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Iranian Journal of Fuzzy Systems Vol. 6, No. 4, (2009) pp. 1-9

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FUZZY HV -SUBSTRUCTURES IN A TWO DIMENSIONAL EUCLIDEAN VECTOR SPACE A. DRAMALIDIS AND T. VOUGIOUKLIS

Abstract. In this paper, we study fuzzy substructures in connection with Hv -structures. The original idea comes from geometry, especially from the two dimensional Euclidean vector space. Using parameters, we obtain a large number of hyperstructures of the group-like or ring-like types. We connect, also, the mentioned hyperstructures with the theta-operations to obtain more strict hyperstructures, as Hv -groups or Hv -rings (the dual ones).

1. Introduction A set H, equipped with at least one multivalued map · : H × H → P(H) called hyperoperation, is said to be hyperstructure. Marty, in 1934, introduced the notion of a hypergroup as a hyperstructure (H, ·) where, the following two axioms hold: (i) x · H = H · x = H, ∀x ∈ H, (ii) (x · y) · z = x · (y · z), ∀x, y, z ∈ H. Since then, many researchers have been worked on this area. Vougiouklis in 1990 introduced [12] the concept of Hv -structures which are generalizations of the classical hyperstructures. One can find, definitions and results on Hv -structures in the books [2], [13]. We recall some definitions from [13]: Itemize. Let H be a set equipped with the hyperoperation (·), then the weak associativity is given by the relation (x · y) · z ∩ x · (y · z) 6= ∅,

∀x, y, z ∈ H.

The hyperstructure (H, ·), is called Hv -semigroup, if the weak associativity is valid. The Hv -group is defined to be a Hv -semigroup, where the reproductivity axiom is valid, i.e. x · H = H · x = H, ∀x ∈ H. The Hv -ring is defined to be the triple (H,+,·), where in both (+) and (·) the weak associativity is valid, the weak distributivity of (·) with respect to (+) is also valid, i.e. x · (y + z) ∩ (x · y + x · z) 6= ∅, (x + y) · z ∩ (x · z + y · z) 6= ∅,

∀x, y, z ∈ H

and (+) is reproductive. An Hv -ring (R, +, ·, +) is called dual Hv -ring if the hyperstructure (R, +, ·, +) is an Hv -ring, too [5]. In [15] a hyperoperation denoted by ∂ is defined as follows: Key words and phrases: Hv -structures, Hv -group, Fuzzy sets, Fuzzy Hv -group.

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A. Dramalidis and T. Vougiouklis

Let (H, ·) be a groupoid and f : H → H is any map. We define a hyperoperation (∂) , called theta-operation, on H as follows x∂y = {f (x) · y, x · f (y)},

∀x, y ∈ H

or in case where (·) is hyperoperation or f is multivalued map we have x∂y = (f (x) · y) ∪ (x · f (y)),

∀x, y ∈ H.

A very useful proposition in this paper is the following one: Proposition 1.1. [14] If a hyperoperation (·) is weak associative or weak commutative then every greater hyperoperation than (·), will be weak associative or weak commutative. If a hyperoperation (·) is weak distributive with respect to (+), then every greater hyperoperation than (·) or (+), will be weak distributive with respect to (+). For a categorical aspect of Hv -rings one can see [8]. The concept of a fuzzy subgroup of a group G was introduced in [11]. If G is a group and m : G → [0, 1] is a fuzzy set, then m is called fuzzy subgroup of G if it satisfies: (i) min{m(x), m(y)} 6 m(xy), ∀x, y ∈ G (ii) m(x) 6 m(x−1 ), ∀x ∈ G. Davvaz in [3], [4] has given the following definitions: Let H be a hypergroup (or Hv -group) and let m be a fuzzy subset of H. Then m is said to be a fuzzy subhypergroup (or fuzzy Hv -subgroup) of H, if the following axioms hold: (i) min{m(x), m(y)} 6 inf a∈x·y {m(a)}, ∀x, y ∈ H (ii) For all x, a ∈ H there exists y ∈ H such that x ∈ a · y and min{m(a), m(x)} 6 m(y). Let (R, +, ·) be an Hv -ring and m be a fuzzy subset of R. Then m is said to be a fuzzy Hv -subring of R, if the following axioms hold: (i) min{m(x), m(y)} 6 infa∈x+y {m(a)}, ∀x, y ∈ R (ii) For all x, a ∈ R there exists y∈R such that x ∈ a + y and min{m(a), m(x)} 6 m(y) (iii) For all x, a ∈ R there exists z ∈ R such that x ∈ z + a and min{m(a), m(x)} 6 m(z) and (iv) min{m(x), m(y)} 6 infa∈x·y {m(a)}, ∀x, y ∈ R. Let (R, +, ·) be an Hv -ring and let m be a fuzzy subset of R. Then a fuzzy Hv -subring of R is said to be a right (resp. left) fuzzy Hv -ideal of R, if the following axiom hold: m(x) 6 infa∈x·y {m(a)} (resp. m(y) 6 infa∈x·y {m(a)}), ∀x, y ∈ R. The concept of a dual fuzzy Hv -subring of a dual Hv -ring, is introduced in [7] as follows: Let (R, +, ·) be a dual Hv -ring and m be a fuzzy subset of R. Then m is said to be a dual fuzzy Hv -subring of R, if m is fuzzy Hv -subring of both Hv -rings (R, +, ·) and (R, ·, +).

