Abstract: After the introduction of fuzzy sets by Zadeh, several researchers were conducted on the generalization of fuzzy sets. The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. An intuitionistic fuzzy set for a given universal set is defined by a degree of membership function and a degree of non-membership function. The notion of ternary polygroup is a generalization of the notion of polygroup in the sense of Comer. In this research, we study the concept of intuitionistic fuzzy ternary subpolygroups of a ternary polygroup.
INTRODUCTION
The theory of algebraic hyperstructures which is a generalization of the concept
of algebraic structures first was introduced by Marty (1934)
and had been studied in the following decades and nowadays by many mathematicians
and many papers concerning various hyperstructures have appeared in the literature.
The basic definitions of the object can be found (Corsini,
1993; Corsini and Leoreanu, 2003). A hyperstructure
is a non-empty set H together with a map
| • | (x · y)· z = x ·(y · z) for all x, y, z∈H |
| • | a · H = H · a = H for all a∈H |
If x∈H and A, B are non-empty subsets of H, then by A.B, A.x and x.B we mean:
The concept of fuzzy sets was introduced by Zadeh (1965). Let X be a set. A fuzzy subset A of X is characterized by a membership function μA:X→[0, 1], which associates with each point x∈X its grade or degree of membership μA(x)∈[0, 1]. Let A and B be two fuzzy subsets of X. Then:
| • | A = B if and only if μA(x) = μB(x), for all x∈X |
| • | A⊆B if and only if μA(x)≤μB(x), for all x∈X |
| • | C = A∪B if and only if μC(x) = max {μA(x), μB(x)}, for all x∈X |
| • | D = A∩B if and only if μD(x) = min{μA(x), μB(x)} for all x∈X |
| • | The complement of A, denoted by Ac is defined by μAc(x) = 1-μA(x). for all x∈X |
Fuzzy set theory and its applications in several branches of science are growing day by day. These applications can be found in various fields such as computer science, artificial intelligence, operation research, management science, control engineering, expert systems and many others, for example (Ali, 2011; Ayanzadeh et al., 2012; Dehini et al., 2012; Fatemi, 2011; Saad et al., 2007; Yufeng et al., 2011).
Rosenfeld (1971) applied this concept to the theory of groups and studied fuzzy subgroups of a group (Ersoy et al., 2002; Fathi and Salleh, 2009; Massadehss, 2011). Davvaz (1999) applied fuzzy sets to the theory of algebraic hyperstructures and studied their fundamental properties. Further investigations are contained in many papers, for example see the list of references.
Definition 1: Let (H,·) be a hypergroup and let μ a fuzzy subset of H. Then, μ is said to be a fuzzy subhypergroup of H if the following axioms hold:
| • | |
| • | for all x, a∈H there exists y∈H such that x ∈ a · y and min{μ(a), μ(x)}≤μ(y) |
| • | for all x, a∈H there exists z∈H such that x ∈ z · a and min{μ(a), μ(x)}≤μ(z) |
POLYGROUPS AND TERNARY POLYGROUPS
Application of hypergroups have mainly appeared in special subclasses. For
example, polygroups which are certain subclasses of hypergroups are studied
by Comer (1984) and are used to study color algebra.
Quasi-canonical hypergroups (called polygroups by Comer) were introduced by
Bonansinga and Corsini (1982), as a generalization of
canonical hypergroups introduced by Mittas (1972). Some
algebraic and combinatorial properties were developed by Comer. We recall the
following definition from Comer (1984). A polygroup
is a multi-valued system
| • | (x*y)*z = x*(y*z) for all x, y, z∈P |
| • | e*x = x*e = x |
| • | x∈y*z implies y∈x*z-1 and z∈y-1*x |
Zahedi et al. (1995) defined the concept of
fuzzy subpolygroups of a polygroup which is a generalization of the concept
of Rosenfeld's fuzzy subgroups and special case of Davvaz’s for fuzzy subhypergroups.
