On normalizers in fuzzy groups
In an arbitrary fuzzy group we define the normalizer of fuzzy subgroup and study some its properties. In particular, the characterization of nilpotent fuzzy group has been obtained.
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irk-123456789-1522612019-06-10T01:25:24Z On normalizers in fuzzy groups Kurdachenko, L.A. Grin, K.O. Turbay, N.A. In an arbitrary fuzzy group we define the normalizer of fuzzy subgroup and study some its properties. In particular, the characterization of nilpotent fuzzy group has been obtained. 2013 Article On normalizers in fuzzy groups / L.A. Kurdachenko, K.O. Grin, N.A. Turbay // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 23–36. — Бібліогр.: 6 назв. — англ. 1726-3255 2010 MSC:Primary 20N25, 08A72, 03E72. http://dspace.nbuv.gov.ua/handle/123456789/152261 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In an arbitrary fuzzy group we define the normalizer of fuzzy subgroup and study some its properties. In particular, the characterization of nilpotent fuzzy group has been obtained. |
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Kurdachenko, L.A. Grin, K.O. Turbay, N.A. On normalizers in fuzzy groups Algebra and Discrete Mathematics |
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On normalizers in fuzzy groups |
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On normalizers in fuzzy groups |
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On normalizers in fuzzy groups |
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On normalizers in fuzzy groups |
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On normalizers in fuzzy groups |
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on normalizers in fuzzy groups |
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On normalizers in fuzzy groups / L.A. Kurdachenko, K.O. Grin, N.A. Turbay // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 23–36. — Бібліогр.: 6 назв. — англ. |
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Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 15 (2013). Number 1. pp. 23 – 36
c© Journal “Algebra and Discrete Mathematics”
On normalizers in fuzzy groups
L. A. Kurdachenko, K. O. Grin, N. A. Turbay
Abstract. In an arbitrary fuzzy group we define the normal-
izer of fuzzy subgroup and study some its properties. In particular,
the characterization of nilpotent fuzzy group has been obtained.
Let G be a group with a multiplicative binary operation denoted by
juxtaposition and identity e. We recall that a fuzzy subset γ : G → [0, 1]
is said to be a fuzzy group on G ( see, for example, [1, S 1.2]), if it satisfies
the following conditions:
(FSG 1) γ(xy) ≥ γ(x) ∧ γ(y) for all x, y ∈ G,
(FSG 2) γ(x−1) ≥ γ(x) for every x ∈ G.
Here and everywhere we adopt the usual convention on the operator
wedge ∧( and on the operator vee ∨ ). If W is a subset of [0, 1], then
denote by
∧
W the greatest lower bound of W and denote by
∨
W the
least upper bound of W . If W = {a, b}, then, as usual, instead of
∧
W
we will write a ∧ b, and instead of
∨
W we will write a ∨ b. We assume
that the least upper bound of the empty set is 0, and the greatest lower
bound of the empty set is 1.
However we remark that we deliberately replace the standard expres-
sion a fuzzy subgroup of G by a fuzzy group on G in order to avoid
2010 MSC: Primary 20N25, 08A72, 03E72.
Key words and phrases: Fuzzy group; fuzzy subgroup; lower central series;
nilpotent fuzzy group; normalizer condition; ascendant fuzzy subgroup; characteristic
function.
24 On normalizers in fuzzy groups
possible misunderstanding in the sequel and to emphasize that a fuzzy
group is in fact a function defined on a group G. For example, if γ, κ are
the fuzzy groups on G and γ ⊆ κ, occurs, we will say that γ is a fuzzy
subgroup of κ and denote this by γ 4 κ.If γ is a fuzzy subgroup of κ, then
γ(e) ≤ κ(e). Fuzzy subgroup γ of κ is called unitary, if γ(e) = κ(e). If γ is
an arbitrary fuzzy subgroup of κ, then clearly γ∗ = γ ∪χ(e, κ(e)) is a fuzzy
subgroup of κ. Moreover, γ∗(x) = γ(x) for all x 6= e and γ∗(e) = κ(e).
Therefore, in the future we will focus on the unitary fuzzy subgroups of
κ only. More precisely, when referring to that γ is a fuzzy subgroup of
κ we will assume that the mentioned term means γ is an unitary fuzzy
subgroup of κ.
Recall the following definition. If X is a set, for every subset Y of X
and every a ∈ [0, 1] we define a fuzzy subset χ(Y, a) as follows:
χ(Y, a) =
{
a, x ∈ Y,
0, x /∈ Y.
Clearly χ(H, a) is a fuzzy group on G for every subgroup H of G. If
Y = {y}, then instead of χ({y}, a) we will write shorter χ(y, a). A fuzzy
subset χ(y, a) is called a fuzzy point (or fuzzy singleton).
