On Q Fuzzy R- Subgroups of Near - Rings

Transcription

1 International Mathematical Forum, Vol. 8, 2013, no. 8, On Q Fuzzy R- Subgroups of Near - Rings Mourad Oqla Massa'deh Department of Applied Science, Ajloun College Al Balqa' Applied University Jordan Moradoqla2000@yahoo.com Abstract In this paper, we shall study Q fuzzy characteristic right ( resp. left ) R- subgroups and the Q fuzzy same type right (resp. left) R subgroups of a near ring R. And give its characterizations. Mathematics Subject Classification: 20N25, 03E72 Keywords: Fuzzy sets, Q fuzzy set, Q fuzzy groups, Q fuzzy right (resp. left) R- subgroups, Q fuzzy characteristic right (resp. left) R subgroup, Q fuzzy same type right (resp. left) R subgroups 1. INTRODUCTION R subgroups of a near ring is introduced by S. Abou Zaid [8], and present authors [3,4] investigated further properties of fuzzy right (resp.left) R- subgroups of near ring R. In [9], considered the intuitionistic fuzzification of a right (resp.left) R subgroup in a near ring. A. Solairaju and R. Nagarajan [1] introduced the new structures of Q fuzzy groups and then they investigate the notion upper Q fuzzy index with upper Q- fuzzy subgroups in [7]. On the other hand Osman Kazanci in [6] the notion of intuitionistic Q- fuzzy R- subgroup in a near ring is introduced and related properties are investigated. The aim of this paper is to formulate Q- fuzzy characteristic right (resp.left ) R- subgroups and the Q- fuzzy same type right (resp.left) R subgroups of a near ring R. Characterizations of Q fuzzy R- subgroups are given. 2. PRELIMINARY 2.1Definition:[2] A near ring is defined to be a non empty set R with two binary operations "+" and "." satisfying the following axioms:

2 388 Mourad Oqla Massa'deh i- ( R, +) is a group. ii- (R,.) is a semi group. iii- x. (y + z) = x.y + x. z ; for all x, y,z R. It is called a left near ring by (iii). In this paper, we will use the word "near ring". 2.2Remark: 1- We denote xy instead of x.y 2- x0 = 0 and x ( -y ) = -xy but in general 0x 0 for some x R. 2.3Definition: A two sided R subgroup of a near ring R is a subset H of R such that: 1- (H, +) is a subgroup of (R, +) 2- RH H 3- HR H If H satisfies 1 and 2 then it is called left R- subgroup of R, while if H satisfies 1 and 3 then it is called a right R subgroup of R. 2.4Definition:[5] A fuzzy set μ in a set R is just a function μ : R [ 0, 1]. 2.5Definition: Let Q and G be a set and a group respectively. A mapping λ : G Q [0,1] is called Q fuzzy set in G. 2.6Definiton: Let ϕ be a mapping from a set R 1 to a set R 2 and let μ be a Q fuzzy set in R 1. Then ϕ(μ) the image of μ is a Q fuzzy set in R 2 defined by ϕ( μ)( y, = For all y R 2. sup 1 x ϕ ( y) 0 μ( x, ; 1 ϕ ( y) Φ ; otherwise 2.7Definition: A Q fuzzy set λ is called Q- fuzzy subgroup of G if the following conditions are satisfied: 1- λ (xy, min { λ(x,, λ(y, } 2- λ(x -1, = λ(x, for all x, y G and q Q. 2.8Definition: Let Im( λ) denote the image set of λ. Let λ be a Q- fuzzy set in a set R. For t [0,1] the set λ t = { x R, q Q; λ(x, t} is called Q- level subset of λ. 2.9Definition: Let R be a near ring. A fuzzy set λ in R is called Q- fuzzy sub near ring in R if. 1- λ(x-y, min{ λ(x,, λ(y, } 2- λ(xy, min { λ(x,, λ(y,} for all x, y R.

