intuitionistic m-fuzzy groups - Soochow Journal of Mathematics

SOOCHOW JOURNAL OF MATHEMATICS
Volume 30, No. 1, pp. 85-90, January 2004
INTUITIONISTIC M-FUZZY GROUPS
BY
JIANMING ZHAN AND ZHISONG TAN
Abstract. In this paper, we introduce the concept of intuitionistic M-fuzzy groups
and obtain some results.
1. Introduction and Preliminaries
After the introduction of the concept of fuzzy sets by Zadeh, several researches
were conducted on the generalizations of the notion of fuzzy sets. The
idea of “intuitionistic fuzzy set” was first published by Atanassov[1,2], as a generalization
of the notion of fuzzy sets. Rosenfeld[7] introduced the concept of a
fuzzy group and Wu[6] studied the fuzzy normal subgroup. Gu[3] put forward the
notion of the fuzzy groups with operators. In this paper, we introduce the concept
of intuitionistic fuzzy group with operators and obtain some related results.
For the sake of convenience, we set out the former concepts.
Definition 1.1.([7]) A fuzzy set A of a group G is called a fuzzy subgroup
if (1) A(xy) ≥ A(x) ∧ A(y); (2) A(x −1 ) ≥ A(x) for all x, y ∈ G.
Definition 1.2.([4]) A group with operators is an algebraic system consisting
of a group G ,asetM and a function defined in the product set M × G and
having values in G such that, if ma denotes the element in G determined by the
element a of G and the element m of M, thenm(ab) =(ma)(mb) holdsforall
a, b ∈ G and m ∈ M. We shall use the phrases “G is an M-group” to a group
with operators.
Received February 25, 2003; revised May 22, 2003.
AMS Subject Classification. 22F05, 06F10, 03E72, 04A72.
Key words. intuitionistic M-fuzzy groups, homomorphism, M-homomorphism.
85

86 JIANMING ZHAN AND ZHISONG TAN
Definition 1.3.([4]) A subgroup A of an M-group G is said to be an M-
subgroup if mx ∈ A for all m ∈ M and x ∈ A.
Definition 1.4.([4]) Let G and G ′ be M-groups, f be a homomorphism from
G onto G ′ . If f(mx) =mf(x) for all m ∈ M and x ∈ G, thenf is called an
M-homomorphism.
An intuitionistic fuzzy set (briefly, IFS) A in a nonempty set X is an object
having the form
A = {(x, α A (x),β A (x))|x ∈ X}
where the functions α A : X → [0, 1] and β A : X → [0, 1] denote the degree of
membership and the degree of nonmembership, respectively, and
0 ≤ α A (x)+β A (x) ≤ 1, ∀x ∈ X.
An intuitionistic fuzzy set A = {(x, α A (x),β A (x))|x ∈ X} in X can be identified
to an ordered pair (α A ,β A )inI X × I X . For the sake of simplicity, we shall use
the symbol A =(α A ,β A )fortheIFS A = {(x, α A (x),β A (x))|x ∈ X}.
2. Intuitionistic M-Fuzzy Subgroups
Definition 2.1. An IFS A=(α A ,β A ) in a group G is called an intuitionistic
fuzzy subgroup of G if
(i) α A (xy) ≥ α A (x) ∧ α A (y) andβ A (xy) ≤ β A (x) ∨ β A (y)
(ii) α A (x −1 ) ≥ α A (x) andβ A (x −1 ) ≤ β A (x)
for all x, y ∈ G.
Definition 2.2. Let G be an M-group and IFS A=(α A ,β A )beanintuitionistic
fuzzy subgroup of G. If α A (mx) ≥ α A (x) andβ A (mx) ≤ β A (x) for all
x ∈ G and m ∈ M, thenIFS A=(α A ,β A )issaidtobeanintuitionisticfuzzy
subgroup with operators of G. We use the phrase “A =(α A ,β A )isanintuitionistic
M-fuzzy subgroup of G” instead of an intuitionistic fuzzy subgroup with
operators of G.
Example 2.3. Let A be an M-subgroup of an M-group G and let IFS
A=(α A ,β A ) be an intuitionistic fuzzy set in G defined by
⎧
⎧
⎨0.7 if x ∈ A
⎨0.2 if x ∈ A
α A (x) =
β
⎩
A (x) =
0.2 otherwise
⎩
0.6 otherwise