Let
| • | min{μ(x), μ(y)}≤μ(z) for all x, y∈P and for all z∈x*y |
| • | μ(x)≤μ(x-1) for all x∈P |
The concept of n-ary hypergroup is defined by Davvaz and Vougiouklis (2006), which is a generalization of the concept of hypergroup in the sense of Marty and a generalization of n-ary group, too. Davvaz and Corsini (2007) introduced the notion of a fuzzy n-ary subhypergroup of an n-ary hypergroup. Then this concept studied (Davvaz et al., 2009; Davvaz and Leoreanu-Fotea, 2010a, b; Ghadiri and Waphare, 2009; Kazanci et al., 2010, 2011; Zhan et al., 2010). A ternary hypergroup is a particular case of an n-ary hypergroup for n = 3. Davvaz and Leoreanu-Fotea (2010b) studied the ternary hypergroups associated with a binary relations. Davvaz et al. (2011) provided examples of ternary hyperstructures associated with chain reactions in chemistry (also, see, Davvaz (2009)).
Let H be a non-empty set and f:HxHxH→P*(H). Then f is called a ternary hyperoperation on H and the pair (H, f) is called a ternary hypergroupoid. If A, B, C are non-empty subsets of H, then we define:
The ternary hypergroupoid (H, f) is called a ternary semihypergroup if for every a1,...a5∈H, we have:
A ternary semihypergroup (H, f) is called a ternary hypergroup if for all a, b, c∈H there exist x, y, z∈H such that:
Notice that a ternary semigroup (S, f) is said to be a ternary group if it satisfies the following property that for all a, b, c∈S, there exist unique x, y, z∈S such that:
A ternary polygroup is a multi-valued system
| (1) |
for every α1,..., α5εP, e is a unique element such that f (x, e, e) = f (e, x, e) = x for every xεP and e-1 = e, zε f (x1,x2, x3) implies x1ε f (z, x2-1, x3-1), x2ε f (x1-1, z, x3-1) and x3ε f (x1-1, x2-1, z).
INTUITIONISTIC FUZZY SETS
We recollect some relevant basic preliminaries and in particular, the study of Atanassov (1986).
Let X be a fixed set. An intuitionistic fuzzy set A in X is an object having the form:
A = {<x, μA(x), λA(x)>|xεX} |
where, the functions μA: X→[0,1] and λA: X→[0,1] are the degree of membership and the degree of non-membership of the element x ε X to the set A, respectively; moreover, 0≤μA(x)+λA(x)≤1 must hold. Note that a Zadeh fuzzy set, written down as an intuitionistic one, is of the form:
A = {<x, μA(x), 1-μA(x)>|xεX} |
Let X be a non-empty set and let A = {<x, μA(x), λA(x)>|xεX} and B = {<x, μB(x), λB(x)>|xεX} be two intuitionistic fuzzy sets. Then:
| • | A⊆B iff μA(x)≤μB(x) and λA(x)≥λB(x) for all x ε X |
| • | A = B iff A⊆B and B⊆A |
| • | Ac = {<x, λA(x), μA(x)>|x ε X} |
| • | A∩B = {<x, min {μA(x), μB(x)},max {λA(x), λB(x)}>|x ε X} |
| • | A∪B = {<x max {μA(x), μB(x)},min {λA(x), λB(x)}>|x ε X} |
| • | □A = {<x, μA(x), 1-μA(x)>|xεX} |
| • | ○A = {<x, 1-λA(x),λA(x)>|xεX} |
For the sake of simplicity, we shall use the symbol A = (μA, λA) or A = (μ, λ) for intuitionistic fuzzy set A = {<x, μA(x), λA(x)>|xεX}.
The concept of intuitionistic fuzzy subgroup of a group is introduced by Biswas (1989). Let G be an ordinary group. An intuitionistic fuzzy A = (μA, λA) set in G is called an intuitionistic fuzzy subgroup of G if:
| • | min {μA(x), μA (y)}≤μA(xy) for all x, yεG |
| • | μA(x)≤μA(x-1) for all xεG |
| • | λA(xy)≤max {λA(x), λA(y)} for all x, yεG |
| • | λA(x-1)≤λA(x) for all xεG |
INTUITIONISTIC FUZZY TERNARY SUBPOLYGROUPS
Definition 2: Let
| • | |
| • | |
| • | |
| • |
For any fuzzy set μ of H and any tε[0, 1] we define two sets:
which are called an upper and lower t-level cut of μ and can be used to the characterization of μ.