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Fuzzy HV -substructures in a Two Dimensional Euclidean Vector Space

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2. A Geometric-like Hyperoperation Let IR2 be the two dimensional real vector space over the field of the real numbers IR, where we consider a coordinate system with origin O. We shall refer to the elements X, Y, Z, . . . , . . . of the set IR2 by their position vectors x, y, z, . . . respectively. We consider an order in IR2 according to the measure of the position vectors, i.e. x 6 y if |x| 6 |y| for every x, y ∈ IR2 . Definition 2.1. In IR2 , for a given λ ∈ [0, 1], we define a commutative hyperoperation (◦), as follows: For every x, y ∈ IR2 x ◦ y = [min{x, y}, min{x, y} + λ max{x, y}) which is a left closed, right opened segment. For simplicity, let us assume that min{x, y} = x, then the above hyperoperation is of the form x ◦ y = [x, x + λy) = {x + µy/ µ ∈ [0, λ) } . Using the above hyperoperation into the plane, one can easily combine abstract algebraic properties with geometrical figures. The same procedure is appeared in [1]. From geometrical point of view, in that sense, the above hyperoperation is the side of the parallelogram with vertices O, X, Y, Z, where Z corresponds to the position vector z = x + y. Proposition 2.2. The hyperstructure (IR2 , ◦) is a commutative Hv -group. Proof. For every x ∈ IR2 : [ [ (x ◦ r0 ) = x ◦ IR2 = (x ◦ r) ∪ x>r 0

x6r

=

[

[ {x + µr/µ ∈ [0, λ)} ∪ {r0 + µx/µ ∈ [0, λ)} = IR2 = IR2 ◦ x. x>r 0

x6r

Now, let x < y < z. Then (x ◦ y) ◦ z = {x + µy/µ ∈ [0, λ)} ◦ z =

[

[ (w ◦ z) ∪ (w ◦ z) =

w∈x◦y w6z

=

[ w∈x◦y w6z 0

w∈x◦y w>z

[ {w + µz / µ ∈ [0, λ)} ∪ {z + µw / µ ∈ [0, λ)} = w∈x◦y w>z

= {x + µ y + µz /µ ∈ [0, λ), µ0 ∈ [0, λ0 ] where λ0 < λ, z = x + λ0 y} ∪ ∪ {z + µx + µµ00 y /µ ∈ [0, λ) , µ00 ∈ (λ0 , λ) where λ0 < λ, z = x + λ0 y}.

(1)

Similarly, since x < y + z we get the equation x ◦ (y ◦ z) = {x + µy + µ2 z /µ ∈ [0, λ)}.

(2)

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By setting in (1) and (2) µ = µ0 = 0 we get that x, z ∈ (x◦y)◦z and x ∈ x◦(y◦z). That means (x ◦ y) ◦ z ∩ x ◦ (y ◦ z) 6= ∅. Checking the associativity, similarly, for the rest five cases, i.e.: x < z < y, y < x < z, y < z < x, z < x < y, z < y < x we get that (x ◦ y) ◦ z ∩ x ◦ (y ◦ z) 6= ∅, ∀x, y, z ∈ IR2 .

Now, we define a new hyperoperation (◦0 ), depending on a parameter λ0 , similar to the hyperoperation (◦) which depends on the parameter λ, where λ0 6= λ, assuming for simplicity that x < y, as follows: Definition 2.3. In IR2 , for a given λ0 ∈ [0, 1], we define a commutative hyperoperation (◦0 ), x ◦0 y = [x, x + λ0 y) = {x + µ0 y / µ0 ∈ [0, λ0 ) }. Proposition 2.4. The hyperstructure (IR2 , ◦, ◦0 ) is a commutative dual Hv -ring. Proof. The only axiom we have to prove is that of the weak distributivity of (◦) with respect to (◦0 ) and vice versa, since the rest axioms have been proven in Proposition 3. Let x < y < z then [ [ x ◦ (y ◦0 z) = x ◦ {y + µ0 z /µ0 ∈ [0, λ0 )} = (x ◦ w) = {x + µw / µ ∈ [0, λ)} 0

w∈y◦0 z 0

w∈y◦0 z 0

= {x + µy + µµ z/ µ ∈ [0, λ), µ ∈ [0, λ )}.

