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^{1}Department of Mathematics, St. Thomas College, Thrissur, Kerala, India, **E-mail:** [email protected],

^{2}Department of Mathematics, St. Joseph's College, Irinjalakuda, Kerala, India, **E-mail:** [email protected]

**Received** July 14, 2009 **Revised** September 15, 2009

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

The concept of (; )-fuzzy Lie algebras over an (; )-fuzzy eld is introduced. We provide characterizations of an (2;2 _q)-fuzzy Lie algebra over an (2;2 _q)-fuzzy eld.

Zadeh [12] formulated the notion of fuzzy sets and after that many scholars developed fuzzy
system of dierent algebraic structures. The idea of quasi-coincidence of a fuzzy point with a
fuzzy set, which is mentioned in [10], has played a vital role in generating some dierent types
of fuzzy subgroups. Using the belong-to relation (∈) and quasi-coincidence with relation (q)
between fuzzy points and fuzzy sets, the concept of (*α, β*)-fuzzy subgroup was introduced
by Bhakat and Das [4]. Akram [1] introduced (*α, β*)-fuzzy Lie subalgebras and investigated
some of its properties. Nanda [9] introduced fuzzy algebra over fuzzy field. It is natural
to investigate similar types of generalization of the existing fuzzy subsystem. In [3], we
introduced fuzzy Lie algebra over a fuzzy field and some properties were discussed.

In this paper, we introduce the concept of (*α, β*)-fuzzy Lie algebra over an (*α, β*)-fuzzy
eld and investigate some of its properties.

In this section, we present some definitions needed for our study. We denote a complete distributive lattice with the smallest element 0 and the largest element 1 by I. By a fuzzy subset of a nonempty set X, we mean a function from X to I.

**Definition 2.1** (see [5]). Let X be a field and let F be a fuzzy subset of X. Then F is called
a fuzzy field of X if

**Remark 2.2.** It is seen that if F is a fuzzy field of X, then

**Definition 2.3.** Let A be a fuzzy subset of a Lie algebra L. Then A is called a *fuzzy Lie
algebra of L over a fuzzy field F*, if for all

Let L be a Lie algebra over a field *X*, let A : *L* → [0; 1] be a fuzzy set on L, and let
F : *X* → [0; 1] be a fuzzy set on *X*. The support of fuzzy set A is the crisp set that contains
all elements of L that have nonzero membership grades in A.

**Definition 3.1** (see [10]). A fuzzy set *A* : *L* → [0; 1] of the form

is said to be a fuzzy point with support *x* and value *t* and is denoted by *x _{t}*.

For a fuzzy point *x _{t}* and a fuzzy set

A fuzzy point *x _{t}* is said to

For a fuzzy set we denote

The following notations are used in this paper.

1. means that either belong to or *quasi-coincident with*,

2. means that α does not hold.

Let min be denoted by m(t; s) and let maxft; sg be denoted by M(t; s). Take I = [0; 1] and ^ = min, V = max with respect to the usual order in Definitions 2.1 and 2.3.

**Lemma 3.2.** *A fuzzy subset F of a field X is a fuzzy field if and only if it satisfies the
following conditions*:

**Lemma 3.3.** *Let L be a Lie algebra over a field X. Then a fuzzy subset A of Lie algebra
L is a fuzzy Lie algebra over a fuzzy field F of X if and only if it satisfies the following
conditions*:

Let α and β denote any one of unless otherwise specified.

**Definition 4.1.** Let X be a field and let be a fuzzy subset of X. Then *F* is
called an (*α, β*)-fuzzy field of X, if it satisfies the following conditions:

**Definition 4.2.** Let L be a Lie algebra over a field X, and let be an fuzzy field of X. Then a fuzzy subset is called an (*α, β*)-fuzzy Lie algebra of L
over an (*α, β*)fi)-fuzzy field F of X, if it satisfies the following conditions:

**Example 4.3.** In the real vector space where is cross product
of vectors for all Then is a Lie algebra over the field

Define

and define

(i) Then by actual computation, it follows that F is an -fuzzy field of and *A* is
an -fuzzy Lie algebra of over the -fuzzy field *F* of . Also it can be verified
that *A* is an -fuzzy Lie algebra of over an -fuzzy field *F* of .

(ii) Let *x* = (1; 0; 0), *y* = (2; 0; 0), *t* = 0:4, *s* = 0:3. Then and m(t, s) = 0.3. Hence *A* is not an -fuzzy Lie algebra.

(iii) Let x = (0; 0; 0), y = (2; 0; 0) be elements in and *t* = 0:4, *s* = 0:6. Then and This shows that Hence A is
not a (*q, q*)-fuzzy Lie algebra.

**Theorem 4.4.** *Let X be a field. Then a fuzzy subset is a fuzzy field if and
only if F is an -fuzzy field of X.*

**Proof. **The result follows immediately from Lemma 3.2.

**Theorem 4.5.** *Let L be a Lie algebra over a field X. Then a fuzzy subset A of L is a fuzzy
Lie algebra over a fuzzy field F of X if and only if A is an -fuzzy Lie algebra of L over
an -fuzzy field F of X*.

**Proof.** The result follows immediately from Lemmas 3.2 and 3.3.

**Theorem 4.6.** *Let X be a field and let be a fuzzy subset of X. Then F is an-fuzzy field of X if and only if*

**Proof.** Suppose that *F* is an -fuzzy field of X. It is clear that

We consider two possibilities.

*Case* 1. Let Then, If possible, let Let be such that and so Also shows that shows that Therefore, a contradiction.

