝑟𝑞𝜇}, [𝜇]_𝑟={𝑥∈𝐿∣𝑥_𝑟∈∨𝑞𝜇}, 𝜈_𝑠={𝑥∈𝐿∣𝜇(𝑥)<𝑠}, and [𝜈]_𝑠={𝑥∈𝐿∣𝑥_𝑠∈∨𝑞𝜈}. Clearly, 𝐴^(𝑟,𝑠)=𝜇_𝐴𝑟̂∩𝜇_𝐴𝑠 for all 𝑟,𝑠∈[0,1].

We state here a nice characterization without proof.

Theorem 23. Let 𝐿 be a Lie algebra and 𝐴 a bifuzzy set in 𝐿. Then (a)𝐴 is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 if and only if nonempty subsets 𝜇_𝐴𝑟̂ and 𝜈_𝐴𝑠 are Lie subalgebras of 𝐿 for all 𝑟∈[0,0.5) and 𝑠∈(0.5,1];(b)𝐴 is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 if and only if nonempty subsets 𝜇_𝐴𝑟̂ and [𝜈_𝐴]_𝑠 are Lie subalgebras of 𝐿 for all 𝑟∈[0,0.5) and 𝑠∈(0,1];(c)𝐴 is an (∈,∈∨𝑞)-intuitionistic fuzzy Lie subalgebra of 𝐿 if and only if nonempty subsets ⟨𝜇_𝐴⟩_𝑟 and 𝜈_𝐴𝑠 are 𝐾-subalgebras of 𝒦 for all 𝑟,𝑠∈(0.5,1];(d)𝐴 is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 if and only if nonempty subsets ⟨𝜇_𝐴⟩_𝑟 and [𝜈_𝐴]_𝑠 are Lie subalgebras of 𝐿 for all 𝑟∈(0.5,1] and 𝑠∈(0,1];(e)𝐴 is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 if and only if nonempty subsets [𝜇_𝐴]_𝑟 and 𝜈_𝐴𝑠 are Lie subalgebras of 𝐿 for all 𝑟∈(0,1] and 𝑠∈(0.5,1];(f)𝐴 is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 if and only if nonempty subsets [𝜇_𝐴]_𝑟 and [𝜈_𝐴]_𝑠 are ideals of 𝐿 for all 𝑟,𝑠∈(0,1].

4. Bifuzzy Soft Lie Algebras

In this section, we introduce bifuzzy soft Lie subalgebras and investigate some of their properties.

Definition 24. Let 𝐿 be a Lie algebra and let (𝑓,𝐴) be a bifuzzy soft set over 𝐿. Then (𝑓,𝐴) is said to be a bifuzzy soft Lie subalgebra over 𝐿 if 𝑓(𝑥) is a bifuzzy Lie subalgebra of 𝐿 for all 𝑥∈𝐴; that is, a bifuzzy soft set (𝑓,𝐴) on 𝐿 is called a bifuzzy soft Lie subalgebra of 𝐿 if (a)𝜇_𝑓𝜀(𝑥+𝑦)⩾min{𝜇_𝑓𝜀(𝑥),𝜇_𝑓𝜀(𝑦)},(b)𝜈_𝑓𝜀(𝑥+𝑦)⩽max{𝜈_𝑓𝜀(𝑥),𝜈_𝑓𝜀(𝑦)},(c)𝜇_𝑓𝜀(𝑚𝑥)⩾𝜇_𝑓𝜀(𝑥),(d)𝜈_𝑓𝜀(𝑚𝑥)⩽𝜇_𝑓𝜀(𝑥),(e)𝜇_𝑓𝜀([𝑥,𝑦])⩾min{𝜇𝑓_𝜀(𝑥),𝜇_𝑓𝜀(𝑦)},(f)𝜈_𝑓𝜀([𝑥,𝑦])⩽max{𝜈𝑓_𝜀(𝑥),𝜈_𝑓𝜀(𝑦)}
hold for all 𝑥,𝑦∈𝐿 and 𝑚∈𝐾.