Fuzzy group theory, as well as other fuzzy algebraic structures, was
introduced very soon after the beginning of fuzzy set theory. Many basic
results of this theory were collected in the book [1]. But they are not
systemized since it just a set of particular results that belong to different
objects. There are a lot results on the structure of the largest fuzzy group
χ(G, 1) on G. However one of the main goals of fuzzy group theory is
the study of algebraic properties of an arbitrary fuzzy group defined on
an abstract group G. There is essential difference with the case of fuzzy
group χ(G, 1). Lets specify the next case. Consider arbitrary fuzzy group
γ as an union the fuzzy point χ(x, γ(x)), x ∈ G. By multiplication it is
a semigroup. A fuzzy group χ(G, 1) has many invertible elements (all
fuzzy points χ(g, 1), g ∈ G, are invertible), and this makes possible to use
essentially impact on χ(G, 1) the group G. At the same time, an arbitrary
fuzzy group defined on a group G may have very few invertible elements,
and as a consequence, we have very little tangible results on arbitrary
fuzzy group defined on a group G.
Our goal is to begin a systematic study of the properties of an arbitrary
fuzzy group defined on a group G. One of the important concept not only
in group theory, but also in the whole algebra is the notion of nilpotency.
It was introduced for fuzzy groups too ( see, [1, Chapters 3.2] and the
L. A. Kurdachenko, K. O. Grin, N. A. Turbay 25
papers [2], [3], [4]). In the paper [5] has been introduced the concept of
upper central series in fuzzy group and considered some properties of
hypercentral fuzzy group. In this paper we continue the investigation
of generalized nilpotent fuzzy group. This study based on the concept
of normalizer of fuzzy subgroup, which we introduce here. We consider
also some properties of the class of fuzzy group satisfying the normalizer
condition.
We will start from following useful
Proposition 1. Let G be a group and γ, κ be a fuzzy subsets of G. Then
(γ ⊚ κ) = ∪y∈Supp(γ),z∈Supp(κ)χ(y, γ(y)) ⊚ χ(z, κ(z)).
Proof. By definition we have
(γ ⊚ κ)(x) =
∨
y,z∈G,yz=x(γ(y) ∧ κ(z)).
If y /∈ Supp(γ), then γ(y) = 0 and γ(y) ∧ κ(z) = 0. Similarly, if z /∈
Supp(κ), then κ(z) = 0 and again γ(y) ∧ κ(z) = 0. It follows that
(γ ⊚ κ)(x) =
∨
y∈Supp(γ),z∈Supp(κ),yz=x(γ(y) ∧ κ(z)).
On the other hand, consider a fuzzy subset
ξ = ∪y∈Supp(γ),z∈Supp(κ)χ(y, γ(y)) ⊚ χ(z, κ(z)).
By Proposition 1 of paper [5] χ(y, γ(y)) ⊚ χ(z, κ(z)) = χ(yz, γ(y) ∧ κ(z)).
If x ∈ G and x = yz, then χ(yz, γ(y) ∧ κ(z))(x) = γ(y) ∧ κ(z), otherwise
χ(yz, γ(y) ∧ κ(z))(x) = 0. Therefore
ξ(x) =
∨
y∈Supp(γ),z∈Supp(κ)(χ(yz, γ(y) ∧ κ(z)))(x) =
=
∨
y∈Supp(γ),z∈Supp(κ),yz=x(γ(y) ∧ κ(z)) = (γ ⊚ κ)(x).
Since it is true for each x ∈ G,
(γ ⊚ κ) = ∪y∈Supp(γ),z∈Supp(κ)χ(y, γ(y)) ⊚ χ(z, κ(z)).
Corollary 1. Let G be a group.
(i) If γ, λ, κ are the fuzzy subsets of G such that λ ⊆ κ, then γ⊚λ ⊆ γ⊚κ
and λ ⊚ γ ⊆ κ ⊚ γ;
(ii) If γ, λa are the fuzzy subsets of G, a ∈ A, then γ ⊚ (∪a∈Aλa) =
∪a∈A(γ ⊚ λa) and (∪a∈Aλa) ⊚ γ = ∪a∈A(λa ⊚ γ).
Proof. (i) We have
λ = ∪x∈Supp(λ)χ(x, λ(x)), κ = ∪x∈Supp(κ)χ(x, κ(x)).
26 On normalizers in fuzzy groups
Since λ ⊆ κ, λ(x) ≤ κ(x) for each x ∈ G. We have χ(x, λ(x))(x) =
λ(x) ≤ κ(x) = χ(x, κ(x))(x) and χ(x, λ(x))(y) = 0 ≤ 0 = χ(x, κ(x))(y)
whenever y 6= x. It follows that χ(x, λ(x)) ⊆ χ(x, κ(x)). By Proposition 1
γ ⊚ λ = ∪y∈Supp(γ),z∈Supp(λ)χ(y, γ(y)) ⊚ χ(z, λ(z)) ⊆
∪y∈Supp(γ),z∈Supp(κ)χ(y, γ(y)) ⊚ χ(z, κ(z)) = γ ⊚ κ.
The second inclusion proved in a similar way.
(ii) Put λ = ∪a∈Aλa. By Proposition 1
γ ⊚ λ = ∪y∈Supp(γ),z∈Supp(λ)χ(y, γ(y)) ⊚ χ(z, λ(z)).
