International Journal of Fuzzy Mathematics and Systems.
ISSN 2248-9940 Volume 2, Number 2 (2012), pp. 171-177
© Research India Publications
http://www.ripublication.com
Anti Q-Fuzzy Right R -Subgroup of Near-Rings with
Respect to S-Norms
1
G.Subbiah and 2R.Balakrishnan
1
Associate Professor, Department of Mathematics, Sri K.G.S Arts College,
Srivaikuntam-628619.Tamilnadu, India
2
Associate Professor, PG & Research Department of Mathematics V.O.C College
Thoothukudi-628008.Tamilnadu India
E-mail: [email protected], [email protected]
Abstract
In this paper, we introduce the notion of anti Q- fuzzification of right Rsubgroups in a near-ring and investigate some related properties.
Characterization of anti Q- fuzzy right R-subgroups and onto homomorphic
image of anti Q- fuzzy right R- subgroup with the inf property with respect to
a s-norm are given.
Index Terms: Q- fuzzy set, Imaginable, inf property, anti Q- fuzzy right Rsubgroups, s-norm.
Mathematics Subject Classification: 03F055, 03E72.
Introduction
The theory of fuzzy sets which was introduced by Zadeh [6] is applied to many
mathematical branches. Abou-zoid [1] , introduced the notion of a fuzzy sub near-ring
and studied fuzzy ideals of near-ring. This concept discussed by many researchers
among cho, Davvaz, Dudek, Jun, Kim [2],[3],[4]. In [5], considered the intuitionistic
fuzzification of a right (resp left ) R- subgroup in a near-ring. Also cho.at.al in [4] the
notion of normal intuitionistic fuzzy R- subgroup in a near-ring is introduced and
related properties are investigated. The notion of intuitionistic Q- fuzzy semi primality
in a semi group is given by Kim [3]. A.Solairaju and R.Nagarajan introduced the
concept of Structures of Q- fuzzy groups [7] . In this paper, We introduce the notion
of anti Q- fuzzification of right R- subgroups in a near ring and investigate some
related properties. Characterization of anti Q- fuzzy right- subgroups with respect to
S-norm are given.
172
G. Subbiah and R. Balakrishnan
Preliminaries
Definition 2.1: A non empty set with two binary operations ‘+’ and ‘.’ is called a
near-ring if it satisfies the following axioms
• ( R,+ ) is a group.
• ( R,. ) is a semi group.
• x . (y+z) = x .y + x . z for all x,y,z ε R. Precisely speaking it is a left near-ring.
Because it satisfies the left distributive law
•
•
•
As R – subgroup of a near- ring ‘R’ is a subset ‘H’ of ‘R’ such that
( H , + ) is a subgroup of ( R, + ).
RH ⊂ H
HR ⊂ H. If ‘H’ satisfies (i) and (ii) then it is called left R- subgroup of ‘R’ and if
‘H’ satisfies (i) and (iii) then it is called a right R- subgroup of ‘R’. A map f :
R→ S is called homomorphism if f(x+y) = f(x) + f (y) for all x,y in R.
Definition 2.2: Let Q and G be a set and a group respectively. A mapping
μ: G × Q → [0,1] is called a Q – fuzzy set.
Definition 2.3: Let ‘R’ be a near ring. A fuzzy set ‘μ’ in R is called anti fuzzy sub
near ring in ‘R’ if (i) μ(x+y) ≤ Max { μ(x) , μ(y) } (ii) μ(xy) ≤ Max { μ(x) , μ(y) } for
all x,y in R.
Definition 2.4: A ‘Q’-fuzzy set ‘μ’ is called a anti Q-fuzzy right R- subgroup of R
over Q if ‘μ’ satisfies (AQFS1) μ(x+y,q ) ≤ S { μ(x,q ) , μ(y,q) } (AQFS2) μ(xr, q )
≤ μ(x q). (AQFS3) A(0,q ) = 1
Definition 2.5: By a s- norm ‘S’ , we mean a function S: [0,1]× [0,1]→ [0,1]
satisfying the following conditions
(S1) S(x ,0) = x
(S2) S(x,y) ≤ S(x,z) if y ≤ z
(S3) S(x,y) = S(y,x)
(S4) S(x, S(y,z) ) = S(S(x,y),z), for all x,y,z ε [0,1].
