Southeast Asian Bulletin of Mathematics (2007) 31: 843–855
Southeast Asian Bulletin of Mathematics c SEAMS. 2007 °
Intuitionistic Fuzzy Lie Algebras M. Akram∗ Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore-54000, Pakistan E-mail:
[email protected]
K. P. Shum† Faculty of Science, The Chinese University of Hong Kong, Shatin, Hong Kong, P.R. China (SAR) E-mail:
[email protected]
AMS Mathematics Subject Classification (2000): 17B99, 03E72, 20N25 Abstract. We introduce the concept of intuitionistic fuzzy Lie subalgebras of a Lie algebra and investigate some of their properties. The Cartesian product of intuitionistic fuzzy Lie subalgebras will be discussed. In particular, the homomorphisms between the Lie subalgebras of a Lie algebra and their relationship between the domains and the co-domains of the fuzzy subalgebras under these homomorphisms will be investigated. Some recent results obtained by Akram [1, 2] will be extended and strengthened. Keywords: Lie algebras; Intuitionistic fuzzy sets; Congruence relation; Cartesian product; Lie homomorphism.
1. Introduction Lie algebras were first discovered by Sophus Lie (1842-1899) when he attempted to classify certain smooth subgroups of general linear groups. The groups he considered are now called Lie groups. By taking the tangent space at the identity element of such a group, he obtained the Lie algebra and hence the problems on groups can be reduced to problems on Lie algebras so that it becomes more tractable. There are many applications of Lie algebras in many branches of mathematics and physics. ∗ The
research work of the first author is supported by PUCIT. research of the second author is partially supported by a CUHK Mathematics research project grant #7103084(2006/2007). † The
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After introducing of the concept of fuzzy sets by Zadeh in 1965, there are many generalizations of this fundamental concept. Among these new concepts, the concept of of intuitionistic fuzzy sets given by Atanassov [3, 4] in 1983 is the most important and interesting one because it is simply an extension of fuzzy sets. The elements of the intuitionistic fuzzy 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. There have numerous applications in various areas of sciences, for instance, computer science, mathematics, medicine, chemistry, economics, astronomy etc. In this paper, we shall apply intuitionistic fuzzy sets to Lie algebras. The Cartesian product of intuitionistic fuzzy Lie subalgebras will be discussed. In particular, we will study the homomorphisms between the Lie subalgebras of a Lie algebra and the investigate the relationship between the domains and the co-domains of these fuzzy subalgebras under the above homomorphisms. Some recent results obtained by Akram [1, 2] will be extended and strengthened.
2. Preliminaries In this section, we first review some elementary aspects 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). In this paper, we will always use L to denote a Lie algebra. We note that the multiplication in a Lie algebra is not associative, i.e., it is not true in general that [[x, y], z] = [x, [y, z]]. But it is anti commutative, i.e., [x, y] = −[y, x]. We call a subspace H of L closed under [·, ·] a Lie subalgebra. A subspace I of L with the property [I, L] ⊆ I is called a Lie ideal of L. Obviously, any Lie ideal is a subalgebra. Let µ be a fuzzy set on L, i.e., a map µ : L → [0, 1]. Then, we call a fuzzy set µ : L → [0, 1] a fuzzy Lie subalgebra [13] of L if the following conditions are satisfied: (a) µ(x + y) ≥ min{µ(x), µ(y)}, (b) µ(αx) ≥ µ(x), (c) µ([x, y]) ≥ min{µ(x), µ(y)},
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for all x, y ∈ L and α ∈ F . A fuzzy subset µ : L → [0, 1] satisfying the above conditions (a), (b) and the following condition (d) µ([x, y]) ≥ µ(x) is called a fuzzy Lie ideal [1] of L. A complex mapping A = (µA , λA ) : L → [0, 1]× [0, 1] is called an intuitionistic fuzzy set (IF set, in short) in L if µA (x) + λA (x) ≤ 1, for all x ∈ L, where the mappings µA : L → [0, 1] and λA : L → [0, 1] denote the degree of membership (namely µA (x)) and the degree of non-membership (namely λA (x)) of each element x ∈ L to A respectively. In particular, we use 0∼ and 1∼ to denote the intuitionistic fuzzy empty set and the intuitionistic fuzzy whole set in a set L such that 0∼ (x) = (0, 1) and 1v (x) = (1, 0), for each x ∈ L respectively.
3. Intuitionistic Fuzzy Lie Algebras
Definition 3.1. An intuitionistic fuzzy set A = (µA , λA ) on L is called an intuitionistic fuzzy Lie subalgebra if the following conditions are satisfied: (1) µA (x + y) ≥ min(µA (x), µA (y)) and λA (x + y) ≤ max(λA (x), λA (y)), (2) µA (αx) ≥ µA (x) and λA (αx) ≤ λA (x), (3) µA ([x, y]) ≥ min{µA (x), µA (y)} and λA ([x, y]) ≤ max{λA (x), λA (y)} for all x, y ∈ L and α ∈ F .
Definition 3.2. An intuitionistic fuzzy set A = (µA , λA ) on L is called an intuitionistic fuzzy Lie ideal if it satisfies the conditions (1), (2) and the following additional condition: (4) µA ([x, y]) ≥ µA (x) and λA ([x, y]) ≤ λA (x) for all x, y ∈ L. From (2). it follows that: (5) µA (0) ≥ µA (x), (6) µA (−x) ≥ µA (x),
λA (0) ≤ λA (x), λA (−x) ≤ λA (x).