Clearly L = Supp(∪a∈Aλa) = ∪a∈ASupp(λa). We have
∪a∈A(γ ⊚ λa) = ∪a∈A(∪y∈Supp(γ),z∈Supp(λa)χ(y, γ(y)) ⊚ χ(z, λa(z))) =
∪y∈Supp(γ),z∈L(∪a∈Aχ(y, γ(y)) ⊚ χ(z, λa(z))).
Using Proposition 1 of paper [5] we obtain χ(y, γ(y)) ⊚ χ(z, λa(z)) =
χ(yz, γ(y) ∧ λa(z)). By the definition of the union of fuzzy subsets,
(∪a∈Aχ(yz, γ(y) ∧ λa(z)))(g) =
∨
a∈A(χ(yz, γ(y) ∧ λa(z))(g))
for every g ∈ G. In particular,
(∪a∈Aχ(yz, γ(y) ∧ λa(z)))(yz) =
∨
a∈A(χ(yz, γ(y) ∧ λa(z))(yz)) =
∨
a∈A(γ(y) ∧ λa(z)) = γ(y) ∧ (
∨
a∈A λa(z)) = γ(y) ∧ λ(z),
(∪a∈Aχ(yz, γ(y) ∧ λa(z)))(g) =
∨
a∈A(χ(yz, γ(y) ∧ λa(z))(g)) = 0
whenever g 6= yz.
In particular, ∪a∈Aχ(yz, γ(y) ∧ λa(z)) = χ(yz, γ(y) ∧ λ(z)). Hence
∪a∈A(γ ⊚ λa) = ∪y∈Supp(γ),z∈L(∪a∈Aχ(y, γ(y)) ⊚ χ(z, λa(z))) =
∪y∈Supp(γ),z∈Lχ(yz, γ(y) ∧ λ(z)) = γ ⊚ λ.
Using the similar arguments, we can prove a second equation.
Let γ, κ be the fuzzy groups on G and κ 4 γ. We define a normalizer
Nγ(κ) of κ in γ as an union of all fuzzy points χ(x, a) ⊆ γ, satisfying the
following condition χ(x−1, a) ⊚ κ ⊚ χ(x, a) = κ.
Since κ = ∪g∈Gχ(g, κ(g)), an application of Proposition 1 gives a
following equation
χ(x−1, a) ⊚ κ ⊚ χ(x, a) = ∪g∈Gχ(x−1, a) ⊚ χ(g, κ(g)) ⊚ χ(x, a),
which implies a following
L. A. Kurdachenko, K. O. Grin, N. A. Turbay 27
Proposition 2. Let G be a group and γ, κ be the fuzzy groups on G and
κ 4 γ. Then a normalizer Nγ(κ) includes a fuzzy point χ(x, a) ⊆ γ if and
only if for each g ∈ G there are the elements z, u ∈ G such that
χ(x−1, a) ⊚ χ(g, κ(g)) ⊚ χ(x, a) ⊆ χ(z, κ(z))
and
χ(g, κ(g)) ⊆ χ(x−1, a) ⊚ χ(u, κ(u)) ⊚ χ(x, a).
Lemma 1. Let G be a group and γ, κ be the fuzzy groups on G and κ 4 γ.
Then a normalizer Nγ(κ) includes a fuzzy point χ(x, a) ⊆ γ if and only
if for each g ∈ G there are the elements z, u ∈ G such that
χ(g, κ(g)) ⊚ χ(x, a) ⊆ χ(x, a) ⊚ χ(z, κ(z))
and
χ(x, a) ⊚ χ(g, κ(g)) ⊆ χ(u, κ(u)) ⊚ χ(x, a).
Proof. Suppose that χ(x, a) ⊆ Nγ(κ) and g be an arbitrary element of G.
By Proposition 2 there are the elements z, u ∈ G satisfying the following
inclusions
χ(x−1, a) ⊚ χ(g, κ(g)) ⊚ χ(x, a) ⊆ χ(z, κ(z))
and
χ(g, κ(g)) ⊆ χ(x−1, a) ⊚ χ(u, κ(u)) ⊚ χ(x, a).
First inclusion implies
χ(x, a) ⊚ χ(x−1, a) ⊚ χ(g, κ(g)) ⊚ χ(x, a) ⊆ χ(x, a) ⊚ χ(z, κ(z)).
Further, using Proposition 1 of paper [5], we obtain
χ(x, a) ⊚ χ(x−1, a) ⊚ χ(g, κ(g)) ⊚ χ(x, a) =
χ(e, a) ⊚ χ(g, κ(g)) ⊚ χ(x, a) = χ(gx, a ∧ κ(g)).
On the other hand,
χ(g, κ(g)) ⊚ χ(x, a) = χ(gx, a ∧ κ(g)) =
χ(x, a) ⊚ χ(x−1, a) ⊚ χ(g, κ(g)) ⊚ χ(x, a).
So we obtain that χ(g, κ(g)) ⊚ χ(x, a) ⊆ χ(x, a) ⊚ χ(z, κ(z)).
