The concept of fuzzy homomorphisms of groups in the setting of the most gener- alized form ... abstract algebra, usually called fuzzy algebra, thereby sprang up.
Weak and Strong Fuzzy Homomorphisms of Groups B.K. Sarmaa and Tazid Alib,∗ a
Department of Mathematics, Indian Institute of Technology Guwahati, 781039, Assam, India b
Department of Mathematics, Dibrugarh University, Dibrugarh, Assam, India
Abstract The concept of fuzzy homomorphisms of groups in the setting of the most generalized form of fuzzy mappings is introduced and the invariance of fuzzy groups under fuzzy homomorphisms is established. Keywords: Fuzzy relation, Fuzzy subgroup, Fuzzifying mapping, Fuzzy homomorphism.
1
Introduction
The notion of a fuzzy set was initiated by Zadeh in [7]. Rosenfeld introduced the concept of fuzzy groups on the structure of fuzzy sets in his classical paper [5] in 1971. A new branch of abstract algebra, usually called fuzzy algebra, thereby sprang up. Rosenfeld’s fuzzy groups caught the imagination of the algebraist like wildfire and there seems to be no end to its ramifications. A fuzzifying function f from X into Y is an ordinary mapping from X into I Y , the class of all fuzzy subsets of Y [2]. This definition fuzzifies the notion of a mapping to the greatest extent and the concept of a fuzzifying function is mathematically equivalent to the notion of a fuzzy relation. Some discussion on fuzzy homomorphism can be seen in [1] and [3]. In this paper, we study the concept of fuzzy homomorphism of groups with fuzzifying functions as fuzzy mappings and discuss their actions on fuzzy groups. ∗ Corresponding
author
1
2
Preliminaries
According to Zadeh [7], a mapping from a nonempty set X into [0, 1] is a fuzzy set of X. If λ and µ are fuzzy subsets of X, the equality λ = µ and the inclusion λ ⊆ µ are defined pointwise. By I X we mean the class of all fuzzy subsets of X. Definition 2.1 Let f be a mapping from a set X into a set Y , and let µ be a fuzzy subset of X and ν be a fuzzy subset of Y . Then the fuzzy subset f (µ) of Y and the fuzzy subset f −1 (ν) of X are defined as follows: ( sup{µ(x) : x ∈ X, f (x) = y} if f −1 (y) 6= ∅, f (µ)(y) = 0 otherwise, y ∈ Y, and f −1 (ν)(x) = ν(f (x)), x ∈ X. The fuzzy subset f (µ) of Y is called the image of µ under f and the fuzzy subset f −1 (ν) of X is called the pre-image (or the inverse image ) of ν under f . Definition 2.2 [6] A fuzzy relation µ on X × Y is a fuzzy subset of X × Y . The inverse of a fuzzy relation µ on X × Y is the fuzzy relation on Y × X, denoted by µ−1 , defined as follows: µ−1 (y, x) = µ(x, y), (y, x) ∈ Y × X. According to Rosenfeld [5], a fuzzy subset λ of a group G is a fuzzy subgroup of G if for all a, b ∈ G, min(λ(a), λ(b)) ≤ λ(ab) and λ(a−1 ) = λ(a). In that case, λ(x) ≤ λ(e) for all x ∈ G, where e is the identity element of G. A fuzzy subgroup is normal, if λ(ab) = λ(ba) for all a, b ∈ G. It is well known that images and preimages of fuzzy subgroups under group homomorphisms are fuzzy subgroups. Moreover, the preimage of a normal fuzzy subgroup under a surjective group homomorphism is a normal fuzzy subgroup. Definition 2.3 A fuzzifying function f from X into Y is an ordinary mapping from X into I Y , the class of all fuzzy subsets of Y . Fuzzifying functions have been discussed by Dubois and Prade [2]. The concept of a fuzzifying function and that of a fuzzy relation are mathematically equivalent; a fuzzifying function f of X into Y is associated with a fuzzy relation µf on X ’ Y defined by µf (x, y) = f (x)(y), x ∈ X, y ∈ Y . An ordinary mapping f1 of X into Y always induces a fuzzifying function f for which the associated fuzzy relation µf on X × Y is defined as follows: ( 1, if y = f1 (x), µf (x, y) = 0, otherwise, (x, y) ∈ X × Y. 2