INTUITIONISTIC M-FUZZY GROUPS 87
for all x ∈ X. Then it is easy to verify that IFS A=(α A ,β A ) is an intuitionistic
M-fuzzy subgroup of G.
Example 2.4. Let G be an M-group while A is a nonempty subset of
G. If χ A is the characteristic function of A, thenA is an M-subgroup of G if
and only if IFS A=(χ A , χ A ) is an intuitionistic M-fuzzy subgroup of G, where
χ A (x) =1− χ A (x) for all x ∈ G.
Proposition 2.5. If IFS A=(α A ,β A ) is an intuitionistic M-fuzzy subgroup
of an M-group G, then for any x, y ∈ G and m ∈ M.
(i) α A (m(xy)) ≥ α A (mx) ∧ α A (my) and β A (m(xy)) ≤ β A (mx) ∨ β A (my)
(ii) α A (mx −1 ) ≥ α A (x) and β A (mx −1 ) ≤ β A (x).
Theorem 2.6. Let A be an M-subgroup of an M-group G and let IFS
A=(α A ,β A ) be an intuitionistic fuzzy set in G defined by
⎧
⎨
α A (x) =
⎩
t 1
t 2
⎧
if x ∈ A
⎨s 1 if x ∈ A
β A (x) =
otherwise
⎩
otherwise
for all x ∈ X, t 1 > t 2 and s 1 < s 2 in [0, 1]. Then IFS A=(α A ,β A ) is an
intuitionistic M-fuzzy subgroup of G.
Proof. Assume that A is an M-subgroup of G. Letx, y ∈ G. Ifx and y are
in A, thenα A (x) =t 1 = α A (y) andxy ∈ A. Hence α A (xy) =t 1 = α A (x)∧α A (y).
If either x ∉ A or y ∉ A, then either α A (x) =t 2 or α A (y) =t 2 . It follows that
α A (x∗y) ≥ t 2 ≥ α A (x)∧α A (y). Similarly, we have β A (x∗y) ≤ β(x)∨β A (y). Now,
let x ∈ G. Ifx ∈ A, thenα A (x) =t 1 and x −1 ∈ A. Thusα A (x) =t 1 = α A (x −1 ).
If x ∉ A, thenα A (x) =t 2 and x −1 ∉ A. Thusα A (x −1 )=t 2 = α A (x). Similarly,
we have β A (x −1 )=β A (x). Hence IFS A=(α A ,β A ) is an intuitionistic fuzzy
subgroup of G. Since A is an M-subgroup of G, wehavemx ∈ A for all x ∈ A
and m ∈ M. Therefore α A (mx) =t 1 = α A (x) andβ A (mx) =s 1 = β A (x). If
x ∉ A, thenα A (mx) ≥ t 2 = α A (x) andβ A (mx) ≤ s 2 = β A (x). Consequently,
IFS A=(α A ,β A ) is an intuitionistic M-fuzzy subgroup of G. This completes the
proof.
For any α ∈ [0, 1], the set U(α A ; α) ={x ∈ G|α A (x) ≥ α} (resp. L(α A ; α) =
{x ∈ G|α A (x) ≤ α}) is called an upper (resp. lower) α -level cut of α A .
s 2

88 JIANMING ZHAN AND ZHISONG TAN
Theorem 2.7. Let IFS A=(α A ,β A ) be an IFS in an M-group G. ThenIFS
A=(α A ,β A ) is an intuitionistic M-fuzzy subgroup of G if and only if the nonempty
sets U(α A ; α) and L(β A ; α) are M-subgroups of G for every α ∈ Imα A ∩ Imβ A .
Proof. Let α ∈ Imα A ∩Imβ A ⊆ [0, 1] and x, y ∈ U(α A ; α). Then α A (x) ≥ α
and β A (y) ≥ α. It follows that α A (xy) ≥ α A (x) ∧ α A (y) ≥ α, sothatxy ∈
U(α A ; α). Moreover, if x ∈ U(α A ; α), then α A (x) ≥ α. It follows that α A (x −1 ) ≥
α A (x) ≥ α, and that x −1 ∈ U(α A ; α). Hence U(α A ; α) is a subgroup of G.
Now, let x, y ∈ L(β A ; α), then β A (x) ≤ α and β A (y) ≤ α. It follows that
β A (xy) ≤ β A (x) ∨ β A (y) ≤ α, andthatxy ∈ L(β A ; α). Moreover, if x ∈ L(β A ; α),
then β A (x) ≤ α. It follows that β A (x −1 ) ≤ β A (x) ≤ α, andthatx −1 ∈ L(β A ; α).
Hence L(β A ; α) is a subgroup of G. For any x ∈ U(α A ; α) andm ∈ M, wehave
α A (mx) ≥ α A (x) ≥ α, andthatmx ∈ U(α A ; α). If x ∈ L(β A ; α) andm ∈ M,
we have β A (mx) ≤ β A (x) ≤ α ,andthatmx ∈ L(β A ; α). Thus U(α A ; α) and
L(β A ; α) areM-subgroups of G.
Conversely, assume that the nonempty sets U(α A ; α) andL(β A ; α) areMsubgroups
of G. Ifthereexistx 0 ,y 0 ∈ G such that α A (x 0 y 0 )