Definition 2.6: Define
Sn (x1,x2, , xn) = S (xi, Sn-1(x1,x2, xi-1, xi+1,xn)) for all 1 ≤i ≤n, n ≥ 2, S1 = S . Also
define S∞ (x1,x2, ) = lim Sn (x1,x2, ,xn) as n → ∞.
Definition 2.7: By the union of Q-fuzzy subsets A1 and A2 in a set X with respect to
s-norm S we mean the Q-fuzzy subset A = A1U A2 in the set X such that for any x ε X
A(x,q) = (A1UA2) (x,q) = S (A1(x,q), A2(x,q)). By the union of a collection of Q-fuzzy
subsets { A1,A2,… } in a set X with respect to a s- norm S we mean the Q-fuzzy
subset UAi such that for any x ε A, (UAi)(x,q) = S∞(A1(x,q), A2(x,q),}.
Anti Q-Fuzzy Right R -Subgroup of Near-Rings
173
Definition 2.8: By the direct product of fuzzy sets { A1,A2,} with respect to s- norm S
we mean the Q-fuzzy subset A = ∏Ai such that
A(x1,x2,.xn}q = (∏Ai) {(x1,x2,xn )q} = Sn(A1(x1,, q), A2(x2, q) . An(xn, q)}
Proposition 2.1: For a S-norm , then the following statement holds S(x,y) ≥
max{x,y}, for all x,y ε [0,1]
Properties of Q- Fuzzy Left R- Subgroups
Proposition 3.1: Let ‘S’ be a s- norm. Then every imaginable anti Q- fuzzy right Rsubgroup ‘μ’ of a near ring ‘ R’ is anti Q- fuzzy right R-subgroup of R.
Proof: Assume ‘μ’ is imaginable Q- fuzzy right R- subgroup of ‘R’, then we have
μ (x+y , q) ≤ S { μ(x,q), μ(y,q) }and μ (xr, q) ≤ μ (x,q) for all x,y in R.
Since ‘μ’ is imaginable, we have
max { μ(x,q) , μ(y,q) } = S { max { μ(x,q) , μ(y,q), max { μ(x,q) , μ(y,q) } }
≥ S ( μ(x,q) , μ(y,q) )
≥ max { μ(x,q) , μ(y,q) }
And so S( μ(x,q) , μ(y,q) ) = max { μ(x,q) , μ(y,q) } . It follows that
μ(x+y, q ) ≤ S( μ(x,q) , μ(y,q) )= max { μ(x,q) , μ(y,q) } for all x,y ε R.
Hence ‘μ’ is anti Q- fuzzy right R- subgroup of R
Proposition 3.2: If ‘μ’ is anti Q- fuzzy right R- subgroups of a near ring ‘R’ and ‘Ө’
is an endomorphism of R, then μ[Ө] is anti Q- fuzzy right R- subgroup of ‘R’.
Prof: For any x,y ε R, we have
μ[Ө] ( x+y , q )
= μ ( Ө(x+y , q ) )
= μ ( Ө(x,q ) , Ө(y,q) )
≤ S { μ( Ө(x,q)), μ( Ө(y,q) ) }
= S { μ[Ө] (x,q), μ[Ө] (y,q ) }
μ[Ө] (xr , q )
= μ( Ө (xr, q )
≤ μ ( Ө(x,q) )
≤ μ [Ө] (x,q)
Hence μ[Ө] is anti Q- fuzzy right R- subgroup of R
Proposition 3.3: An onto homomorphism’s of anti Q- fuzzy right R- subgroup of
near ring ‘R’ is anti Q- fuzzy right R- subgroup.