Example 25. Let ℜ²={(𝑥,𝑦)∶𝑥,𝑦∈ℝ} be the set of all 2-dimensional real vectors. Then ℜ² with [𝑥,𝑦]=𝑥×𝑦 is a real Lie algebra. Let ℕ and ℤ denote the set of all natural numbers and the set of all integers, respectively. Define 𝑓∶ℤ→([0,1]×[0,1])^ℜ2 by 𝑓(𝑛)=𝑓_𝑛∶ℜ²→[0,1]×[0,1] for all 𝑛∈ℤ, 𝜇_𝑓𝑛(𝑥)=0.6if𝑥=(0,0)=0,()𝜈0.2if𝑥=0,𝑎,𝑎≠0,0otherwise,_𝑓𝑛(𝑥)=0.0if𝑥=(0,0)=0,0.4if𝑥=(0,𝑎),𝑎≠0,0.6otherwise. (14) By routine computations, we can easily check that (𝑓,ℤ) is a bifuzzy soft Lie subalgebra of ℜ².

The following proposition is obvious.

Proposition 26. Let (𝑓,𝐴) be a bifuzzy soft Lie subalgebra of 𝐿; then (i)𝜇_𝑓𝜀(0)⩾𝜇_𝑓𝜀(𝑥),(ii)𝜈_𝑓𝜀(0)⩽𝜈_𝑓𝜀(𝑥)
for all 𝑥∈𝐿.

Definition 27. Let (𝑓,𝐴) be a bifuzzy soft set over 𝑈. For each 𝑠,𝑡∈[0,1], the set (𝑓,𝐴)^(𝑠,𝑡)=(𝑓^(𝑠,𝑡),𝐴) is called an (𝑠,𝑡)-level soft set of (𝑓,𝐴), where 𝑓_𝜀^(𝑠,𝑡)={𝑥∈𝑈∣𝜇_𝑓𝜀(𝑥)≥𝑠,𝜈_𝑓𝜀(𝑥)≤𝑡} for all 𝜀∈𝐴.

Theorem 28. Let (𝑓,𝐴) be a bifuzzy soft set over 𝐿. (𝑓,𝐴) is a bifuzzy soft Lie subalgebra if and only if (𝑓,𝐴)^(𝑠,𝑡) is a soft Lie subalgebra over 𝐿 for each 𝑠,𝑡∈[0,1].