3 On Q fuzzy R- subgroups of near rings Remark: If a Q- fuzzy set μ in R satisfies the condition (1) of Definition 2.8, then μ(0, μ(x, for all x R. 3. Q FUZZY CHARACTERISTIC AND Q FUZZY SAME TYPE R SUBGROUPS. 3.1Definition: A Q fuzzy set μ in R is called Q fuzzy right (resp. Q left) R- subgroup of R if. 1- μ is a Q- fuzzy subgroup of (R, +). 2- μ(xr, μ(x, q ) ( resp. μ(rx, q ) μ(x, q )) for all r, x R. 3.2Remark: 1- A Q- fuzzy set μ in R is a Q- fuzzy right (resp. Q- left) R- subgroup of R iff the level subset μ t is a right (resp. left) R subgroup of R, which is called a Q- level right ( resp. left) R- subgroup of R, where t Im(μ). 2- If μ is a Q- fuzzy set in R and ϕ is an endomorphism of R, we define a new fuzzy set μ ϕ in R by: μ ϕ (x, = μ( ϕ(x, ) = μ(ϕ(x), for all x R. 3.3Proposition: Let ϕ be an endomorphism of R. If μ is a Q fuzzy right (resp. left) R subgroup, the so isμ ϕ. Let x, y R. Then μ ϕ (x-y, = μ(ϕ(x-y), = μ(ϕ(x) - ϕ(y), min {μ(ϕ(x),, μ(ϕ(y), } = min {μ ϕ (x,, μ ϕ (y, } and for every r, x R we have. μ ϕ (xr, =μ(ϕ(xr), = μ(ϕ(x) ϕ(r), μ(ϕ(x), = μ ϕ (x,, resp,( μ ϕ (rx, =μ(ϕ(rx), = μ(ϕ(r) ϕ(x), μ(ϕ(x), = μ ϕ (x,. Hence μ ϕ is Q fuzzy right (resp. left) R subgroup of R. 3.4Definition: A Q fuzzy right (resp. left) R subgroup μ of R is said to be Q fuzzy characteristic if μ ϕ (x, = μ (x, for all x R, q Q and ϕ Aut(R). For a family of fuzzy sets { μ i, i Δ}, then μ is defined by Δ i μi ( x) = inf { μi ; i Δ } 3.5Propostion: If { μ i, i Δ} is a family of Q- fuzzy characteristic right is a Q fuzzy μ (resp. left) R subgroups of a near ring R, then i Δ Characteristic right (resp. left) R subgroup of R. i i Δ i

4 390 Mourad Oqla Massa'deh is Q fuzzy right (resp. left) R subgroup of R its clear μi i Δ Since Now, let x R, q Q and ϕ Aut (R). Then ϕ μi ( x, = μi ( ϕ( x, ) = μi ( ϕ( x), i Δ i Δ i Δ = inf { μ i (ϕ(x),, i Δ} = inf { μ ϕ i (x,, i Δ} = inf { μ i (x,, i Δ} = μi ( x, i Δ is a Q- fuzzy characteristic Hence μ i i Δ right (resp. left) R subgroup of R. 3.6Definition: A right ( resp. left) R subgroup H of R is called characteristic right (resp. left) R subgroup of R if ϕ(h) = H for all ϕ Aut (R). 3.7Lemma: Let H be a non empty subset of a near ring R and let μ be a Q fuzzy set in R defined by: α ; x H μ ( x, = β ; x H such that α > β in [0,1]. Then μ is a Q fuzzy right (resp. left) R subgroup iff H is a right (resp.left) R subgroup. 3.8Corollary: If H is a characteristic right (resp.left) R subgroup of R and μ is a Q fuzzy set defined in Lemma 3.7, then μ is a Q fuzzy characteristic right (resp.left) R subgroup of R. Let x R, q Q and ϕ Aut (R). If x H, then ϕ(x) ϕ(h) = H and so μ ϕ (x, = μ (ϕ(x), = α = μ (x,. If x H, we have μ ϕ (x, = μ (ϕ(x), = β = μ (x,, therefore μ is Q fuzzy characteristic. 3.9Lemma: Let μ be a Q- fuzzy right (resp. left) R subgroup of R and let x R, q Q. Then μ (x, = t iff x μ t and x μ s for all s > t in [0,1]. Straight forward ϕ