Definition 2.4 Let f be a fuzzifying function of X into Y . The inverse of f , denoted by f −1 , is the fuzzifying function of Y into X for which the associated fuzzy relation is the inverse of µf , that is, µ−1 f (y, x) = µf (x, y), x ∈ X, y ∈ Y . Definition 2.5 [4] Let f be a fuzzifying function of X into Y . The domain Df and the range Rf of f are fuzzy subsets of X and Y , respectively, which are defined as follows: Df (x) = sup{µf (x, y) : y ∈ Y }, x ∈ X, and Rf (y) = sup{µf (x, y) : x ∈ X}, y ∈ Y. We note that Df−1 = Rf and Rf−1 = Df . Definition 2.6 [2] Let f be a fuzzifying function of X into Y , and g be a fuzzifying function of Y into Z. The composition of f and g, written as g ◦ f , is a fuzzy subset of X × Z defined as follows: µg◦f f (x, z) = sup{min(µf (x, y), µg (y, z))} : y ∈ Y }, (x, z) ∈ X × Z. We propose to introduce the following definitions. Definition 2.7 Let f be a fuzzifying function of X into Y . The image of a fuzzy subset λ of X under f is the fuzzy subset of Y , denoted by f (λ), and defined by f (λ)(y) = sup{min(λ(x), µf (x, y)) : x ∈ X}, y ∈ Y. The preimage of a fuzzy subset ν of Y under f is the fuzzy subset of X, denoted by f −1 (ν), and defined by f −1 (ν)(x) = sup{min(µf (x, y), ν(y)) : y ∈ Y }, x ∈ X. We note that the preimge of ν under f coincide with the image of ν under the inverse fuzzifying function f −1 of f . For any λ ∈ I X and νI Y , we have f (λ) ⊆ Rf and f −1 (ν) ⊆ Df . One can easily see that if f is an ordinary mapping f1 of X into Y , then f (λ) (resp. f −1 (ν)) coincides with f1 (λ) (resp. f1−1 (ν)), according to Definition 2.1.
3
Fuzzy homomorphisms of groups
Definition 3.1 A fuzzifying function f of G1 into G2 is called a weak fuzzy homomorphism if the following two conditions are satisfied: 3
(3.1) ∀x1 , x2 ∈ G1 and ∀y1 , y2 ∈ G2 , µf (x1 x2 , y1 y2 ) ≥ min(µf (x1 , y1 ), µf (x2 , y2 )); (3.2) ∀x ∈ G1 and ∀y ∈ G2 , µf (x−1 , y −1 ) = µf (x, y). Definition 3.2 A fuzzifying function f of G1 into G2 is called a strong fuzzy homomorphism if the following two conditions are satisfied: (3.3) ∀x1 , x2 ∈ G1 and ∀y1 , y2 ∈ G2 , µf (x1 x2 , y) = sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : y1 , y2 ∈ G2 , y = y1 y2 }; (3.4) ∀x ∈ G1 and ∀y ∈ G2 , µf (x−1 , y −1 ) = µf (x, y). It follows from the definitions that a strong fuzzy homomorphism is also a weak fuzzy homomorphism. However a weak fuzzy homomorphism may not be a strong fuzzy homomorphism as can be seen from Example 3.7. It can also be seen that a fuzzifying function f of G1 into G2 is a weak fuzzy homomorphism if and only if f −1 is a weak fuzzy homomorphism of G2 into G1 . The following theorem shows that Definition 3.1 generalizes the notion of an ordinary group homomorphism. Theorem 3.3 Let f1 be a group homomorphism of G1 into G2 . Then the induced fuzzifying