Example 3.3. Let t. Since f is surjective, there exists y ∈ L1 such that f (y) = x and y = i=1 [ai bi ] for some ai ∈ f −1 (yi ), bi ∈ f −1 (zi ) with f (ai ) = yi , f (bi ) = zi , A(ai ) > t and B(bi ) > t. Since n n n n X X X X f ( [ai bi ]) = [ f ([ai bi ]) = [ [f (ai )f (bi )] = [ [yi zi ] = x, i=1
i=1
i=1
i=1
f (¿ AB À)(x) > t. This is a contradiction. Similarly, for the case f (¿ AB À) > ¿ f (A)f (B) À, we can also obtain a contradiction. Hence, f (¿ AB À) = ¿ f (A)f (B) À.
Definition 4.5. Let A = (µA , λA ) and B = (µB , λB ) be intuitionistic fuzzy subalgebras of L. Then A is said to be of the same type of B if there exists f ∈ Aut(L) such that A = B ◦ f , i.e., µA (x) = µB (f (x)), λA (x) = λB (f (x)) for all x ∈ L.
Theorem 4.6. Let A = (µA , λA ) and B = (µB , λB ) be two intuitionistic fuzzy subalgebras of L. Then A is an intuitionistic fuzzy subalgebra having the same type of B if and only if A is isomorphic to B. Proof. We only need to prove the necessity part because the sufficiency part is trivial. Let A = (µA , λA ) be an intuitionistic fuzzy subalgebra having the same type of B = (µB , λB ). Then there exists φ ∈ Aut(L) such that µA (x) = µB (φ(x)), λA (x) = λB (φ(x)) ∀x ∈ L. Let f : A(L) → B(L) be a mapping defined by f (A(x)) = B(φ(x)) for all x ∈ L, that is, f (µA (x)) = µB (φ(x)), f (λA (x)) = λB (φ(x)) ∀x ∈ L. Then, it is clear that f is surjective. Also, f is injective because if f (µA (x)) = f (µA (y)) for all x, y ∈ L, then µB (φ(x)) = µB (φ(y)) and hence µA (x) = µB (y). Likewisely, we have f (λA (x)) = f (λA (y)) =⇒ λA (x) = λB (y) for all x ∈ L. Finally, f is a homomorphism because for x, y ∈ L, f (µA (x + y)) = µB (φ(x + y)) = µB (φ(x) + φ(y)), f (λA (x + y)) = λB (φ(x + y)) = λB (φ(x) + φ(y)),
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f (µA (αx)) = µB (φ(αx)) = αµB (φ(x)), f (λA (αx)) = λB (φ(αx)) = αλB (φ(x)), f (µA ([x, y])) = µB (φ([x, y])) = µB ([φ(x), φ(y)]), f (λA ([x, y])) = λB (φ([x, y])) = λB ([φ(x), φ(y)]). Hence A = (µA , λA ) is isomorphic to B = (µB , λB ). This completes the proof.
Acknowledgement. The second author wants to dedicate this paper in memory of the late Professor Wong Yung Chow, Hong Kong University, Hong Kong.
References [1] M. Akram : Anti fuzzy Lie ideals of Lie algebras, Quasigroups and Related Systems 14, 123-132 (2006). [2] M. Akram : Intuitionistic fuzzy Lie ideals of Lie algebras, Int. Journal of Fuzzy Math. (To appear). [3] K. T. Atanassov : Intuitionistic fuzzy sets, VII ITKRs Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84), (in Bulgarian)(1983). [4] K. T. Atanassov : Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20, 87-96 (1986). [5] P. Coelho and U. Nunes : Lie algebra application to mobile robot control: a tutorial, Robotica 21, 483-493 (2003). [6] B. Davvaz : Fuzzy Lie algebras, Intern. J. Appl. Math. 6, 449-461 (2001). [7] W. A. Dudek : Fuzzy subquasigroups, Quasigroups and Related Systems 5, 81-98 (1998). [8] C. G. Kim and D.S. Lee : Fuzzy Lie ideals and fuzzy Lie subalgebras, Fuzzy Sets and Systems 94, 101-107 (1998). [9] J. E. Humphreys : Introduction to Lie Algebras and Representation Theory, Springer, New York 1972. [10] Q. Keyun, Q. Quanxi and C. Chaoping : Some properties of fuzzy Lie algebras, J. Fuzzy Math. 9, 985-989 (2001). [11] A. K. Katsaras and D. B. Liu : Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58, 135-146 (1977). [12] A. Rosenfeld : Fuzzy groups, J. Math. Anal. Appl. 35, 512-517 (1971). [13] S. E. Yehia : Fuzzy ideals and fuzzy subalgebras of Lie algebras, Fuzzy Sets and Systems 80, 237-244 (1996). [14] L. A. Zadeh : The concept of a lingusistic variable and its application to approximate reasoning, Part 1, Information Sci. 8 (1975), 199-249. [15] L. A. Zadeh : Fuzzy sets, Information and Control 8, 338-353 (1965).