5 On Q fuzzy R- subgroups of near rings Theorem: Let μ be a Q- fuzzy right (resp. left) R- subgroup of R, then μ is a Q fuzzy characteristic iff each Q level right (resp. left) R subgroup of μ characteristic. ( ) Let t Im(μ),ϕ Aut (R) and x μ t then μ ϕ (x, = μ (x, t, this means that μ (ϕ(x), μ t which show that ϕ (μ t ) μ t on the other hand let x μ t and y R such that ϕ(y) = x. Then μ (y, = μ ϕ (y, = μ (ϕ(y), = μ (x, t, showing that y μ t. Thus x = ϕ(y) ϕ (μ t ), we get μ t ϕ (μ t ) and hence ϕ (μ t ) = μ t and μ t is characteristic. ( ) Let x R, q Q, ϕ Aut (R) and μ (x, = t then by Lemma 3.9 x μ t and x μ s for all s > t in [0,1]. It's follows from hypothesis that ϕ(x) ϕ(μ t ) = μ t so that μ ϕ (x, = μ (ϕ(x), t. Let s = μ ϕ (x, and suppose that s > t. Then ϕ(x) μ s = ϕ(μ s ), which implies from injoctivity of ϕ that x μ s, a contradiction. Hence μ ϕ (x, = μ (ϕ(x), = t =μ (x, showing that μ is Q fuzzy characteristic. We construct a new Q fuzzy right (resp.left) R subgroup by using a given Q- fuzzy right (resp. left) R subgroup. Let i 0 be a real number. If u [0,1], u i shall mean the non negative i-th root in case i < 1. We define μ i : R [0,1] by μ i (x) = ( μ(x)) i if μ is a Q fuzzy right (resp. left) R subgroup of R, then μ i is also a Q- fuzzy right(resp. left) R- subgroup of R Proposition: If μ is a Q- fuzzy characteristic right (resp. left) R- subgroup of R, then μ i is also a Q- fuzzy right (resp. left) R subgroup of R for all i > 0. For any ϕ Aut (R), x Rand q Q we have ( μ i ) ϕ (x, = μ i (ϕ(x), = μ(ϕ(x), i = ( μ(x, ) i = μ i (x, for all i > Definition: Let μ and λ be Q fuzzy right (resp. left) R subgroup of R. Then μ is said to be Q fuzzy same type with λ if there exist ϕ Aut(R) such that μ(x, = λ(ϕ(x), for all x R, q Q. Since all Q fuzzy right (resp. left) R subgroups μ, λ and δ in R, (1) μ is Q fuzzy same type with μ it self (2) if μ is Q- fuzzy same type with λ, then λ is Q- fuzzy same type with μ, and (3) if μ is Q fuzzy same type with λ and λ is Q fuzzy same type with δ, then μ is Q- fuzzy same type with δ.

6 392 Mourad Oqla Massa'deh 3.13 Theorem: Let μ, δ be two Q- fuzzy right (resp. left) R subgroups in R such that μ is Q- fuzzy same type with δ. Then μ is isomorphic to δ. Since μ is Q- fuzzy same type with δ, there exists ϕ Aut(R) such that μ (x, = δ(ϕ(x), for all x R. Let ψ: μ(r) δ(r) such that ψ( μ (x, ) = δ(ϕ(x), for all x R. Then for every x,y R, we have ψ( μ ((x+y), ) = δ(ϕ(x+y), = δ((ϕ(x)+ϕ(y)), and ψ( μ ((x.y), ) = δ(ϕ(xy), = δ((ϕ(x).ϕ(y)),. If ψ( μ (x, ) = ψ( μ (y, ) for all x, y R, then δ(ϕ(x), =δ(ϕ(y), and hence μ (x, = μ (y, showing that ψ is injective Theorem: If μ, δ are two Q- fuzzy right (resp. left) R subgroups in R. If μ ο ϕ = δ then ψ(μ) = δ for some ϕ, ψ Aut(R). Since μ ο ϕ = δ for some ϕ Aut (R). Thus δ(ϕ(x), = δ(x, for all x R, q Q. It follows that ϕ -1 (μ)(x) = sup μ( y, = μ(ϕ(x), = δ(x, y ϕ ( x) for all x R, q Q, put ψ = ϕ -1 then ψ Aut (R) and ψ(μ) = δ Corollary: Let μ, δ be two Q- fuzzy right (resp. left) R subgroups in R, if there exist ϕ Aut (R) such that μ t = ϕ(δ t ) for all t [0,1]. Then μ is Q fuzzy same type withδ. Suppose that there exists ϕ Aut (R) such that μ t = ϕ(δ t ) for all t [0,1]. Let x R and μ(x, = t. Then ϕ -1 (x) ϕ -1 (μ t ) = δ t and so δ(ϕ -1 (x), t = μ(x,. Putting δ(ϕ -1 (x), = s, then ϕ -1 (x) δ s and hence x ϕ(δ s ) = μ s. It follows that μ(x, s = δ(ϕ -1 (x),. Hence μ(x, = δ(ϕ -1 (x), for all x R. Since ϕ -1 Aut (R), then μ is fuzzy same type withδ. REFERENCES [1] A. Solairaju and R. Nagarajan, A New structure and constructions of Q fuzzy groups, Advances in fuzzy mathematics. 4 (2009),

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