Proof: Let f : R→ R1 be an onto homomorphism of near rings and let ‘λ’ be anti Qfuzzy right R- subgroup of R1 and ‘μ’ be the pre image of ‘λ’ under ‘f’, then we have
μ(x+y , q) = λ ( f(x+y , q ))
= λ ( f(x,q) , f(y,q) )
≤ S ( λ (f(x,q)) , λ(f(y,q)) )
174
G. Subbiah and R. Balakrishnan
μ(xr,q)
≤ S (μ(x,q) , μ(y,q) )
= λ(f (xr,q))
≤ λ (f(x,q) )
= μ(x,q).
Proposition 3.4: An onto homomorphic image of anti Q- fuzzy right R- subgroup
with the inf property is anti Q- fuzzy right R- subgroup.
Prof: Let f: R→R1 be an onto homomorphism of near rings and let ‘μ’ be a inf
property of anti Q-fuzzy right R- subgroups of ‘R’
Let x1, y1 ε R1 , and x 0 ε f-1(x1) , y 0 ε f-1(y1) be such that
μ(x 0, q ) = inf μ(h,q)
, μ(y 0,q) = inf μ(h,q)
(h,q)εf-1(x1)
(h,q)εf-1(y1)
Respectively, then we can deduce that
μ ( x1+y1, q ) = inf μ(z,q)
(z,q) ε f-1(x1+y1,q)
≤ max { μ(x0,q) , μ(y0,q)
= max
inf μ(h ,q) ,
(h,q)εf-1(x1,q)
f
inf μ(h,q)
(h,q)εf-1(y1,q)
= max { μf(x1,q) , μf(y1,q) }
μf(xr,q) = inf μ(z,q)
(z,q)εf-1(x1r1, q)
≥
=
μ(y 0 , q)
inf μ(h,q)
(h,q)εf-1(y1,q)
= μf(y1,q) . Hence ‘μf’ is anti Q- fuzzy right R- subgroup of R1
Proposition 3.5: Let ‘S’ be a continuous s-norm and let ‘f’ be a homomorphism on a
near ring ‘R’ . If ‘μ’ is anti Q- fuzzy right R- subgroup of R, then μf is anti Q- fuzzy
right R- subgroup of f(S)
Proof: Let A1 = f -1(y1,q) , A2 = f -1(y2,q) and A12 = f -1(y1+y2 , q) where y1,y2 ε f(S),
qε Q consider the set
A1-+A2 = { x ε S / (x,q) = (a1,q) + (a2,q) } for some (a1,q) εA1 and (a2,q) ε A2
If (x,q) ε A1+A2 , then (x,q) = (x1,q) + (x2,q) for some (x1,q) ε A1 and (x2,q) ε A2
so that we have
f (x,q) = f(x1,q) - f(x2,q)
= y1- y2
(ie) (x,q) ε f-1((y1,q) + (y2,q)) = f-1(y1+y2, q) = A12. Thus A1+A2 c A12.
Anti Q-Fuzzy Right R -Subgroup of Near-Rings
It follows that
μf(y1+y2, q) =
=
≤
≤
≤
175
inf { μ(x,q)/ (x,q) ε f-1((y1,q)+(y2,q))}
inf{ μ(x,q) / (x,q) ε A12 }
inf { μ(x,q)/ (x,q) ε A1-A2}
inf { μ((x1,q)- (x2,q) ) / (x1,q) ε A1 and (x2,q) ε A2}
inf { T(μ(x1,q) , μ(x2,q))/ (x1,q) ε A1 and (x2,q) ε A2}
Since ‘S’ is continuous. For every ε > 0 , we see that if
inf { μ(x1,q) / (x1,q) ε A1} + (x1*, q) } ≥ δ and
inf { μ(x2,q) / (x2,q) ε A2} + (x2*,q) } ≥ δ
S{inf{μ(x1,q) / (x1,q) ε A1} , inf { μ(x2,q) / (x2,q) ε A2 } + S ((x1*,q), (x2*,q) ≥ ε
Choose (a1,q) ε A1 and (a2,q) ε A2 such that
inf { μ(x1,q) / (x1.q) ε A1 } + μ(a1,q) ≥ δ and
inf { μ(x2,q) / (x2,q) ε A2} + μ(a2,q) ≥ δ. Then we have
S{inf{ μ(x1,q) / (x1,q) ε A1}, inf { μ(x2,q) / (x2,q) ε A2 } + S(μ(a1,q), μ(a2,q) ≥ ε
consequently, we have μf(y1+y2, q ) ≤ inf { S(μ(x1,q), μ(x2,q)) / (x1,q) ε A1 ,(x2,q) ε
A2}
≤ S (inf{μ(x1,q) / (x1,q) ε A1}, inf{μ(x2,q) / (x2,q)εA2}
≤ S (μf(y1,q) , μf(y2,q) }
Similarly we can show μf(xr,q) ≤ μf(y,q). Hence ‘μf’ is anti Q- fuzzy right Rsubgroup of ‘f(R)’.