Using the similar arguments, we obtain an inclusion
χ(x, a) ⊚ χ(g, κ(g)) ⊆ χ(u, κ(u)) ⊚ χ(x, a).
28 On normalizers in fuzzy groups
We can prove inverse assertion also using the similar arguments.
We can reformulate Lemma 1 in a following form
Lemma 2. Let G be a group and γ, κ be the fuzzy groups on G and κ 4 γ.
Then a normalizer Nγ(κ) includes a fuzzy point χ(x, a) ⊆ γ if and only
if for each g ∈ G there are the elements z, u ∈ G such that
χ(g, κ(g)) ⊚ χ(x−1, a) ⊆ χ(x−1, a) ⊚ χ(z, κ(z))
and
χ(x−1, a) ⊚ χ(g, κ(g)) ⊆ χ(u, κ(u)) ⊚ χ(x−1, a).
Lemma 3. Let G be a group and γ be a fuzzy group on G. If λ, κ ⊆ γ,
then λ ⊚ κ ⊆ γ, in particular, γ ⊚ γ ⊆ γ.
Proof. Let x be an arbitrary element of G. We have
(λ ⊚ κ)(x) =
∨
y,z∈G,yz=x(λ(y) ∧ κ(z)).
The inclusions λ, κ ⊆ γ imply λ(y) ∧κ(z) ≤ γ(y) ∧γ(z). Since γ is a fuzzy
subgroup, γ(y) ∧ γ(z) ≤ γ(yz), thus
(λ ⊚ κ)(x) =
∨
y,z∈G,yz=x
(λ(y) ∧ κ(z)) ≤
∨
y,z∈G,yz=x
γ(yz) = γ(x).
Thus we get a criterion of being a fuzzy group needed in the future.
Proposition 3. Let G be a group and γ be a fuzzy subset of G. Then γ
is a fuzzy group if and only if the following assertion holds:
(FSG 3) χ(x, γ(x)) ⊚ χ(y, γ(y)) ⊆ γ for all x, y ∈ Supp(γ),
(FSG 4) χ(x−1, γ(x)) ⊆ γ for all x ∈ Supp(γ).
Proof. Suppose first that γ is a fuzzy group on G. Since γ includes the
function χ(x, γ(x)) and χ(y, γ(y)) for every elements x, y ∈ Supp(γ),
using Lemma 3, we obtain that χ(x, γ(x)) ⊚ χ(y, γ(y)) ⊆ γ.
Let x be an arbitrary element of Supp(γ). So (χ(x−1, γ(x))(x−1) =
γ(x). Since γ is a fuzzy group, γ(x) ≤ γ(x−1). We note that if y 6= x−1,
then (χ(x−1, γ(x))(y) = 0, so that (χ(x−1, γ(x))(y) ≤ γ(y) for every
y ∈ G. This means that χ(x−1, γ(x)) ⊆ γ.
Conversely, suppose that γ satisfies the both conditions (FSG 3),
(FSG 4). Let x,y be the arbitrary elements of G. If, for example, x /∈
Supp(γ), then γ(x) = 0. It follows that γ(x) ∧ γ(y) = 0, and hence
γ(xy) ≥ 0 = γ(x) ∧ γ(y). Therefore assume that x, y ∈ Supp(γ). Then
(FSG 3) shows that χ(x, γ(x)) ⊚ χ(y, γ(y)) ⊆ γ. By Proposition 1 of
paper [5]
L. A. Kurdachenko, K. O. Grin, N. A. Turbay 29
γ(x) ∧ γ(y) = (χ(x, γ(x)) ⊚ χ(y, γ(y)))(xy) ≤ γ(xy),
thus we obtain γ(x) ∧ γ(y) ≤ γ(xy), and γ satisfies (FSG 1).
Let x ∈ G. Since χ(x−1, γ(x)) ⊆ γ, (χ(x−1, γ(x)))(y) ≤ γ(y) for every
y ∈ G. In particular,
(χ(x−1, γ(x)))(x−1) = γ(x) ≤ γ(x−1),
so that γ satisfies (FSG 2).
Theorem 1. Let G be a group, γ, κ be the fuzzy groups on G and κ 4 γ.
Then a normalizer Nγ(κ) is a fuzzy subgroup of γ.
Proof. Put ν = Nγ(κ). Let x, y are the arbitrary elements of G. Con-
sider a product χ(x, ν(x)) ⊚ χ(y, ν(y)). By Proposition 1 of paper [5]
χ(x, ν(x)) ⊚ χ(y, ν(y)) = χ(xy, ν(x) ∧ ν(y)). Let g ∈ G. Consider the
products χ(g, κ(g))⊚χ(xy, ν(x)∧ν(y)) and χ(xy, ν(x)∧ν(y))⊚χ(g, κ(g)).
We have
χ(g, κ(g)) ⊚ χ(xy, ν(x) ∧ ν(y)) = χ(g, κ(g)) ⊚ (χ(x, ν(x)) ⊚ χ(y, ν(y))) =
(χ(g, κ(g)) ⊚ χ(x, ν(x))) ⊚ χ(y, ν(y)).