(3)

(x ◦ y) ◦0 (x ◦ z) = {x + µy / µ ∈ [0, λ)} ◦0 {x + µz / µ ∈ [0, λ)} = [ [ = (w◦0 w0 ) = {w + µ0 w0 /µ0 ∈ [0, λ0 )} = w∈x◦y w0 ∈x◦z

w∈x◦y w0 ∈x◦z 0

= {(1 + µ0 )x + µy + µµ z / µ ∈ [0, λ), µ0 ∈ [0, λ0 )}.

(4)

Setting in (3) and (4) µ0 = 0, we get that {x + µy / µ ∈ [0, λ)} ⊂ [x ◦ (y ◦0 z)] ∩ [(x ◦ y) ◦0 (x ◦ z)]

(5)

Setting in (4) µ = 0, then when µ0 6= 0 we get that {(1 + µ0 )x / µ0 ∈ [0, λ0 )} ⊂ (x ◦ y) ◦0 (x ◦ z)

(6)

{(1 + µ0 )x / µ0 ∈ [0, λ0 )} 6⊂ x ◦ (y ◦0 z).

(7)

but obviously

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From (5), (6) and (7), we get that (x ◦ (y ◦0 z)) ∩ ((x ◦ y) ◦0 (x ◦ z)) 6= ∅. Similarly, it can be proved that (x ◦0 (y ◦ z)) ∩ ((x ◦0 y) ◦ (x ◦0 z)) 6= ∅. Checking, similarly, the distributivity for the rest cases of x, y, z ∈ IR2 we can prove that, for every x, y, z ∈ IR2 (x ◦ (y ◦0 z)) ∩ ((x ◦ y) ◦0 (x ◦ z)) 6= ∅,

(x ◦0 (y ◦ z)) ∩ ((x ◦0 y) ◦ (x ◦0 z)) 6= ∅.

3. Constructing a Fuzzy Set In order to use fuzzy sets, we define a map f from the set of positive real numbers to the closed interval I = [0, 1], i.e. f : IR+ → I. Construction 3.1. Let be a Cartesian system. In the first quad √ √ coordinate Oxy √ rant, take the points A 0, 22 and B 22 , 22 , then obviously |OB| = 1. Let C(c, 0) be the point on the x-axis, then CA intersects OB in D(p, w). The equation of the line passes through the points C and D is of the form x−p y−w = c−p 0−w and since the point A belongs to the above line, we get that √ √ 2cw + 2p = 2c.

(8)

Obviously, p = w.

(9) √

2c √ 2c+ 2

From (8) and (9) that we get that p = w = and then v !2 u √ √ u c(2c − 2) 2c t √ . = |OD| = 2 2c2 − 1 2c + 2 √

2) That means, every a ∈ IR+ is mapped to 0 6 a(2a− 6 1. Thus we have 2a2 −1 |x| 6 |y| ⇒ m(x) 6 m(y), ∀x, y ∈ IR2 . Therefore, the measure map, we defined, together with the map f constructed above, define a fuzzy set m : IR2 → [0, 1].

4. Reducing IR2 or Enlarging (◦) In this paper, our aim is to combine geometrical hyperoperations with fuzzy sets. By taking x, y ∈ IR2 , x < y, such that x and y being into different quadrants of the Cartesian plane, we get inf {m(a)} = m(w) 6= min{m(x), m(y)}

a∈x◦y

(10)