mapping is a strong fuzzy homomorphism of G1 into G2 . Proof. Let f1 be a group homomorphism of G1 into G2 . Let f be the induced fuzzy homomorphism of G1 into G2 and µf the fuzzy relation associated to f . Let x1 , x2 ∈ G1 and y ∈ G2 . If µf (x1 x2 , y) = 1, then y = f1 (x1 x2 ) = f1 (x1 )f1 (x2 ) and min{µf (x1 , f1 (x1 )), µf (x2 , f1 (x2 ))} = 1. On the other hand, if µf (x1 x2 , y) = 0, then f1 (x1 x2 ) 6= y. For any decomposition y = y1 y2 of y in G2 , we have either f1 (x1 ) 6= y1 or f1 (x2 ) 6= y2 , and therefore min{µf (x1 , f1 (x1 )), µf (x2 , f1 (x2 ))} = 0. Consequently, for all x1 , x2 ∈ G1 and y ∈ G2 , we have µf (x1 x2 , y) = sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : y = y1 y2 }. Moreover, for x ∈ G1 and y ∈ G2 , we have µf (x−1 , y −1 ) = µf (x, y) because f1 (x) = y if and only if f1 (x−1 ) = y −1 . Thus f is a strong fuzzy homomorphism of G1 into G2 . This completes the proof. Theorem 3.4 Let f1 : G1 → G2 be an ordinary mapping. If the induced fuzzifying mapping f is a weak fuzzy homomorphism from G1 into G2 , then f1 is a group homomorphism. Proof. Let f1 be an ordinary mapping of G1 into G2 , and suppose that the induced fuzzifying mapping f is a weak fuzzy homomorphism. For x1 , x2 ∈ G1 , we have 1 ≥ µf (x1 x2 , f1 (x1 )f1 (x2 )) ≥ min{µf (x1 , f1 (x1 )), µf (x2 , f1 (x2 ))} = 1, and, consequently, f1 (x1 x2 ) = f1 (x1 )f1 (x2 ). Hence, f1 is an ordinary homomorphism. 4
Theorem 3.5 The domain and the range of a weak fuzzy homomorphism of G1 into G2 are fuzzy subgroups of G1 and G2 , respectively. Proof. Let f be a weak fuzzy homomorphism of G1 into G2 with domain Df and range Rf . For x1 , x2 ∈ G1 , we have Df (x1 x2 )
=
sup{µf (x1 x2 , y) : y ∈ G2 }
= sup{µf (x1 x2 , y1 y2 ) : y = y1 y2 , y1 , y2 ∈ G2 } ≥ sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : y = y1 y2 , y1 , y2 ∈ G2 } = min(sup{µf (x1 , y1 ) : y1 ∈ G1 }, sup{µf (x2 , y2 ) : y2 ∈ G2 }) = min{Df (x1 ), Df (x2 )}. Moreover, for x ∈ G1 , we have Df (x−1 )
= sup{µf (x−1 , y) : y ∈ G2 } =
sup{µf (x−1 , y − 1) : y −1 ∈ G2 }
=
sup{µf (x, y) : y ∈ G2 }
=
Df (x).
Thus Df is a fuzzy subgroup of G1 . Similarly, Rf is a fuzzy subgroup of G2 , since it is the domain of the weak fuzzy homomorphism f −1 of G2 into G1 . Theorem 3.6 Let δ and ρ be fuzzy subgroups of G1 and G2 , respectively. Then the fuzzy mapping f of G1 into G2 defined by µf (x, y) = min(δ(x), ρ(y)), x ∈ G1 , y ∈ G2 , is a weak fuzzy homomorphism of G1 into G2 . Moreover, if δ(e1 ) = ρ(e2 ), then Df = δ and Rf = ρ. Proof. For x1 , x2 ∈ G1 and y1 , y2 ∈ G2 , we have µf (x1 x2 , y1 y2 ) =
min(δ(x1 x2 ), ρ(y1 y2 ))
≥ min{min(δ(x1 ), δ(x2 )), min(ρ(y1 ), ρ(y2 ))} = min{min(δ(x1 ), ρ(y1 )), min(δ(x2 ), ρ(y2 ))} = min(µf (x1 , y1 ), µf (x2 , y2 )). Moreover, for x ∈ G1 , y ∈ G2 we have µf (x−1 , y −1 ) = min(δ(x−1 ), ρ(y −1 )) = min(δ(x), ρ(y)) = µf (x, y). 5
Hence, f is a weak fuzzy homomorphism of G1 into G2 . Next, let δ(e1 ) = ρ(e2 ). For x ∈ G1 , we have Df (x) =
sup{µf (x, y) : y ∈ G2 }
= sup{min(δ(x), ρ(y)) : y ∈ G2 } = min(δ(x), ρ(e2 )) = min(δ(x), δ(e1 )) =
δ(x),
yielding Df = δ. Similarly, Rf = ρ. Example 3.7 Let Z be the cyclic group of integers under addition. For n ≥ 0, let Hn = < 2n > and Kn = < 3n >. Define fuzzy subsets δ and ρ as follows : δ(Hn−1 \ Hn ) = (n − 1)/n, n ≥ 1,