Proof. Suppose that (𝑓,𝐴) is a bifuzzy soft Lie subalgebra. For each 𝑠,𝑡∈[0,1], 𝜀∈𝐴, and 𝑥₁, 𝑥₂∈(𝑓,𝐴)_𝜀^(𝑠,𝑡), then 𝜇_𝑓𝜀(𝑥₁)≥𝑠, 𝜇_{𝑓𝜀(𝑥2)}≥𝑠 and 𝜈_𝑓𝜀(𝑥₁)≤𝑡, 𝜈_{𝑓𝜀(𝑥2)}≤𝑡. From Definition 27, it follows that (𝑓,𝐴)_𝜀^(𝑠,𝑡) is a bifuzzy Lie subalgebra over 𝐿. Thus 𝜇_𝑓𝜀(𝑥₁+𝑥₂)≥min(𝜇_𝑓𝜀(𝑥₁),𝜇_𝑓𝜀(𝑥₂)), 𝜇_𝑓𝜀(𝑥₁+𝑥₂)≥𝑠, 𝜈_𝑓𝜀(𝑥₁+𝑥₂)≤max(𝜈_𝑓𝜀(𝑥₁),𝜈_𝑓𝜀(𝑥₂)), 𝜈_𝑓𝜀(𝑥₁+𝑥₂)≤𝑡. This implies that 𝑥₁+𝑥₂∈(𝑓,𝐴)_𝜀^(𝑠,𝑡). Verification for other conditions is similar. Hence we omit the details. This shows that (𝑓,𝐴)_𝜀^(𝑠,𝑡) is a Lie subalgebra over 𝐿. According to Definition 27, (𝑓,𝐴)^(𝑠,𝑡) is a soft Lie subalgebra over 𝐿 for each 𝑠,𝑡∈[0,1].
Conversely, assume that (𝑓,𝐴)^(𝑠,𝑡) is a soft Lie subalgebra over 𝐿 for each 𝑠,𝑡∈[0,1]. For each 𝜀∈𝐴 and 𝑥₁,𝑥₂∈𝐿, let 𝑠=min{𝜇_𝑓𝜀(𝑥₁),𝜇_𝑓𝜀(𝑥₂)} and let 𝑡=max{𝜈_𝑓𝜀(𝑥₁),𝜈_𝑓𝜀(𝑥₂)}; then 𝑥₁,𝑥₂∈(𝑓,𝐴)_𝜀^(𝑠,𝑡). Since (𝑓,𝐴)_𝜀^(𝑠,𝑡) is a Lie subalgebra over 𝐿, then 𝑥₁+𝑥₂∈(𝑓,𝐴)_𝜀^(𝑠,𝑡). This means that 𝜇_𝑓𝜀(𝑥₁+𝑥₂)≥min(𝜇_𝑓𝜀(𝑥₁),𝜇_𝑓𝜀(𝑥₂)), 𝜈_𝑓𝜀(𝑥₁+𝑥₂)≤max(𝜈_𝑓𝜀(𝑥₁),𝜈_𝑓𝜀(𝑥₂)). Verification for other conditions is similar. Hence we omit the details. Thus, (𝑓,𝐴)_𝜀^(𝑠,𝑡) is a bifuzzy Lie subalgebra over 𝐿. According to Definition 24, (𝑓,𝐴) is a bifuzzy soft Lie subalgebra over 𝐿. This completes the proof.

Definition 29. Let 𝜙∶𝑋→𝑌 and 𝜓∶𝐴→𝐵 be two functions, where 𝐴 and 𝐵 are parametric sets from the crisp sets 𝑋 and 𝑌, respectively. Then the pair (𝜙,𝜓) is called a bifuzzy soft function from 𝑋 to 𝑌.

Definition 30. Let (𝑓,𝐴) and (𝑔,𝐵) be two bifuzzy soft sets over 𝐿₁ and 𝐿₂, respectively, and let (𝜙,𝜓) be a bifuzzy soft function from 𝐿₁ to 𝐿₂.(1)The image of (𝑓,𝐴) under the bifuzzy soft function (𝜙,𝜓), denoted by (𝜙,𝜓)(𝑓,𝐴), is the bifuzzy soft set on 𝐿₂ defined by (𝜙,𝜓)(𝑓,𝐴)=(𝜙(𝑓),𝜓(𝐴)), where for all 𝑘∈𝜓(𝐴), 𝑦∈𝐿₂𝜇_{𝜙(𝑓)𝑘}𝑦=_{𝜙(𝑥)=𝑦𝜓(𝑎)=𝑘}𝑓_𝑎(𝑥)if𝑥∈𝜓⁻¹𝑦,𝜈1otherwise,_{𝜙(𝑓)𝑘}𝑦=_{𝜙(𝑥)=𝑦𝜓(𝑎)=𝑘}𝑓_𝑎(𝑥)if𝑥∈𝜓⁻¹𝑦,0otherwise. (15)(2)The preimage of (𝑔,𝐵) under the bifuzzy soft function (𝜙,𝜓), denoted by (𝜙,𝜓)⁻¹(𝑔,𝐵), is the bifuzzy soft set over 𝐿₁ defined by (𝜙,𝜓)⁻¹(𝑔,𝐵)=(𝜙⁻¹(𝑔),𝜓⁻¹(𝐵)), where for all 𝑎∈𝜓⁻¹(𝐴), for all 𝑥∈𝐿₁, 𝜇_𝜙⁻¹_(𝑔)𝑎(𝑥)=𝜇_{𝑔𝜓(𝑎)}𝜙(𝑥),𝜈_𝜙⁻¹_(𝑔)𝑎(𝑥)=𝜈_{𝑔𝜓(𝑎)}𝜙(𝑥). (16)