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where w is the position vector of the element W , such that OW ⊥ XZ and Z corresponds to the position vector z = x + y. Since the relation (4), is not the desirable one for studying fuzzy sets, we choose as a set, any of the four quadrants of the Cartesian plane, working on this, let us say V = IR+ × IR+ ⊂ IR2 . Obviously, for every x, y ∈ V , the angle XOY < 900 and that means that the angle (OXZ) = angle(OY Z) > 900 , so from the obtuse-angle triangle OXZ (or OY Z) we get that max{x, y} < z = x + y. Let A(a1 , a2 ) ∈ V be any point with position vector a. Then we denote by Ga , the following set Ga = {(x, y) ∈ V /x > a1 and y > a2 }. Now, take any a ∈ V and x ∈ V − Ga , such that a < x. We shall try to find y ∈ V such that x ∈ a ◦ y. Suppose, there exists y ∈V, a < y such that x ∈ a ◦ y = {a + µy/µ ∈ [0, λ)}, then there exists µ0 ∈ [0, λ), such that x = a + µ0 y and that means X(a1 + µ0 y1 , a2 + µ0 y2 ). Since X ∈ V − Ga we get, for example, a1 + µ0 y1 > a1 and a2 + µ0 y2 < a2 . These two inequalities leads to y1 > 0 and y2 < 0, so y ∈ / V , which is a contradiction. Notice that, by taking any a ∈ V and x ∈ Ga , such that a < x, the above inequalities are becoming a1 + µ0 y1 > a1 and a2 + µ0 y2 > a2 , which means that y1 , y2 > 0, so in this case, there exists y ∈ V such that x ∈ a ◦ y. Going further, all these y ∈ V are elements of the set {k(x − a) / k > λ1 }. Indeed, let y = k 0 (x − a) , k0 > λ1 , then n 1o , a ◦ y = [a, a + λy) = {a + µy/µ ∈ [0, λ)} = a + µk 0 (x − a)/µ ∈ [0, λ), k 0 > λ but since k 0 > λ1 ⇒ k10 < λ, by taking µ = k10 , we get that x ∈ a ◦ y. The above condition is very important for studying fuzzy sets, so in order to by-pass these difficulties there is two choices, either reduce the set or enlarge the hyperoperation. For example, as we have seen, using the above notation one can get results relating to the concept of fuzzy Hv -subgroup of the set Ga . The hyperoperation (∂) will be used to enlarge the hyperoperation (◦) and its necessity will be shown next. 5. Fuzzy HV -substructures Let V = IR+ × IR+ ⊂ IR2 be the first quadrant of the Cartesian plane with origin O. In the set V , define a commutative hyperoperation (∂) as follows: Definition 5.1. Let f be the multivalued map, such that for a given λ ∈ [0, 1] f : V → V : x → f (x) = [0, λx). Then, for every x, y ∈ V x∂y = (f (x) + y) ∪ (x + f (y)) = [y, y + λx) ∪ [x, x + λy).

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From the above definition we get that (x∂y) ⊃ (x ◦ y) for every x, y ∈ V . That means that (∂) > (◦) and according to the Propositions 1 and 3 the hyperstructure (V, ∂) is an Hv -group. Notice that in [9] Konguetsof defined a hyperoperation (⊥) in a set H 6= ∅ as follows: For every x, y ∈ H, x⊥y = {x, y}, proving that (H, ⊥) is a hypergroup. The hyperoperation (∂) we defined above, is a generalization of the (⊥), since for λ = 0, we get x∂y = {x, y} = x⊥y. In [16], it is proved that if (G, ·) is a group and f (x) = a constant map on G, then (G, ∂)/β ∗ is singleton. In that case f (x) = e, and so x∂y = {x, y} is the smallest incidence hyperoperation. In [6], a hyperstructure (H, ) with xy ⊃ {x, y} for every x, y ∈ H is called containing Hv -group. So, (V, ∂) is a containing Hv -group. Proposition 5.2. Consider the Hv -group (V, ∂) and let m be a fuzzy subset of V , such that x 6 y ⇒ m(x) 6 m(y), ∀x, y ∈ V . Then m is a fuzzy Hv -subgroup of V . Proof. Let us consider the cases: (i) x 6 y, then by Chapter 4, x 6 y < x + y Then, min{m(x), m(y)} = m(x) and inf {m(a)} = m(x). a∈x∂y

(ii) y 6 x, then y 6 x < x + y Then, min{m(x), m(y)} = m(y) and inf {m(a)} = m(y). a∈x∂y

In both cases, the following is valid: min{m(x), m(y)} 6 inf {m(a)}, ∀x, y ∈ V. a∈x∂y