Theorem 1: Let ⟨P, f, e-1⟩ be a ternary polygroup and A = (μ, λ) be an intuitionistic fuzzy subset of P. Then, U(μ; t) and L(λ; t) are subpolygroups of P for every tεIm(μA)∩Im(λA).
Proof: Suppose that A= (μ, λ) is an intuitionistic fuzzy ternary subpolygroup of P. For every x, y, εU (μ; t) we have min {μ (u), μ (y), μ (z)}≥t and so infαεf(z, y, z) {μ(α)}≥t. Thus, for every x, y z, εU (μ; t) we have μ(α)≥t . Therefore, f(x, y, z)⊆U (μ; t). Now, if xε U (μ; t) then t≥μ(x) Since μ(x)≤μ(x-1) we conclude that which implies that x-1εU(μ; t).
Also, for every x, y, z εL (μ; t) we have max {λ(x), λ(y), λ(z)} and so:
Thus, for every αεf(x, y, z) we have λ(α)≤t. Therefore, f (x, y, z)⊆L (μ; t). Now, if xεL (μ; t) then λ (x)≤t. Since λ(x-1)≤λ(x) we conclude that λ(x-1)≤t which implies that x-1 εL (μ; t).
Theorem 2: Let ⟨P, f, e, -1⟩ be a ternary polygroup and A = (μ, λ) be an intuitionistic fuzzy subset of P such that the non-empty sets U (μ; t) and L (λ; t) are ternary subpolygroups of P for all tε[0, 1]. Then, A = (μ, λ) is an intuitionistic fuzzy ternary subpolygroup of P.
Proof: Assume that for every 0≤t≤1, U (μ; t) (≠φ) is a ternary subpolygroup of P. For every x, y, zεP, we put t0 = min {μ (x), μ (y), μ (z)}. Then x, y, zεU (μ; t) and so f (x, y, z)⊆U(μ; t). Therefore, for every αεf (x, y, z) we have μ(α)≥t0 implying that:
and in this way the first condition of Definition 2 is verified. In order to verify the second condition, let xεP. We put t1 = μ(x)} Since U(μ; t) is a ternary subpolygroup, x-1 εU (μ; t1), which implies that μ(x), ≤μ(x-1)
Now, suppose that for every 0≤t≤1, L (λ; t) (≠ø) is a ternary subpolygroup of P. For every x, y, z, εP, we put t0 = max {λ(x), λ(y), λ(z)}. Then x, y, z, ε L (μ; t0) and so f (x, y, z)⊆L (μ; t0). Therefore, for every αεf (, y, z) we have λ(α)≤t0 implying that:
and in this way the third condition of Definition 2 is verified. In order to verify the last condition, let xεP. We put t1 = λ(x)} Since L (λ; t) is a ternary subpolygroup, x-1 εL(λ; t1), which implies that λ(x-1)≤λ(x).
Corollary 1: Let χK be the characteristic function of a ternary subpoly K of P. Then, K = (χK, χck) is an intuitionistic fuzzy ternary subpolygroup of P.
Corollary 2: Let ⟨P, f, e, -1⟩ be a ternary polygroup. Then A = (μ, λ) is an intuitionistic fuzzy ternary subpolygroup of P if and only if ~ A and ◊A are intuitionistic fuzzy ternary subhypergroup of P.
Proof: Suppose that A = (μ, λ) is an intuitionistic fuzzy ternary polygroup of P. For every x, y, z in P, we have:
| • | min {μ(x), μ(y), μ(z)}≤ infαεf (x, y, z) {μ (α)}, or |
| • | min {1-μc (x), 1-μc (y), 1-μc (z)}≤ supαεf (x, y, z) {1-μc (α)}, or |
| • | min {1-μc (x), 1-μc (y), μc (z)}≤ supαεf (x, y, z) {μc (α)}, or |
| • | supαεf (x, y, z) {μc (x)≤ 1-min (1-μc(x ), 1-μc (y), 1-μc (z)}, or |
| • | supαεf (x, y, z) {μc (α)≤ max (μc (x ), μc (y), μc (z)}. |
Since μ is a fuzzy ternary subpolygroup of P, so for every xεP, μ(x)αμ(x-1) or 1-μ(x-1)≤1-μ(x) which implies that. The converse also can be proved similarly.