Definition 31. Let (𝜙,𝜓) be a bifuzzy soft function from 𝐿₁ to 𝐿₂. If 𝜙 is a homomorphism from 𝐿₁ to 𝐿₂, then (𝜙,𝜓) is said to be bifuzzy soft homomorphism. If 𝜙 is a isomorphism from 𝐿₁ to 𝐿₂ and 𝜓 is one-to-one mapping from 𝐴 onto 𝐵 then (𝜙,𝜓) is said to be bifuzzy soft isomorphism.

Theorem 32. Let (𝑔,𝐵) be a bifuzzy soft Lie subalgebra over 𝐿₂ and let (𝜙,𝜓) be a bifuzzy soft homomorphism from 𝐿₁ to 𝐿₂. Then (𝜙,𝜓)⁻¹(𝑔,𝐵) is a bifuzzy soft Lie subalgebra over 𝐿₁.

Proof. Let 𝑥₁,𝑥₂∈𝐿₁; then 𝜙⁻¹𝜇_𝑔𝜀𝑥₁+𝑥₂=𝜇_{𝑔𝜓(𝜀)}𝜙𝑥₁+𝑥₂=𝜇_{𝑔𝜓(𝜀)}𝜙𝑥₁+𝜙𝑥₂⩾min𝜇_{𝑔𝜓(𝜀)}𝜙𝑥₁,𝜇_{𝑔𝜓(𝜀)}𝜙𝑥₂=min𝜙⁻¹𝜇_𝑔𝜀𝑥₁,𝜙⁻¹𝜇_𝑔𝜀𝑥₂,𝜙⁻¹𝜈_𝑔𝜀𝑥₁+𝑥₂=𝜈_{𝑔𝜓(𝜀)}𝜙𝑥₁+𝑥₂=𝜈_{𝑔𝜓(𝜀)}𝜙𝑥₁+𝜙𝑥₂⩽max𝜈_{𝑔𝜓(𝜀)}𝜙𝑥₁,𝜈_{𝑔𝜓(𝜀)}𝜙𝑥₂=max𝜙⁻¹𝜈_𝑔𝜀𝑥₁,𝜙⁻¹𝜈_𝑔𝜀𝑥₂. (17) Verification for other conditions is similar. Hence we omit the details. Hence (𝜙,𝜓)⁻¹(𝑔,𝐵) is a bifuzzy soft Lie subalgebra over 𝐿₁.

Remark 33. Let (𝑓,𝐴) be a bifuzzy soft Lie subalgebra over 𝐿₁ and let (𝜙,𝜓) be a bifuzzy soft homomorphism from 𝐿₁ to 𝐿₂. Then (𝜙,𝜓)(𝑓,𝐴) may not be a bifuzzy soft Lie subalgebra over 𝐿₂.

Theorem 34. Let (𝑓,𝐴) be a bifuzzy soft Lie subalgebra over 𝐿 and let {(ℎ_𝑖,𝐵_𝑖)∣𝑖∈𝐼} be a nonempty family of bifuzzy soft Lie subalgebras of (𝑓,𝐴). Then (a)⋂_𝑖∈𝐼(ℎ_𝑖,𝐵_𝑖) is a bifuzzy soft Lie subalgebra of (𝑓,𝐴);(b)⋀_𝑖∈𝐼(ℎ_𝑖,𝐵_𝑖) is a bifuzzy soft Lie subalgebra of ⋀_𝑖∈𝐼(𝑓,𝐴);(c)if 𝐵_𝑖∩𝐵_𝑗=∅ for all 𝑖,𝑗∈𝐼, then ⋁_𝑖∈𝐼(𝐻_𝑖,𝐵_𝑖) is an bifuzzy soft Lie subalgebra of ⋁_𝑖∈𝐼(𝑓,𝐴).

As a generalization of above theorem, we have the following result.