δ(∩Hn = {0}) = 1,
ρ(Kn−1 \ Kn ) = (n − 1)/n, n ≥ 1,
ρ(∩Kn = {0}) = 1,
where by A\B we mean the usual set difference. Then δ and ρ are fuzzy subgroups of Z. For x, y ∈ Z, let µf (x, y) = min{δ(x), ρ(y)}. By Theorem 3.6, f is a weak fuzzy homomorphism of Z into itself. However, f is not a strong fuzzy homomorphism, as can be seen as follows: δ(2) = δ(6) = 1/2 and therefore for all y ∈ Z we have µf (2, y) ≤ 1/2, µf (6, y) ≤ 1/2. Consequently, µf (8, 27) = 3/4 > 1/2 ≥ sup{min(µf (2, y1 ), µf (6, y2 )) : y1 + y2 = 27}. Theorem 3.8 Let f1 be a homomorphism of G1 into G2 and λ be a normal fuzzy subgroup of G2 . The fuzzy mapping f of G1 into G2 defined by µf (x, y) = λ(f1 (x−1 )y), x ∈ G1 , y ∈ G2 is a strong fuzzy homomorphism of G1 into G2 . Proof. First let x1 , x2 ∈ G1 , y ∈ G2 . For any decomposition y = y1 y2 in G2 , we have µf (x1 x2 , y) =
λ(f1 ((x1 x2 )−1 )y1 y2 )
=
−1 λ(fI (x−1 2 )f1 (x1 )y1 y2 )
=
−1 λ(f1 (x−1 1 )y1 y2 f1 (x2 ))
≥
−1 min{λ(f (x−1 1 )y1 ), λ(y2 f (x2 ))}
=
−1 min(λ(f (x−1 1 )y1 ), λ(f (x2 )y2 ))
(since λ is normal) (since λ is a fuzzy subgroup)
= min(µf (x1 , y1 ), µf (x2 , y2 )). Taking supremum over all decompositions of y in G2 , we have µf (x1 x2 , y) ≥ sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : y = y1 y2 }. 6
On the other hand, taking y1 = f1 (x1 ) and y2 = f1 (x1 − 1)y we have y = y1 y2 and −1 min(λ(e2 ), λ(f1 (x−1 2 )f (x1 )y))
min(µf (x1 , y1 ), µf (x2 , y2 )) = =
λ(f1 (x1 x2 )−1 y)
=
µf (x1 x2 , y).
This gives sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : y = y1 y2 } ≥ µf (x1 x2 , y). Hence, we get µf (x1 x2 , y) = sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : y = y1 y2 }. Again, for x ∈ G1 , y ∈ G2 we have µf (x−1 , y −1 ) = λ(f (x)y −1 ) = λ(yf (x−1 )) = λ(f (x−1 )y) = µf (x, y). Thus f is a strong fuzzy homomorphism of G1 into G2 . Example 3.9 Let K be the Klein’s group {e, h, v, t} with identity e and S3 be the symmetric group of degree 3. Consider the group homomorphism f1 : K → S3 defined by f1 (e) = f1 (h) = (1) and f1 (v) = f1 (t) = (13). Let λ : S3 → [0, 1] be defined as follows: λ((1)) = 1, λ({(123), (132)}) = 1/2 and λ(x) = 1/4 for all other elements x of S3 . Then λ is a normal fuzzy subgroup of S3 . Using Theorem 3.8 we get a strong fuzzy homomorphism f from K into S3 with the following table for µf : µf e h v t
(1) (123) 1 1/2 1 1/2 1/4 1/4 1/4 1/4
(132) 1/2 1/2 1/4 1/4
(12) 1/4 1/4 1/2 1/2
(23) 1/4 1/4 1/2 1/2
(13) 1/4 1/4 1 1
Theorem 3.10 The composition of two weak (resp. strong) fuzzy homomorphisms is a weak (resp. strong) fuzzy homomorphism. Proof. Let f be a weak fuzzy homomorphism of the group G1 into the group G2 and g be a weak fuzzy homomorphism of the group G2 into the group G3 . Then for x1 , x2 ∈ G1 and 7
z1 , z2 ∈ G3 , we have µg◦f (x1 x2 , z1 z2 ) =
sup{min{µf (x1 x2 , y), µg (y, z1 z2 )} : y ∈ G2 }
=
sup{min{µf (x1 x2 , y1 y2 ), µg (y1 y2 , z1 z2 )} : y = y1 y2 , y1 , y2 ∈ G2 }
≥
sup{min{min(µf (x1 , y1 ), µf (x1 , y2 )), min(µg (y1 , z1 ), µg (y2 , z2 ))} : y = y1 y2 , y1 , y2 ∈ G2 }
=
sup{min{min(µf (x1 , y1 ), µg (y1 , z1 )), min(µf (x2 , y2 ), µg (y2 , z2 ))} : y = y1 y2 , y1 , y2 ∈ G2 }
=
min(sup{min(µf (x1 , y1 ), µg (y1 , z1 )) : y1 ∈ G2 }, sup{min(µf (x2 , y2 ), µg (y2 , z2 )) : y2 ∈ G2 })
=
min{µg◦f (x1 , z1 ), µg◦f (x2 , z2 )}.