Since χ(x, ν(x)) ⊆ Nγ(κ), Lemma 1 shows that there is an element z ∈ G
such that χ(g, κ(g)) ⊚ χ(x, ν(x)) ⊆ χ(x, ν(x)) ⊚ χ(z, κ(z)),so that
(χ(g, κ(g)) ⊚ χ(x, ν(x))) ⊚ χ(y, ν(y)) ⊆
(χ(x, ν(x))⊚χ(z, κ(z)))⊚χ(y, ν(y)) = χ(x, ν(x))⊚(χ(z, κ(z))⊚χ(y, ν(y))).
Using again Lemma 1, we obtain the existence of element w ∈ G such
that
χ(z, κ(z)) ⊚ χ(y, ν(y)) ⊆ χ(y, ν(y)) ⊚ χ(w, κ(w)),
so that
χ(g, κ(g)) ⊚ χ(xy, ν(x) ∧ ν(y)) ⊆ χ(x, ν(x)) ⊚ χ(y, ν(y)) ⊚ χ(w, κ(w)) =
χ(xy, ν(x) ∧ ν(y)) ⊚ χ(w, κ(w)).
Using again Lemma 1 and similar arguments we obtain a following inclu-
sion
χ(xy, ν(x) ∧ ν(y)) ⊚ χ(g, κ(g)) ⊆ χ(u, κ(u)) ⊚ χ(xy, ν(x) ∧ ν(y))
for some element u ∈ G. These both inclusion together with Lemma 1
prove that χ(xy, ν(x) ∧ ν(y)) ⊆ Nγ(κ). As we have seen above
χ(xy, ν(x) ∧ ν(y)) = χ(x, ν(x)) ⊚ χ(y, ν(y)),
30 On normalizers in fuzzy groups
thus χ(x, ν(x)) ⊚ χ(y, ν(y)) ⊆ Nγ(κ) and Nγ(κ) satisfies (FSG 3).
Let x ∈ G and consider fuzzy point χ(x, ν(x)). By Lemma 1 for every
element g ∈ G there exist the elements z, u ∈ G such that
χ(g, κ(g)) ⊚ χ(x−1, ν(x)) ⊆ χ(x−1, ν(x)) ⊚ χ(z, κ(z))
and
χ(x−1, ν(x)) ⊚ χ(g, κ(g)) ⊆ χ(u, κ(u)) ⊚ χ(x−1, ν(x)).
But Lemma 1 shows that in this case χ(x−1, ν(x)) ⊆ ν, so that Nγ(κ)
satisfies (FSG 4). Proposition 3 shows that Nγ(κ) is a fuzzy group.
The concept of normalizer is connected with a concept of normal fuzzy
subgroup. Recall that if γ, κ are the fuzzy groups on G and κ 4 γ, then it
is said that κ is a normal fuzzy subgroup of γ, if κ(yxy−1) ≥ κ(x) ∧ γ(y)
for every elements x, y ∈ G [1, 1.4]. We denote this fact by κ E γ. We
need a following criteria of normality, which specifies Proposition 3 of
paper [5].
Proposition 4. Let G be a group, γ, κ be the fuzzy groups on G. Suppose
that κ 4 γ. Then κ is a normal fuzzy subgroup of γ, if and only if
χ(x, γ(x)) ⊚ κ ⊚ χ(x−1, γ(x)) = κ
for every element x ∈ G.
Proof. Suppose first that κ is a normal fuzzy subgroup of γ. Let y ∈ G
and consider a product χ(y, γ(y))⊚κ⊚χ(y−1, γ(y)). Let x be an arbitrary
element of G. From Lemma 2 of paper [5] we obtain
(χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)))(x) = γ(y) ∧ κ(y−1xy).
Put u = y−1xy, then x = y(y−1xy)y−1 = yuy−1, so that
(χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)))(yuy−1) = γ(y) ∧ κ(u).
Since κ(u) ∧ γ(y) ≤ κ(yuy−1), we obtain
(χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)))(yuy−1) ≤ κ(yuy−1),
that is
(χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)))(x) ≤ κ(x).
L. A. Kurdachenko, K. O. Grin, N. A. Turbay 31
Since it is valid for every element x ∈ G, χ(y, γ(y))⊚κ⊚χ(y−1, γ(y)) 4 κ.
We have κ = ∪g∈Supp(κ)χ(g, κ(g)). Since κ is a normal fuzzy subgroup
of γ, Supp(κ) is a normal subgroup of Supp(γ) ( see, for example, [1,
Theorem 1.4.4 ] ), so that there exists an element x ∈ Supp(κ) such that
g = yxy−1. Then x = y−1gy and κ(x) ≥ κ(y) ∧ κ(g). Consider a product
χ(y, γ(y)) ⊚ χ(x, κ(x)) ⊚ χ(y−1, γ(y)) = χ(yxy−1, γ(y) ∧ κ(x)) =
χ(g, γ(y) ∧ κ(x)).