Theorem 35. Let (𝑓,𝐴) be an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra over 𝐿 and let {(ℎ_𝑖,𝐵_𝑖)∣𝑖∈𝐼} be a nonempty family of (∈,∈∨𝑞)-bifuzzy soft 𝐾-subalgebras of (𝑓,𝐴); then (a)⋂_𝑖∈𝐼(ℎ_𝑖,𝐵_𝑖) is an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra of (𝑓,𝐴);(b)⋀_𝑖∈𝐼(ℎ_𝑖,𝐵_𝑖) is an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra of ⋀_𝑖∈𝐼(𝑓,𝐴);(c)if 𝐵_𝑖∩𝐵_𝑗=∅ for all 𝑖,𝑗∈𝐼, then ⋁_𝑖∈𝐼(𝐻_𝑖,𝐵_𝑖) is an (∈,∈∨𝑞)- bifuzzy soft Lie subalgebra of ⋁_𝑖∈𝐼(𝑓,𝐴).

Theorem 36. Let (𝑓,𝐴) and (𝑔,𝐵) be two (∈,∈∨𝑞)-bifuzzy soft 𝐾-subalgebras over a Lie algebra 𝐿. Then (𝑓,𝐴)∩(𝑔,𝐵) is an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra over 𝐿.

Proof. By Definition 12, we can write (𝑓,𝐴)∩(𝑔,𝐵)=(ℎ,𝐶), where 𝐶=𝐴∪𝐵 and ℎ(𝛼)=𝑓(𝛼)𝑔(𝛼)𝑓(𝛼)(𝛼)if𝛼∈𝐴−𝐵,if𝛼∈𝐵−𝐶,∩𝑔if𝛼∈𝐴∩𝐵, (18) for all 𝛼∈𝐶.
Now for any 𝛼∈𝐶, we consider the following cases.
Case 1. Consider (𝛼∈𝐴−𝐵). Then ℎ(𝛼)=𝑓(𝛼) is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 since (𝑓,𝐴) is an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra over 𝐿.
Case 2. Consider (𝛼∈𝐵−𝐴). Then ℎ(𝛼)=𝑔(𝛼) is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 since (𝑔,𝐵) is an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra over 𝐿.
Case 3. Consider (𝛼∈𝐴∩𝐵). Then ℎ(𝛼)=𝑓(𝛼)∩𝑔(𝛼) is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿 by the assumption. Thus, in any case, ℎ(𝛼) is an (∈,∈∨𝑞)-bifuzzy Lie subalgebra of 𝐿. Therefore, (𝑓,𝐴)∩(𝑔,𝐵) is an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra over 𝐿.

Theorem 37. Let (𝑓,𝐴) and (𝑔,𝐵) be two (∈,∈∨𝑞)-bifuzzy soft 𝐾-subalgebras over a Lie algebra 𝐿. If 𝐴 and 𝐵 are disjoint, then (𝑓,𝐴)∪(𝑔,𝐵) is an (∈,∈∨𝑞)-bifuzzy softLie subalgebra over 𝐿.

Lemma 38. Let 𝐿 be a Lie algebra and (𝑓,𝐴) and (𝑔,𝐵) intuitionistic bifuzzy soft sets on 𝐺. If (𝑓,𝐴) and (𝑔,𝐵) are (∈,∈∨𝑞)-intuitionistic bifuzzy soft Lie subalgebra on 𝐿, then so are (𝑓,𝐴)∩(𝑔,𝐵) and (𝑓,𝐴)∩(𝑔,𝐵). Moreover, if (𝑓,𝐴) and (𝑔,𝐵) are an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra on 𝐿 and an (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra on 𝐿, then (𝑓,𝐴)⊙_ℎ(𝑔,𝐵)⊂(𝑓,𝐴)∩(𝑔,𝐵).

Lemma 39. Let 𝐿 be a Lie algebra and (𝑓,𝐴) and (𝑔,𝐵) bifuzzy soft sets on 𝐿. If (𝑓,𝐴) and (𝑔,𝐵) are (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra on 𝐿, then so are (𝑓,𝐴)∪(𝑔,𝐵) and (𝑓,𝐴)∪(𝑔,𝐵).