Proposition 3.6: If R is a near ring with identity and a s-norm S for all x ε [0,1]
satisfies the condition S(x,x)=x.
Then condition (AQFS2) in Definition (2.4) for any r ε R is equivalent to the
condition
A(xr,q) = A(x,q).
(AQFS21)
Proof: Let condition (AQFS1 ) and (AQFS2 ) be fulfilled and 1 be the identity
element in the near ring R. Then
A(x,q) = A(xr + (1- xr),q ) ≤ S (A(xr,q) , A(1-xr,q)) ≤ S (A(xr, S(A(x,q), A(-xr,q)))
≤ S(A(x,q), S(A(x,q), A(-xr))). Taking into consideration conditions (S2) and (S5)
for the s-norm S and again applying ( AQFS2), we obtain S(A(x,q), S(A(,q), A(xr,q))) = S(S(A(x,q), A(x,q),A(-xr,q)) = S (A(x,q), A(-xr,q)) ≤ S (A(x,q), A(xr,q)).
Thus we have A(x,q) = S(A(x,q), A(x,q)) ≤ S (A(x,q), A(xr,q)).From here, using
condition (S4), we conclude that A(x,q) ≤ A(xr,q)
(1)
From (1) and condition (AQFS2) we obtain (AQFS21).
Proposition 3.7: Let A : B→ [0,1] be the characteristic function of a subset B is
contained in R and R be an R-subgroup. Then A is anti Q-fuzzy right R-subgroup of
R with respect to a s-norm S if and only if B is a subgroup of R.
176
G. Subbiah and R. Balakrishnan
Proof: Let A be anti Q-fuzzy R- sub group of R with respect to S. Then, according to
(AQFS2) , A(xr,q) ≤ A(x,q) = 1. Hence xr ε B. Finally, according to condition
(AQFS3), A(0,q) = 1. Therefore, 0ε B. Thus, B is a subgroup of R.
Conversely, Let B be a sub group of R. Then for any x,y ε R,
A(x+y, q) ≤ S(A(x,q), A(y,q)}.
Indeed, for any x,y ε B, A(x+y,q) = 1 ≤ 1 = S(1,1) = S (A(x,q), A(y,q)}
For any x ε B, and y is not in B, S(A(x, q), A(y, q)) = S(1.0) = 0 ≤ A(x+y, q)
For any x is not in B, and y ε B, S(A(x,q), A(y,q)) = S(0,1) = 0 ≤ A(x+y, q)
Finally, for any x,y does not belong to B, S(A(x, q), A(y, q)) = S(0,0) = 0 ≤ A(x+y, q)
Further for all x ε R, and rε R, we have A(xr, q) ≤ A(x, q). Indeed, for all x ε B we
have xr ε B, hence A(xr, q) = 1 ≤ A(x, q), and for all x does not belong to B we have
A(x,q) = 0 ≥ A(x, q).
Finally, since 0 ε B, we have A(0,q) = 1. Therefore, A is anti Q-fuzzy right Rsubgroup of R with respect to S.
Proposition 3.8: The Union of any collection of anti Q-fuzzy R- sub group of R is
anti Q-fuzzy R-subgroup of R.