INTUITIONISTIC M-FUZZY GROUPS 89
Lemma 2.8. Let G and G ′ be M-groups and f a homomorphism from G
onto G ′ .
(1) If IFS A=(α A ,β A ) is an intuitionistic fuzzy subgroup of G ′ ,thenIFS A f =
(α f A ,βf A
) is an intuitionistic fuzzy subgroup of G.
(2) If IFS A f =(α f A ,βf A
) is an intuitionistic fuzzy subgroup of G, thenIFS
A=(α A ,β A ) is an intuitionistic fuzzy subgroup of G ′ .
Proof. (1) Let x, y ∈ G, wehave
α f A (xy) =α A(f(xy)) = α A (f(x)f(y)) ≥ α A (f(x)) ∧ α A (f(y)) = α f A (x) ∧ αf A (y)
β f A (xy) =β A(f(xy)) = β A (f(x)f(y)) ≤ β A (f(x)) ∨ β A (f(y)) = β f A (x) ∨ βf A (y)
α f A (x−1 )=α A (f(x −1 )) = α A (f(x) −1 ) ≥ α A (f(x)) = α f A (x)
β f A (x−1 )=β A (f(x −1 )) = β A (f(x) −1 ) ≤ β A (f(x)) = β f A (x).
Hence IFS A f =(α f A ,βf A
) is an intuitionistic fuzzy subgroup of G.
(2) For any x, y ∈ G ′ ,thereexista, b ∈ G such that f(a) =x and f(b) =y.
α A (xy) =α A (f(a)f(b)) = α A (f(ab)) = α f A (ab) ≥ αf A (a) ∧ αf A (b) =α A(f(a)) ∧
α A (f(b)) = α A (x) ∧ α A (y); β A (xy) =β A (f(a)f(b)) = β A (f(ab)) = β f A (ab) ≤
β f A (a) ∨ βf A (b) =β A(f(a)) ∨ β A (f(b)) = β A (x) ∨ β A (y); α A (x −1 )=α A (f(a) −1 )=
α A (f(a −1 )) = α f A (a−1 ) ≥ α f A (a)=α A(f(a))=α A (x) andβ A (x −1 )=β A (f(a) −1 )=
β A (f(a −1 )) = β f A (a−1 ) ≤ β f A (a) =β A(f(a)) = β A (x). Hence IFS A=(α A ,β A )is
an intuitionistic fuzzy subgroup of G ′ .
Theorem 2.9. Let G and G ′ be M-groups and f an M-homomorphism from
G onto G ′ .IfIFS A=(α A ,β A ) is an intuitionistic M-fuzzy subgroup of G ′ ,then
IFS A f =(α f A ,βf A
) is an intuitionistic M-fuzzy subgroup of G.
Proof. By Lemma 2.8, IFS A f =(α f A ,βf A
) is an intuitionistic fuzzy subgroup
of G. For any x ∈ G and m ∈ M,
α f A (mx)=α A(f(mx)) = α A (mf(x)) ≥ α A (f(x)) = α f A (x);
β f A (mx)=β A(f(mx)) = β A (mf(x)) ≤ β A (f(x)) = β f A (x).
Hence IFS A f =(α f A ,βf A
) is an intuitionistic M-fuzzy subgroup of G.

90 JIANMING ZHAN AND ZHISONG TAN
Theorem 2.10. Let G and G ′ be M-groups and f an M-homomorphism
from G onto G ′ .IfIFS A f =(α f A ,βf A
) is an intuitionistic M-fuzzy subgroup of
G, thenIFS A=(α A ,β A ) is an intuitionistic M-fuzzy subgroup of G ′ .
Proof. By Lemma 2.8, IFS A=(α A ,β A ) is an intuitionistic fuzzy subgroup
of G ′ . For any y ∈ G ′ and m ∈ M,
α f A (my)=
and
β f A (my)=
sup α A (z) ≥ sup α A (mx) (wherex ∈ f −1 (y))
z∈f −1 (my)
mx∈f −1 (my)
= sup α A (mx) =
f(mx)=my
inf β A(z) ≤
z∈f −1 (my)
= inf
f(mx)=my β A(mx) =
sup α A (mx) ≥
mf(x)=my
inf β A(mx)
mx∈f −1 (my)
inf β A(mx) ≤
mf(x)=my
sup α A (x) =α f A (y)
f(x)=y
(where x ∈ f −1 (y))
inf β A(x) =β f A (y).
f(x)=y
Hence IFS A=(α A ,β A ) is an intuitionistic M-fuzzy subgroup of G ′ .
Acknowledgments
The authors are very grateful to the referees for their advice.
References
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Systems, 61(1994), 137-142.
[3] W.X.Gu,S.Y.LiandD.G.Chen,Fuzzy groups with operators, Fuzzy Sets and Systems,
66 (1994), 363-371.
[4] N. Jacobson, Lectures in Abstract Algebras, East-West Press, 1951.
[5] Y.B.Jun,Fuzzy BCK/BCI-algebras with operators, Sci. Math., 3(2000), 283-287.
[6] W.M.Wu,Normal fuzzy subgroups, Fuzzy Math., 1 (1981), 21-23.
[7] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35(1971), 512-517.
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Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei Province, 445000,
P.R. China.
E-mail: zhanjianming@hotmail.com