*Case* 2. Let Then, If possible, let a contradiction. Therefore, it follows that Similarly, (ii) is proved.

Conversely, suppose that conditions (i) and (ii) are satisfied by a fuzzy set *F* of *X*.
Let and so Since *F* satisfies condition (i),

Now consider the possibilities then then and hence Therefore, it follows that if Similarly, if Hence *F* is an -fuzzy field of X.

**Theorem 4.7.** Let *L* be a Lie algebra over a field *X*. Then a fuzzy subset *A* of *L* is an -fuzzy Lie algebra of L over an -fuzzy field *F* of *X* if and only if

**Proof.** Suppose that *A* is an -fuzzy Lie algebra over an -fuzzy field *F* of *X*. It is clear that We consider two possibilities.

*Case* 1. Let If possible,
let be such that But This shows that a contradiction.

*Case* 2. Let If possible, let Then
we have a contradiction. Therefore, it follows that Thus, (ii) is proved. Similarly, (i) and (iii) are proved.

Conversely, suppose that the conditions (i), (ii), and (iii) are satisfied by a fuzzy set *A* of *L*. Let and so Since *A* satisfies condition (iii),

Now consider the possibilities then, then, and
hence Therefore, it follows that if Similarly, if Hence, *A* is an -fuzzy Lie algebra of L over an -fuzzy field *F* of *X*.

**Proposition 4.8.** *Let L be a Lie algebra over a field X. Then every -fuzzy Lie algebra of L over an -fuzzy field of X is an -fuzzy Lie algebra of L over an -fuzzy field of X.*

**Proof.** Suppose *A* is an -fuzzy Lie algebra of *L* over an -fuzzy field *F* of *X*. Let Since *F* is an -fuzzy field of then shows thatSimilarly, for all in *X*. So *F* is an -fuzzy field of *X*. Since *A* is an -fuzzy Lie algebra, for Then by definition Similarly, andHence *A* is an -fuzzy Lie algebra of L over an -fuzzy field F of *X*.

**Remark 4.9.** The converse of this proposition may not be true as seen in the following
example.

**Example 4.10.** Let is cross product for all Then *L* is a Lie algebra over the field

and define

Then by Theorem 4.7, it follows that *A* is an -fuzzy Lie algebra of over an -fuzzy field F of .

But this is not an -fuzzy Lie algebra of over an -fuzzy field of . Let But It is clear that and so Therefore *A* is not an -fuzzy Lie algebra of over an -fuzzy
eld F of .

**Theorem 4.11.** *Let A be an -fuzzy Lie algebra of L over an -fuzzy field F of X such that for all x ∈ L and for all λ ∈ X. Then A is an - fuzzy Lie algebra of L over an -fuzzy field F of X*.

**Proof.** Suppose that *A* is an -fuzzy Lie algebra of *L* over an -fuzzy field F of X. Let be such that Then, and so It follows from Theorem 4.6 that Given that for all for all

Let and so From Theorem 4.7, and from the given condition we get and so, be such that Then By Theorem 4.7,

So Similarly, Therefore, *A* is an -fuzzy Lie algebra of *L* over an -fuzzy field F of *X*.

**Proposition 4.12. ***If A is an -fuzzy Lie algebra of L over an -fuzzy field F, then*

**Proof. **Let Then, from Theorem 4.7, the following hold.

**Theorem 4.13.** *Let A be an -fuzzy Lie algebra of L over an -fuzzy field F of X. Then, for t ∈ *(0; 0:5]*, At is a Lie subalgebra over F _{t} when F_{t} contains at least two
elements.*

**Proof.** For * t ∈ *(0; 0:5], suppose F_{t} contains at least two elements.

Let This shows that Therefore,

and hence Thus, Similarly, Therefore, *F _{t}* is a subfield of X.

Suppose So Then and so

Similarly, for Therefore, A_{t} is a Lie subalgebra over the field F_{t}.

Let be a function. If A and B are fuzzy subsets of L and L', respectively, then are defined using Zadeh's extension principle [6]. If α is one of it follows that if and only iffor all x ∈ L and for all t ∈ (0; 1].

**Theorem 4.14.** *Let L and L' be Lie algebras over a field X and let be a homo-
morphism. If B is an -fuzzy Lie algebra of L' over an -fuzzy field F of X,
then f ^{-1}(B) is an -fuzzy Lie algebra of L over the -fuzzy field F of X.*

**Proof.** Let be such thatThen Since *B* is an -fuzzy Lie algebra of L' over an fuzzy field *F* of *X*,

So we have

Let and so

and hence

Therefore, fuzzy Lie algebra of *L* over the -fuzzy field *F* of *X*.

**Definition 4.15.** A fuzzy set *μ* of a set Y is said to possess *sup property* if for every
nonempty subset S of Y , there exists such that

**Theorem 4.16.** *Let L and L' be Lie algebras over a field X and let be an onto
homomorphism. Let A be an -fuzzy Lie algebra of L over an -fuzzy field F of X, which satisfies the sup property. Then f(A) is an -fuzzy Lie algebra of L' over the -fuzzy field F of X.*

**Proof.** Let and and so

Since *f* is onto, are nonempty subsets of L and by the *sup property of A*,
there exists such that

then Since *A* is an -fuzzy Lie algebra of L over an -fuzzy field F of X, we haveNow Therefore,

and so Also

Let Then it follows that ButThis shows that

Therefore,* f*(A) is an -fuzzy Lie algebra of L' over the fuzzy field *F* of *X*.

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