Suppose first that κ(x) 6= κ(y), then κ(g) = κ(x) ∧ κ(y) ( see, for exam-
ple, [1, p.7 ]). If κ(y) > κ(x), then κ(g) = κ(x), γ(y) ≥ κ(y) > κ(x) and
γ(y) ∧ κ(x) = κ(x). Hence in this case
χ(g, κ(g)) = χ(g, γ(y) ∧ κ(x)) = χ(y, γ(y)) ⊚ χ(x, κ(x)) ⊚ χ(y−1, γ(y)) ⊆
χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)).
Assume that κ(y) < κ(x), then κ(g) = κ(y). If κ(x) ≤ γ(y), then γ(y) ∧
κ(x) = κ(x) > κ(g) and
χ(g, κ(g)) ⊆ χ(g, γ(y) ∧ κ(x)) = χ(y, γ(y)) ⊚ χ(x, κ(x)) ⊚ χ(y−1, γ(y)) ⊆
χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)).
If κ(x) > γ(y), then γ(y) ∧ κ(x) = γ(y) ≥ κ(y) = κ(g) and again
χ(g, κ(g)) ⊆ χ(g, γ(y) ∧ κ(x)) = χ(y, γ(y)) ⊚ χ(x, κ(x)) ⊚ χ(y−1, γ(y)) ⊆
χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)).
Suppose now that κ(x) = κ(y), then κ(x) = κ(x) ∧ κ(y) ≤ κ(x) ∧ γ(y).
Since x = y−1gy, κ(x) ≥ κ(g), and we have again κ(g) ≤ κ(x) ∧ γ(y),
which follows that
χ(g, κ(g)) ⊆ χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)).
Hence we obtain an inclusion κ 4 χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)), so that
κ = χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)).
Conversely, suppose that χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)) = κ for each
y ∈ G. Let x be an arbitrary element of G. Put z = yxy−1, then x = y−1zy.
We have
(χ(y, γ(y)) ⊚ κ ⊚ χ(y−1, γ(y)))(z) ≤ κ(z).
Lemma 2 of paper [5] shows that (χ(y, γ(y))⊚κ⊚χ(y−1, γ(y)))(z) = γ(y)∧
κ(y−1zy). Then γ(y) ∧ κ(y−1zy) ≤ κ(z), that is γ(y) ∧ κ(x) ≤ κ(yxy−1).
32 On normalizers in fuzzy groups
This Proposition shows that κ is a normal fuzzy subgroup of Nγ(κ).
Let γ be the fuzzy group on G, µ be a fuzzy set of G and suppose that
µ ⊆ γ. Recall that a fuzzy subgroup, generated by µ is an intersection of
all fuzzy subgroups, including µ. We denote this subgroup by < µ >. We
have µ = ∪g∈Supp(µ)χ(g, µ(g)). Let κ be a fuzzy group on G, including
µ. Then µ(g) ≤ κ(g) for each element g ∈ Supp(µ). It follows that
χ(g, µ(g)) ⊆ κ for each element g ∈ Supp(µ). Proposition 3 shows that
χ(gt1
1 , µ(g1)) ⊚ χ(gt2
2 , µ(g2)) ⊚ . . . ⊚ χ(gtk
k , µ(gk)) ⊆ κ
for every elements g1, . . . , gk ∈ Supp(µ) where tj ∈ {1, −1}, 1 ≤ j ≤
k. Since it is true for each fuzzy group κ, including µ, χ(gt1
1 , µ(g1)) ⊚
χ(gt2
2 , µ(g2)) ⊚ . . . ⊚ χ(gtk
k , µ(gk)) ⊆< µ >. Put
λ = ∪g1,...,gk∈Supp(µ),k∈N,t1,...,tk∈{1,−1}χ(gt1
1 , µ(g1)) ⊚ χ(gt2
2 , µ(g2)) ⊚ . . . ⊚
χ(gtk
k , µ(gk)).
From an equation
χ(gt1
1 , µ(g1)) ⊚ χ(gt2
2 , µ(g2)) ⊚ . . . ⊚ χ(gtk
k , µ(gk)) =
χ(gt1
1 gt2
2 . . . gtk
k , µ(g1) ∧ µ(g2) ∧ . . . ∧ µ(gk))
follows that λ satisfies the both conditions (FSG 3), (FSG 4) and
Proposition 3 shows that λ is a fuzzy group. By definition of λ,
χ(g, µ(g)) ⊆ λ
for each element g ∈ Supp(µ). It follows that µ ⊆ λ and therefore
< µ >4 λ. On the other hand, we could see above that λ 4< µ >, so
that λ =< µ >. We can see that Supp(< µ >) =< Supp(µ) >.
Let G be a group, x, y ∈ G, a, b ∈ [0, 1]. Then a product χ(x−1, a) ⊚
χ(y−1, b) ⊚ χ(x, a) ⊚ χ(y, b) is called a commutator of χ(x, a) and χ(y, b)
and will denoted by [χ(x, a), χ(y, b)].