Denote by 𝕊𝕀(𝐺,𝐸) the set of all (∈,∈∨𝑞)-bifuzzy soft Lie subalgebras on 𝐿.

Theorem 40. (𝔽𝕊𝕀(𝐺,𝐸),∪,∩) is a complete distributive lattice under the ordering relation ⊂.

Proof. For any (𝑓,𝐴),(𝑔,𝐴)∈𝔽𝕊𝕀(𝐺,𝐸), by above Lemmas, (𝑓,𝐴)∪(𝑔,𝐴)∈𝔽𝕊𝕀(𝐺,𝐸) and (𝑓,𝐴)∩(𝑔,𝐴)∈𝔽𝕊𝕀(𝐺,𝐸). It is obvious that (𝑓,𝐴)∪(𝑔,𝐴) and (𝑓,𝐴)∩(𝑔,𝐴) are the least upper bound and the greatest lower bound of (𝑓,𝐴) and (𝑔,𝐵), respectively. There is no difficulty in replacing {(𝑓,𝐴),(𝑔,𝐴)} with an arbitrary family of 𝔽𝕊𝕀(𝐺,𝐸) and so (𝔽𝕊𝕀(𝐺,𝐸),∪,∩) is a complete lattice. Now we prove that the following distributive law ∩∪()=∩∪∩()𝑓,𝐴𝑔,𝐴ℎ,𝐶𝑓,𝐴𝑔,𝐴𝑓,𝐴ℎ,𝐶 (19) holds for all (𝑓,𝐴),(𝑔,𝐴),(ℎ,𝐶)∈𝔽𝕊𝕀(𝐺,𝐸). Suppose that ∩∪()=((,∩∪∩()=(()∪(=((.𝑓,𝐴𝑔,𝐴ℎ,𝐶𝐼,𝐴∩𝐵∪𝐶))𝑓,𝐴𝑔,𝐵𝑓,𝐴ℎ,𝐶𝐽,𝐴∩𝐵𝐴∩𝐶))𝐽,𝐴∩𝐵∪𝐶)) (20) Now for any 𝜀∈𝐴∩(𝐵∪𝐶), it follows that 𝜀∈𝐴 and 𝜀∈𝐵∪𝐶. We consider the following cases.
Case 1. Consider (𝜀∈𝐴, 𝜀∉𝐵 and 𝜀∈𝐶). Then 𝐼(𝜀)=𝑓(𝜀)∩ℎ(𝜀)=𝐽(𝜀).
Case 2. Consider (𝜀∈𝐴, 𝜀∈𝐵 and 𝜀∉𝐶). Then 𝐼(𝜀)=𝑓(𝜀)∩𝐺(𝜀)=𝐽(𝜀).
Case 3. Consider (𝜀∈𝐴, 𝜀∈𝐵 and 𝜀∈𝐶). Then 𝐼(𝜀)=𝑓(𝜀)∩(𝑔(𝜀)∪ℎ(𝜀))=(𝑓(𝜀)∩𝑔(𝜀))∪ (𝑓(𝜀)∩ℎ(𝜀))=𝐽(𝜀).
Therefore, 𝐼 and 𝐽 are the same operators, and so (𝑓,𝐴)∩((𝑔,𝐴)∪(ℎ,𝐶))= ((𝑓,𝐴)∩(𝑔,𝐵))∪((𝑓,𝐴)∩(ℎ,𝐶)). It follows that (𝑓,𝐴)∩((𝑔,𝐴)∪(ℎ,𝐶))= ((𝑓,𝐴)∩(𝑔,𝐵))∪((𝑓,𝐴)∩(ℎ,𝐶)). This completes the proof.