Now, let x, a ∈ V . Consider the cases: (i) x 6 a, then m(x) 6 m(a) Since a∂x = [a, a + λx) ∪ [x, x + λa) we get that x ∈ a∂x. So, for x = y, there exists y ∈ V such that x ∈ a∂y and min{m(x), m(a)} = m(x) 6 m(y). (ii) a 6 x, then m(a) 6 m(x), again x ∈ a∂x, so for x = y, there exists y ∈ V such that x ∈ a∂y and min{m(x), m(a)} = m(a) 6 m(x) 6 m(y). One can realise the necessity of the hyperoperation ∂, combining the above part of the proof together with the Chapter 4. Now, as before and for a given λ0 6= λ, λ, λ0 ∈ [0, 1], we define the commutative hyperoperation (∂ 0 ) as follows: Definition 5.3. We define the hyperoperation (∂ 0 ), for every x, y ∈ V , x∂ 0 y = [y, y + λ0 x) ∪ [x, x + λ0 y). From the above definition we get that (x∂ 0 y) ⊃ (x ◦0 y) for every x, y ∈ V . That means that (∂ 0 ) > (◦0 ) and according to the Propositions 1 and 5 the hyperstructure (V, ∂, ∂ 0 ) is a dual Hv -ring. Proposition 5.4. Let the dual Hv -ring (V, ∂, ∂ 0 ). Then m is a dual fuzzy Hv subring of V .

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Proof. This follows easily, considering both the Proposition 8 and that the hyperoperations (∂) and (∂ 0 ) are commutative. Proposition 5.5. Let the dual Hv -ring (V, ∂, ∂ 0 ). Then the dual fuzzy Hv -subring of V , is either a right or a left fuzzy Hv -ideal of V . Proof. Let us consider the cases: (i) x 6 y Then, min{m(x), m(y)} = m(x) and inf {m(a)} = m(x), so a∈x∂y

m(x) 6 inf {m(a)}, ∀x, y ∈ V a∈x∂y

(ii) y 6 x Then, min{m(x), m(y)} = m(y) and inf {m(a)} = m(y), so a∈x∂y

m(y) 6 inf {m(a)}, ∀x, y ∈ V. a∈x∂y

References [1] N. Antampoufis, Hypergroups and Hb -groups in complex numbers, Proceedings of 9th AHA Congress, Journal of Basic Science, Babolsar, Iran, 4(1) (2008), 17-25. [2] P. Corsini and V. Leoreanu, Applications of hyperstructures theory, Kluwer Academic Publishers, Boston/Dordrecht/London. [3] B. Davvaz, Fuzzy Hv -groups, Fuzzy Sets and Systems, 101 (1999), 191-195. [4] B. Davvaz, T-fuzzy Hv -subrings of an Hv -ring, J. Fuzzy Math., 11(1) (2003), 215-224. [5] A. Dramalidis, Dual Hv -rings, Rivista di Mathematica Pura ed Applicata, 17 (1996), 55-62. [6] A. Dramalidis, On some classes of Hv -structures, Italian Journal of Pure and Applied Mathematics, 17 (2005), 109-114. [7] A. Dramalidis and T. Vougiouklis, Two fuzzy geometric-like hyperoperations defined on the same set, 9th AHA, Iran, 2005. [8] S. Hoskova, Binary hyperstructures determined by relational and transformation systems, Habilitation thesis, Faculty of Science, University of Ostrava, (2008) 90. [9] S. Hoskova and J. Chvalina, Abelizations of proximal Hv -rings using graphs of good homomorphisms and diagonals of direct squares of hyperstructures, 8th AHA, Greece, (2003), 147-158. [10] S. Hoskova and J. Chvalina, Discrete transformation hypergroups and transformation hypergroups with phase tolerance space, Discrete Mathematics, 308(18) (2008), 4133-4143. [11] L. Konguetsof, Sur les hypermonoides, Bulletin de la Societe Mathematique de Belgique, t. XXV, 1973. [12] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517. [13] T. Vougiouklis, The fundamental relation in hyperrings: the general hyperfield, Proc. 4th AHA, World Scientific, (1991), 203-211. [14] T. Vougiouklis, Hyperstructures and their representations, Monographs, Hadronic Press, USA, 1994. [15] T. Vougiouklis, A new class of hyperstructures, J. Comb. Inf. Syst. Sciences, 20 (1995), 229-235. [16] T. Vougiouklis, The ∂ hyperoperation, Proceedings: Structure Elements of Hyperstructures, Alexandroupolis, Greece, (2005), 53-64. [17] T. Vougiouklis, Hv -fields and Hv -vector spaces with ∂-operations, Proceedings of the 6th Panhellenic Conference in Algebra and Number Theory, Thessaloniki, Greece, (2006), 95102.

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Achilles Dramalidis, School of Sciences of Education, Democritus University of Thrace, 681 00 Alexandroupolis, Greece E-mail address: [email protected] Thomas Vougiouklis∗ , School of Sciences of Education, Democritus University of Thrace, 681 00 Alexandroupolis, Greece E-mail address: [email protected] *Corresponding author

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