Let G be a group, γ, η be the fuzzy groups on G. Then a fuzzy com-
mutator subgroup [γ, η] is a fuzzy subgroup generated by all commutators
[χ(x, γ(x)), χ(y, η(y))] where x ∈ Supp(γ), y ∈ Supp(η).
Let γ be a fuzzy group on a group G. We define the lower central
series of γ by the following rule: put g1(γ) = γ, g2(γ) = [γ, γ]. Assume
that we have already construct the terms gβ(γ) for all ordinals β < α. If
α is a limit ordinal, then we put gα(γ) = ∩β<αgβ(γ). Suppose now that α
is a not limit ordinal, that is α − 1 exists. Then put gα(γ) = [gα−1(γ), γ].
Thus, for every ordinal α we constructed the αth term gα(γ) of a lower
central series of γ. The building of a lower central series of γ come to an
L. A. Kurdachenko, K. O. Grin, N. A. Turbay 33
end on some ordinal σ. In other words, this means that gσ(γ) = [gσ(γ), γ].
Then gσ(γ) is called the lower hypocenter of γ and will denoted by g∞(γ).
A fuzzy group γ is called hypocentral, if g∞(γ) 4 χ(e, γ(e)).
A fuzzy group γ is called nilpotent, if there exists a positive integer k
such that gk(γ) 4 χ(e, γ(e)). By Proposition 2 of paper [5] we obtain that
Supp(gj(γ)) is the jth term of a lower central series of Supp(γ). Hence if
a fuzzy group γ is nilpotent, then Supp(γ) is nilpotent. And conversely,
if Supp(γ) is a nilpotent group, then a fuzzy group γ is nilpotent (see,
for example, [1, Theorem 3.2.24]).
A fuzzy group γ is called locally nilpotent, if for every finite set of fuzzy
points χ(g1, γ(g1)), χ(g2, γ(g2)), . . . , χ(gk, γ(gk)) a fuzzy group, generated
by
µ = χ(g1, γ(g1)) ∪ χ(g2, γ(g2)) ∪ . . . ∪ χ(gk, γ(gk))
is nilpotent.
Let L be a subgroup of G and γ be a fuzzy group on G. We define
the function L|γ : G → [0, 1] by the following rule:
L|γ(x) =
{
γ(x), if x ∈ L,
0, if x /∈ L.
Let x, y be the arbitrary elements of G. Then it is easy to check that
L|γ(xy) ≥ L|γ(x) ∧ L|γ(y).
It follows that L|γ is a fuzzy group on G.
Proposition 5. Let G be a group and γ be a fuzzy group on G. Then γ
is a locally nilpotent if and only if Supp(γ) is a locally nilpotent abstract
group.
Proof. Suppose first that γ is locally nilpotent. Consider an arbitrary
finite set of fuzzy points χ(g1, γ(g1)), χ(g2, γ(g2)), . . . , χ(gk, γ(gk)). Let
µ = χ(g1, γ(g1)) ∪ χ(g2, γ(g2)) ∪ . . . ∪ χ(gk, γ(gk)).
Since Supp(µ) = {g1, g2, . . . , gk} and Supp(< µ >) =< Supp(µ) >,
we obtain by above remarked that every finitely generated subgroup of
Supp(γ) is nilpotent. In other words, Supp(γ) is locally nilpotent.
Conversely, suppose that Supp(γ) is locally nilpotent and let M =
{g1, g2, . . . , gk} be an arbitrary finite subset of Supp(γ). Then a subgroup
L =< M > is nilpotent. Consider a function L|γ . As we saw above L|γ
is a fuzzy group on G and Supp(L|γ) = L. Then a fuzzy group L|γ is
nilpotent ( see, for example, [1, Theorem 3.2.24]). Let again
34 On normalizers in fuzzy groups
µ = χ(g1, γ(g1)) ∪ χ(g2, γ(g2)) ∪ . . . ∪ χ(gk, γ(gk))
and λ =< µ >. As we saw above λ is an union of fuzzy points
χ(gt1
1 gt2
2 . . . gtk
k , γ(g1) ∧ γ(g2) ∧ . . . ∧ γ(gk)).
Since γ is a fuzzy group, γ(gt1
1 gt2
2 . . . gtk
k ) ≥ γ(g1) ∧ γ(g2) ∧ . . . ∧ γ(gk).
It follows that χ(gt1
1 gt2
2 . . . gtk
k , γ(g1) ∧ γ(g2) ∧ . . . ∧ γ(gk)) ⊆ L|γ , so that
< µ >4 L|γ . Since L|γ is nilpotent, < µ > is also nilpotent ( see, for
example, [1, Theorem 3.2.26]). This means that γ is locally nilpotent.
Let γ, κ be the fuzzy groups on G and κ 4 γ. We say that γ satisfies
a normalizer condition if Nγ(κ) 6= κ for every subgroup κ of γ.
Theorem 2. Let G be a group, γ be a fuzzy group on G. If γ satisfies a
normalizer condition, then Supp(γ) satisfies normalizer condition.