Moreover, for x ∈ G1 , z ∈ G3 we have µg◦f (x−1 , z −1 ) = sup{min(µf (x−1 , y), µg (y, z −1 )) : y ∈ G2 } = sup{min(µf (x−1 , y −1 ), µg (y −1 , z −1 )) : y −1 ∈ G2 } = sup{min(µf (x, y), µg (y, z)) : y ∈ G2 } =
µg◦f (x, z).
Hence, g ◦ f is a weak fuzzy homomorphism of G1 into G2 . Next, let f be a strong fuzzy homomorphism of the group G1 into the group G2 and g be a strong fuzzy homomorphism of the group G2 into the group G3 . Let x1 , x2 ∈ G1 and z ∈ G3 . We have to show that µg◦f (x1 x2 , z) = sup{min(µg◦f (x1 , z1), µg◦f (x2 , z2)) : z1 , z2 ∈ G3 , z1 z2 = z}. By definition, µg◦f (x1 x2 , z) = sup{min(µf (x1 x2 , y), µg (y, z)) : y ∈ G2 } = k (say). Let ² > 0 be given. Then there exists y ∈ G2 such that min(µf (x1 x2 , y), µg (y, z)) > k − ². Now, f being a strong fuzzy homomorphism, we have k − e < µf (x1 x2 , y) = sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : y = y1 y2 , y ∈ G2 }. Therefore, there exist y1 , y2 ∈ G2 such that y = y1 y2 and k−² < min{µf (x1 , y1 ), µf (x2 , y2 )}. Again, because k − ² < µg (y, z) = µg (y1 y2 , z) = sup{min(µg (y1 , z1), µg (y2 , z2)) : z = z1 z2 } 8
(g being a strong fuzzy homomorphism), there exists z1 , z2 ∈ G3 such that z = z1 z2 and k − ² < min{µg (y1 , z1 ), µg (y2 , z2 )}. Now we have µg◦f (x1 , z1 ) ≥ min(µf (x1 , y1 ), µg (y1 , z1 )) > k − ² and µg◦f (x2 , z2 ) ≥ min(µf (x2 , y2 ), µg (y2 , z2 )) > k − ². This shows that sup{min(µg◦f (x1 , z1 ), µg◦f (x2 , z2 )) : z = z1 z2 } > k − ². Since ² is arbitrary, we have sup{min(µg◦f (x1 , z1 ), µg◦f (x2 , z2 )) : z = z1 z2 } ≥ k = µg◦f (x1 x2 , z). The reverse inequality follows from the fact that g ◦ f is a weak fuzzy homomorphism. Hence, g ◦ f is a strong fuzzy homomorphism of G1 into G3 .