Definition 41. The product of two bifuzzy soft sets (𝑓,𝐴) and (𝑔,𝐴) over a Lie algebra is a bifuzzy soft set over 𝐺, denoted by (𝑓∘𝑔,𝐶), where 𝐶=𝐴∪𝐵 and 𝜀)=𝑓(𝜀)𝑔(𝜀)𝑓(𝜀)(𝜀)𝑓∘𝑔(if𝜀∈𝐴−𝐵,if𝜀∈𝐵−𝐴,∘𝐺if𝜀∈𝐴∩𝐵, (21) for all 𝜀∈𝐶. This is denoted by (𝑓∘𝑔,𝐶)=(𝑓,𝐴)⊙(𝑔,𝐴).

The following results can be easily deduced.

Lemma 42. Let (𝑓₁,𝐴), (𝑓₂,𝐴), (𝑔₁,𝐵), and (𝑔₂,𝐵) be bifuzzy soft sets over a Lie algebra 𝐿 such that (𝑓₁,𝐴)⊂(𝑓₂,𝐴) and (𝑔₁,𝐵)⊂(𝑔₂,𝐵). Then (a)(𝑓₁,𝐴)⊙(𝑔₁,𝐵)⊂(𝑓₂,𝐴)⊙(𝑔₂,𝐵),(b)(𝑓₁,𝐴)∩(𝑔₁,𝐵)⊂(𝑓₂,𝐴)∩(𝑔₂,𝐵) and (𝑓₁,𝐴)∩(𝑔₁,𝐵)⊂(𝑓₂,𝐴)∩(𝑔₂,𝐵),(c)(𝑓₁,𝐴)∪(𝑔₁,𝐵)⊂(𝑓₂,𝐴)∪(𝑔₂,𝐵) and (𝑓₁,𝐴)∪(𝑔₁,𝐵)⊂(𝑓₂,𝐴)∪(𝑔₂,𝐵).

Lemma 43. Let (𝑓,𝐴), (𝑔,𝐴), and (ℎ,𝐶) be bifuzzy soft sets over a Lie algebra 𝐿. Then (𝐹,𝐴)⊙((𝑔,𝐴)⊙(ℎ,𝐶))=((𝑓,𝐴)⊙(𝑔,𝐵))⊙(ℎ,𝐶).

Now we consider the bifuzzy soft sets over a definite parameter set. Let 𝐴⊆𝐸, 𝐿 be a Lie algebra and 𝔽𝕊_𝐴(𝐺)=𝑓,𝐴∈𝔽𝕊𝕀(𝐺,𝐸)∣𝐹∶𝐴⟶𝔽(𝑆) (22) the set of bifuzzy soft sets over 𝐿 and the parameter set 𝐴. It is trivial to verify that (𝑓,𝐴)∪(𝐺,𝐴),(𝑓,𝐴)∩(𝑔,𝐴),(𝑓,𝐴)∪(𝑔,𝐴),(𝑓,𝐴)∩(𝑔,𝐴)∈𝔽𝕊_𝐴(𝐺) for all (𝑓,𝐴),(𝑔,𝐴)∈𝔽𝕊_𝐴(𝐺).

Theorem 44. Let (𝑓,𝐴) and (𝑔,𝐴) be (∈,∈∨𝑞)-bifuzzy soft Lie subalgebra over a Lie algebra 𝐿. Then so is (𝑓,𝐴)⊙(𝑔,𝐴).

Theorem 45. Let 𝐿 be a Lie algebra with an identity 𝑒. Then (𝔽𝕊𝕀(𝐺,𝐸),⊙,∩) is a complete lattice under the relation ⊂.

5. Conclusions

Presently, science and technology are featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty. A large number of these models are based on an extension of the ordinary set theory such as bifuzzy sets and soft sets. In the current paper, we have presented the basic properties on bifuzzy soft Lie subalgebras. The most of these properties can be simply extended to bifuzzy soft Lie ideals. A Lie algebra is known algebraic structure and there are still many unsolved problems in it. In our opinion the future study of Lie algebras can be extended with the study of (i) roughness in Lie algebras and (ii) fuzzy rough Lie algebras.

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 1433/363/124. The authors, therefore, acknowledge with thanks DSR technical and financial support.