eu COMPLETE REDUCIBILITY OF REPRESENTATIONS OF L-FUZZY GROUPS

This paper studies the complete reducibility of representations of L-fuzzy groups. We define the comparable family of L-fuzzy spaces, and prove the following theorem: Let T̃μ(V) be a completely reducible representation of an L-fuzzy group μ in a μ-space V, and {Wi}i∈I be a family of μ-subspaces of V. If the intersection ⋂ i∈I Wi is positive, then every Wi is completely reducible. In addition, we introduce a criterion to validate the complete reducibility of T̃μ(V) as follows: If the representation T : Suppμ → GL(SuppV) is completely reducible, then T̃μ(V) is completely reducible too. However, we prove that the inverse is not necessarily true, but it is satisfied when the set ⋃ i∈I Wi is comparable. Finally, as a result, we show that every representation of L-fuzzy group in any μ-space V over the field of complex numbers is always completely reducible. AMS Subject Classification: 20N25, 20F29


Introduction
In [9], Zadeh introduced the notion of fuzzy set of a nonempty set X as a function µ : X → [0, 1], and also defined the extension principles.Goguen in [1] generalized the construction of fuzzy sets by allowing the range of the extended characteristic functions to be a complete distributive lattice L.
Many researchers are engaged in extending the notions of algebra to the broader framework of the fuzzy setting.We are specifically interested in those which are related to the group theory and linear algebra.The notions of fuzzy groups and fuzzy spaces are carried out by Rosenfeld in [7], and Katsaras and Liu in [2] respectively.
The primary purpose of this paper is to study the properties of representations of L-fuzzy groups, especially, the complete reducibility of these representations.

Preliminaries
Throughout the paper, we assume (L, ≤, ∨, ∧) is a complete distributive lattice with an infinity I and a zero 0, G is a finite group, V is a vector space over a field F , and L X is a family of all L-fuzzy sets of a set X.If µ ∈ L X then we write Supp µ for the support set of µ, where Supp µ = {x ∈ X; µ(x) > 0}.When Supp µ = X then µ is said to be positive, and when µ(x) = 0 for every x ∈ X then µ is said to be an empty L-fuzzy set of X, and denoted by ∅.
In this section, we introduce definitions and elementary results of L-fuzzy groups and L-fuzzy spaces.Definition 2.1 [4] Let X, Y be nonempty sets.Through the extension principle of Zadeh, each function f : X → Y induces a corresponding function for every y ∈ Y .The function f is said to be obtained from f by the extension principle.
•) be a finite group.An L-fuzzy set µ ∈ L G is said to be an L-fuzzy group of G if µ satisfies the properties, for every x, y ∈ G: Theorem 2.4 [8] Let G, Ǵ be two finite groups, and f : G → Ǵ be a group homomorphism.If µ is an L-fuzzy group of G, then f µ is an L-fuzzy group of Ǵ, where f is the extended function of f by the extension principle.Definition 2.5 [2][3] Let V be a vector space.An L-fuzzy set V ∈ L V is said to be an L-fuzzy space of V if V satisfies the property, for every v, u ∈ V, α, β ∈ F : Definition 2.6 [3] Let V ∈ L V be an L-fuzzy space of the vector space V , then V is said to be trivial if V = 0 V(0) for every v ∈ V , where 0 V(0) is an L-fuzzy point defined as 0 for every v ∈ V , and they are said to be equal when Theorem 2.9 [3] Let {V i } i∈I be a family of L-fuzzy spaces of V , then Definition 2.11 Let τ be a subset of a lattice L. We say that τ is comparable set in L when for every pair (α, β) ∈ τ × τ , α and β are comparable in Remark 2.12 Note that if the lattice L is the unit interval, [0, 1], then every family {V i } i∈I of L-fuzzy spaces of a vector space V is a comparable family in L.
Theorem 2.13 Let {V i } i∈I be a family of an L-fuzzy spaces of a vector space V , and from (2.13.1),(2.13.2) we have