4
Images and Pre-images under Fuzzy Homomorphism
In this section we discuss invariance of fuzzy group structures under the action of fuzzy homomorphisms. Theorem 4.1 Let f be a weak fuzzy homomorphism of G1 into G2 . Then (a) If λ is a fuzzy subgroup of G1 , then f (λ) is a fuzzy subgroup of G2 . (b) If ν is a fuzzy subgroup of G2 , then f −1 (ν) is a fuzzy subgroup of G1 . Proof. (a) For y1 , y2 ∈ G2 we have f (λ)(y1 y2 ) = sup{min(λ(x), µf (x, y1 y2 )) : x ∈ G1 } = sup{min(λ(x1 x2 ), µf (x1 x2 , y1 y2 )) : x = x1 x2 , x1 , x2 ∈ G1 } ≥ sup{min{min(λ(x1 ), λ(x2 )), min(µf (x1 , y1 ), µf (x2 , y2 ))} : x = x1 x2 , x1 , x2 ∈ G1 } = sup{min{min(λ(x1 ), µf (x1 , y1 )), min(λ(x2 ), µf (x2 , y2 ))} : x1 , x2 ∈ G1 } = min(sup{min(λ(x1 ), µf (x1 , y1 )) : x1 ∈ G1 }, sup{min(λ(x2 ), µf (x2 , y2 )) : x2 ∈ G1 }) = min{f (λ)(y1 ), f (λ)(y2 )}. 9
Moreover, for y ∈ G2 we have f (λ)(y −1 )
=
sup{min(λ(x), µf (x, y − 1)) : x ∈ G1 }
= sup{min(λ(x−1 ), µf (x−1 , y −1 )) : x−1 ∈ G1 } = sup{min(λ(x), µf (x, y)) : x−1 ∈ G1 } = f (λ)(y). Thus f (λ) is a fuzzy subgroup of G2 . (b) Since f −1 (ν) is the image of ν under the weak fuzzy homomorphism f −1 of G2 onto G1 , in view of (a), f −1 (ν) is a fuzzy subgroup of G1 . Theorem 4.2 Let f be a strong fuzzy homomorphism of G1 into G2 . If ν is a normal fuzzy subgroup of G2 then f −1 (ν) is a normal fuzzy subgroup of G1 . Proof. It is enough to show that f −1 (ν)(x1 x2 ) = f −1 (ν)(x2 x1 ) for all x1 , x2 ∈ G1 . We have f −1 (ν)(x1 x2 )
=
sup{min(µf (x1 x2 , y), ν(y)) : y ∈ G2 }
= sup{min(sup{min(µf (x1 , y1 ), µf (x2 , y2 ))} : y = y1 y2 , ν(y)) : y ∈ G2 } = sup{sup{min{min(µf (x1 , y1 ), µf (x2 , y2 )), ν(y)} : y = y1 y2 } : y ∈ G2 } = sup{min{min(µf (x1 , y1 ), µf (x2 , y2 )), ν(y1 y2 )} : y1 , y2 ∈ G2 }. Similarly, f −1 (ν)(x2 x1 ) = sup{min{min(µf (x2 , y2 ), µf (x1 , y1 )), ν(y2 y1 )} : y1 , y2 ∈ G2 }. The equality follows from the fact that ν(y1 y2 ) = ν(y2 y1 ) for all y1 , y2 ∈ G2 . Definition 4.3 A weak fuzzy homomorphism f of G1 into G2 is said to satisfy the surjective condition if f −1 is a strong fuzzy homomorphism, that is, if for all x ∈ G1 and y1 , y2 ∈ G2 , µf (x, y1 y2 ) = sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : x = x1 x2 }. The strong fuzzy homomorphism f in Example 3.9 does not satisfy the surjective condition. However, its inverse homomorphism is a weak fuzzy homomorphism satisfying the surjective condition. Theorem 4.4 Let f be a homomorphism of G1 into G2 . Then f satisfies the surjective condition (as a fuzzy homomorphism) if and only if f is onto. Proof. First, suppose that f is onto. Let x ∈ G1 and y1 , y2 ∈ G2 . If f (x) = y1 y2 , then there exist x1 , x2 ∈ G1 such that x = x1 x2 and f (xi ) = yi , where i = 1, 2. Consequently, sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : x = x1 x2 } = 1 = µf (x, y1 y2 ). 10
On the other hand, if f (x) 6= y1 y2 , then for any decomposition x = x1 x2 in G1 either f (x1 ) 6= y1 or f (x2 ) = 6 y2 . Consequently, sup{min(µf (x1 , y1 ), µf (x2 , y2 )) : x = x1 x2 } = 0 = µf (x, y1 y2 ). This shows that f satisfies the surjective condition as a fuzzy homomorphism. Conversely, suppose that f is not onto. Let y ∈ G2 \ f (G1 ). Then for any x ∈ G1 , µf (x, y) = 0. Consequently, sup{min(µf (x1 , y), µf (x2 , y −1 )) : e1 = x1 x2 } = 0. However µf (e1 , yy −1 ) = µf (e1 , e2 ) = 1. This shows that f does not satisfy the surjective condition as a fuzzy homomorphism. Theorem 4.5 Let f be a weak fuzzy homomorphism of G1 into G2 satisfying the surjective condition. If λ is a normal fuzzy subgroup of G1 , then f (λ) is a normal fuzzy subgroup of G2 . Proof. Since f satisfy the surjective condition, f −1 is a strong fuzzy homomorphism of G2 into G1 . Because f (λ) is the pre-image of the normal subgroup λ under f −1 , in view of Theorem 4.2, f (λ) is a normal fuzzy subgroup of G2 .
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