Proof: For all x,y ε R, and any r ε R, we have
UAi (x+y)q = S∞(A1(x+y)q, A2(x+y)q, …)
≤ S∞(S∞(A1(x,q), A1(y,q)), S∞(A2(x,q), A2(y,q)), …)
= S∞ (S∞(A1(x,q), A2(x,q), , …), S∞(A1(y,q), A2(y,q), …))
= S∞ ((UAi)(x,q) , (UAi)(y,q));
(UAi)(xr,q) = S∞A1(xr,q),A2(xr,q), , ) ≤ S∞(A1(x,q), A2(x,q),) = (UAi) (x,q);
(UAi) (0,q) = S∞(A1(0,q) , A2(0,q) ,) = S∞ (1,1,) = 1.
Proposition 3.9: Let { R1,R2,,Rn} be a collection of R-subgroups and R = ∏ Ai be its
direct product.Let {A1,A2,,An} be anti Q-fuzzy right subgroups of R –subgroups {
R1,R2,, Rn} with respect to a s-norm S. Then A = ∏Ai is anti Q-fuzzy right Rsubgroup of R with respect to the s-norm S.
Proof: Let x,y ε R, x = (x1,x2,…xn), and y = (y1,y2,…yn). Also let r ε R. Then
A(x+y, q) = A (x1+y1, x2+y2, …, xn+yn)q
= Sn(A1(x1+y1)q, A2(x2+y2)q, , An(xn+yn)q)
≤ Sn( S (A1(x1, q),A1(y1, q)),S(A2(x2, q), A2(y2, q)),S(An(xn, q), An(yn, q)))
= S (Sn(A1(x1, q), A2(x2, q),…An(xn, q)), Sn(A1(y1, q), A2(y2, q), A(yn, q)))
= S (A(x,q), A(y,q)),
A(xr,q) = A(x1r,x2r,…,xnr)q
= Sn(A1(x1r), A2(x2r), …, An(xnr)q)
≤ Sn(A1(x1,q) A2(x2, q) ,… An(xn, q)),
A(0, q) = A (01,02,…, 0n)q = Sn((A1(01,q), A2(02,,q) …, An(0n, q))
= Sn (1,1,…,1) = 1. Therefore, A is anti Q-fuzzy R- subgroup of R with
Anti Q-Fuzzy Right R -Subgroup of Near-Rings
177
respect to S.
Conclusion
Y.U. Cho, Y.B.Jun, investigated the concept On intuitionistic fuzzy R- subgroup of
near rings. Osman kazanci , Sultanyamark and Serifeyilmaz introduced the
intutionistic Q- fuzzy R-subgroups of near rings. In this paper we investigate the
notion of anti Q- fuzzy right R- subgroup of near ring w.r.t S-norm and
characterization of them.
References
[1] S. Abou-Zoid , “ On Fuzzy sub near rings and ideals” , Fuzzy sets. Syst. 44
(1991) , 139-146.
[2] Y.U. Cho, Y.B.Jun, “ On Intuitionistic fuzzy R- subgroup of near rings” , J.
Appl. Math. And computing , 18 (1-2) (2005), 665-677.
[3] K.H.Kim , Y.B. Jun, “On Fuzzy R- subgroups of near rings, J. fuzzy math 8
(3) (2000) 549-558.
[4] K.H.Kim, Y.B.Jun, “ Normal Fuzzy R- subgroups of near rings , J. fuzzy sets.
Syst.121(2001) 341-345.
[5] Osman Kazanci, Sultan Yamark and Serife Yimaz “ On intuitionistic Qfuzzy R- subgroups of near rings, International mathematical forum , 2, 2007 ,
59 , (2899-2910)
[6] A.Solairaju and R.Nagarajan, “Q- fuzzy left R- subgroup of near rings w.r.t
T- norms,” Antarctica Journal of Mathematics,5 , no.2(2008) , 59-63.
[7] A.Solairaju and R.Nagarajan, “A New structure and construction of Q- fuzzy
groups”, Advances in Fuzzy Mathematics, 4, No.1(2009), 23-29.
[8] L.A.Zadeh, Fuzzy set, inform. control 8 (1965) 338-353.