Representations of L-Fuzzy Groups
Throughout the paper, we shall consider the following definitions and the results about representations of L-fuzzy groups which we introduced all in our work [8].Definition 3.1 Let µ ∈ L G ba an L-fuzzy group, and be a linear representation of the group G in the vector space V .Through the extension principle, T is extended to the function: where We call the L-fuzzy group T µ of GL(V ), a representation of L-fuzzy group µ in V .Definition 3.2 An L-fuzzy space V ∈ L V is said to be a µ-space when there exists a representation T µ , so that for every g ∈ Supp µ, v ∈ Supp V Definition 3.3 Let T µ be a representation of the L-fuzzy group µ in the vector space V , and V be a µ-space.The representation of L-fuzzy group µ in V is an L-fuzzy group of GL(V ), denoted by T µ (V), which obtained by extending the representation T of the finite group Supp µ in the vector space Supp V, where T (g) = T g | Supp V for every g ∈ Supp µ.
That means for every f ∈ GL(Supp V) ;otherwise Theorem 3.4 Let V be a µ-space, then Supp V is a Supp µ-space, i.e for every v ∈ Supp V and g ∈ Supp µ, T g (v) ∈ Supp V. Theorem 3.5 If {V i } i∈I is any family of µ-spaces of V , then i∈I V i is a µ-space of V .Definition 3.6 Let T µ (V) be a representation of the L-fuzzy group µ in an L-fuzzy space V, then T µ (V) is said to be irreducible if the µ-space V has no non-trivial µ-space W where W ⊂ V. Otherwise the representation T µ (V) is reducible.Definition 3.7 Let T µ (V) be a representation of the L-fuzzy group µ in an L-fuzzy space V, then T µ (V) is said to be completely reducible, or equivalently V is said to be completely reducible, if there exists for every µ-space W where W ⊂ V, a µ-space U where U ⊂ V, so that W ⊕ U ⊆ V. Remark 3.8 It is easy to verify that every irreducible representation T µ (V) is completely reducible representation.
Throughout this paper, we shall consider the previous difintion of the representation of L-fuzzy groups in µ-spaces.

Completely Reducibile Representations of L-Fuzzy Groups
In this section, we present a study of the completely reducible representations of L-fuzzy groups, and introduce in the theorem 3.9 a criterion to validate the complete reducibility of these representations.Theorem 4.1 Let T µ (V) be a completely reducible representation of an Lfuzzy group µ in a µ-space V, and {W i } i∈I be a family of µ-subspaces of V.If the intersection i∈I W i is positive, then every µ-space W i is completely reducible.
Proof.Let T µ (V) be a completely reducible representation, and W i be a µ-subspace from the family {W i } i∈I .We distinguish two different cases: (i) There is no µ-subspace of V in the family {W i } i∈I included in W i , then W i is irreducible, hence it is completely reducible.
(ii) Let W j ∈ {W i } i∈I so that W j ⊂ W i , and we want to prove that there is a µ-subspace Since T µ (V) is completely reducible, and By the theorem 3.5 and since i∈I W i is positive, W i ∩ X ∈ L V is a µ-space, and satisfies W i ∩ X = ∅.
In addition, for every v ∈ V where Since the operation ∧ is associative, and We know, through the lattice theory, that for every α, β ∈ L, then α ∧ β ≤ α, α ∧ β ≤ β, hence for every v ∈ V we have On the other hand, for any µ-subspaces 2. Since the intersection of any µ-subspaces is positive, for every v ∈ Supp W i ∩ Supp W j means W i (v) > 0 and W j > 0, leads v ∈ Supp (W i ∩ W j ), hence From (4.1.1),(4.1.2),we have Supp W i ∩ Supp X = {0}, and then Supp(W i ∩ X) = {0}, that brings the fact Supp W j ∩ Supp(W i ∩ X) = {0}.So we conclued that the µ-subspace W k we look for is the intersection W i ∩ X, and the µ-subspace W i is completely reducible.Theorem 4.2 Let T µ (V) be a representation of an L-fuzzy group µ in a µ-space V.If the representation T : Supp µ → GL(Supp V) is completely reducible, then T µ (V) is completely reducible too.
Proof.(i) If the representation T µ (V) is irreducible, then by the remark 3.8 it is completely reducible.
(ii) Let T µ (V) be a reducible representation, then there exists a µ-space W ∈ L V such ∅ = W ⊂ V.By the theorem 3.4 Supp W is Supp µ-space and included in the vector space Supp V, and since the representation T is completely reducible, there is a Supp µ-space, U , such Since V is a µ-space, for every u ∈ Supp U and g ∈ Supp µ U (u) = V(u) = V(T g (u)) = U ((T g )(u)).Therefore the L-fuzzy set U is a µ-space.
On the other hand, from (4.2.1) we have Supp W ∩ U = {0}, and for every v ∈ Supp V (W ⊕ U )(v) = W(w) ∧ U (u) where w ∈ Supp W, u ∈ Supp U then Hence W ⊕ U ⊆ V. We show that for every non-trivial µ-space W ⊂ V there exists a µ-space U so that W ⊕ U .Which means T µ (V) is completely reducible representation.
Notice 4.3 The inverse of the previous criterion is not necessarily true, but it is satisfied when the set i∈I W i is comparable, where {W i } i∈I is a family of µ-subspaces of V. J (v) = V(v) ; v ∈ (Supp H) ⊥ 0 ;otherwise Likewise we showed in the proof of the previous theorem, the L-fuzzy set J is a µ-space such H ⊕ J ⊆ V, hence the representation T µ (V) where V is over the field of complex numbers, is completely reducible.