Proof. Let L be an arbitrary subgroup of Supp(γ). Put λ = L|γ , ν =
Nγ(λ). Then ν 6= λ. Suppose that Supp(λ) = Supp(ν). Then for every
element x ∈ Supp(λ) we have γ(x) ≥ ν(x) ≥ λ(x) = γ(x), in particular,
ν(x) = λ(x). Hence if we assume that Supp(λ) = Supp(ν), then λ = ν,
and we obtain a contradiction. This contradiction shows that Supp(λ) 6=
Supp(ν). By above remarked, λ is a normal fuzzy subgroup of ν. It follows
that Supp(λ) is a normal subgroup of Supp(ν) ( see, for example, [1,
Theorem 1.4.4]). Then NSupp(γ)(L) ≥ Supp(ν) 6= L. Thus Supp(γ)
satisfies normalizer condition.
Corollary 2. Let G be a group, γ be a fuzzy group on G. If γ satisfies a
normalizer condition, then γ is locally nilpotent.
Proof. Indeed, Theorem 2 shows that Supp(γ) satisfies a normalizer
condition. Then Supp(γ) is locally nilpotent [6]. An application of Propo-
sition 5 shows that γ is locally nilpotent.
Let G be a group, γ be a fuzzy group on G, and suppose that γ
satisfies a normalizer condition. If κ is a proper fuzzy subgroup of γ,
then κ1 = Nγ(κ) 6= κ. By Proposition 4 κ is a normal fuzzy subgroup
of κ1. Suppose that κ1 6= γ. Since γ satisfies a normalizer condition,
κ2 = Nγ(κ1) 6= κ1, so that κ1 is a proper normal fuzzy subgroup of κ2.
Using the same arguments, we construct an ascending series
κ = κ0 E κ1 E . . . E κα E κα+1 E . . . E κβ = γ
L. A. Kurdachenko, K. O. Grin, N. A. Turbay 35
where κα+1 = Nγ(κα) and κσ = ∪τ<σκτ whenever σ is a limit ordinal,
for all α, σ < β.
Let G be a group, γ be a fuzzy group on G. A fuzzy subgroup κ of γ
is called an ascendant subgroup of γ, if there exists an ascending series
κ = κ0 E κ1 E . . . E κα E κα+1 E . . . E κβ = γ.
The above arguments shows that if γ satisfies a normalizer condition, then
every fuzzy subgroup of γ is ascendant. We can obtain now a following
characterization of fuzzy group satisfying a normalizer condition.
Theorem 3. Let G be a group, γ be a fuzzy group on G. Then γ satisfies a
normalizer condition if and only if every fuzzy subgroup of γ is ascendant.
Proof. Let κ be an arbitrary fuzzy subgroup of γ and assume that κ is
ascendant in γ. Let
κ = κ0 E κ1 E . . . E κα E κα+1 E . . . E κβ = γ.
be an ascending series between κ and γ. In particular, κ is normal in
κ1 6= κ. It follows that there exists an element x such that κ1(x) > κ(x).
Then κ does not include χ(x, κ1(x)). By Proposition 4 χ(x−1, κ1(x))⊚κ⊚
χ(x, κ1(x)) = κ, so that χ(x, κ1(x)) ⊆ Nγ(κ) and Nγ(κ) 6= κ. Conversely,
if γ satisfies a normalizer condition, then we have seen above that every
fuzzy subgroup of γ is ascendant in γ.
References
[1] J.N. Mordeson, K.R. Bhutani, A. Rosenfeld, Fuzzy Group Theory, Springer: Berlin,
2005.
[2] K.C. Gupta, B.K. Sarma, Nilpotent fuzzy groups,Fuzzy Sets and Systems, N.101,
1999, pp.167-176.
[3] J.G. Kim, Commutative fuzzy sets and nilpotent fuzzy groups, Information Sciences,
N.83, 1995, pp.161-174.
[4] M. Asaad, S. Abou - Zaid, Fuzzy subgroups of nilpotent groups, Fuzzy Sets and
Systems, N.24, 1993, pp.321-323.
[5] L.A. Kurdachenko, K.O. Grin, N.A. Turbay, On hypercentral fuzzy groups, Algebra
and Discrete Mathematics, Vol. 13, N.1, 2012, pp.92-106.
[6] B.I. Plotkin, To the theory of locally nilpotent groups, Doklady AN USSR, N.76,
1951, pp.639-642.
36 On normalizers in fuzzy groups
Contact information
L. A. Kurdachenko Department of Algebra, Oles Honchar
Dnipropetrovsk National University, 72 Gagarin
Av., Dnepropetrovsk, Ukraine 49010
E-Mail: lkurdachenko@i.ua
K. O. Grin Department of Algebra, Oles Honchar
Dnipropetrovsk National University, 72 Gagarin
Av., Dnepropetrovsk, Ukraine 49010
E-Mail: k.grin@mail.ru
N. A. Turbay Department of Algebra, Oles Honchar
Dnipropetrovsk National University, 72 Gagarin
Av., Dnepropetrovsk, Ukraine 49010
E-Mail: nadezhda.turbay@gmail.com
Received by the editors: 18.01.2013
and in final form 18.01.2013.
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