Fuzzy geometric spaces associated to fuzzy hyperrings

Abstract

For given two families ${\tilde{B}}_{R}, {\tilde{B}}_{C}$ of fuzzy subsets of a fuzzy hyperring R, we obtain some sufficient conditions such that two fuzzy geometric spaces $(Δ, {\tilde{B}}_{R}), (Δ, {\tilde{B}}_{C})$ are strongly transitive and Δ be a nonzero fuzzy subset of R . Moreover, we show that the relation α and Γ on a fuzzy hyperfield R are transitive and obtain some related basic results.

Keywords

Fuzzy geometric spaces,fuzzy hyperring,fuzzy hyperfield,strongly transitive.

1 Introduction

Fuzzy hyperstructure is an interesting research topic of fuzzy subsets. A hyperoperation assigns a subset of H to every pair of elements of H, that is defined by Marty in [18] as a generalization of a group, while a fuzzy hyperoperation assigns a fuzzy subset of H to every pair of elements of H . This idea was introduced by Corsini and Tofan [10] and studied by Serafimidis, Kehagias and Konstantinidou [22] and they obtained interesting properties in connections with an important hyperstructure, called an interesting paper concerning the join spaces. Recently, Sen, Ameri and Chowdhury introduced and analyzed fuzzy hypersemigroups in [21]. Afterwards these ideas extended to fuzzy hyperrings and fuzzy hypermodules by Leoreanu and Davvaz in [16, 17]. A geometric space is a pair (S, B) such that S is a nonempty set and B is a nonempty family of subsets of S, that are called points and blocks. This concept was initiated by D. Freni in [12] and he investigated some results in order to connection between geometric spaces and its structures. Also D. Freni indicated that the relation β defined by Koskas [15] (studied mainly by Corsini [8] and Vougiouklis [24]) is transitive in hypergroups. In [4] Anvariyeh and Davvaz introduced the concept of strongly transitive geometric spaces associated to hypermodules. In [19] Mirvakili and Davvaz studied on strongly transitive geometric spaces: applications to hyperrings. The concept of fuzzy geometric space was introduced by Ameri et.al. in [3] and they investigate some important structures and relationships between them. In this paper we follow [3], and introduce transitive and strongly transitive relations on a given fuzzy geometric space and obtain some related basic results. In particular, we prove that the fuzzy geometric space associated to a hyperfield is strongly transitive.

2 Preliminaries

A hypergroupoid (H, ∘) is a non-empty set H equipped with a hyperoperation ∘, that is a map ∘ : H × H → P^∗ (H) , where P^∗ (H) denotes the family of all non-empty subset of H . If x, y ∈ H, we will denote by x ∘ y the hyperproduct of x and y . A hypergroupoid (H, ∘) is said to be semi-hypergroup if (x ∘ y) ∘ z = x ∘ (y ∘ z) , for all x, y, z ∈ H . A hypergroup is a semi hypergroup (H, ∘) such that x ∘ H = H ∘ x = H, for all x ∈ H(this condition is called reproducibility). A non-empty setK of a hypergroup H is a subhypergroup of H if x ∘ K = K ∘ x = K, for every x ∈ K .

Definition 2.1. [21] Let S be a non-empty set and F (S) denotes the set of all fuzzy subset of S . A fuzzy hyperoperation on S is the mapping ⊕ : S × S → F (S) written as (a, b) ↦ a ⊕ b . S together with a fuzzy hyperoperation ⊕ is called fuzzy hypergroupoid. A fuzzy hypergroupoid (S, ⊕) is called a fuzzy hypersemigroup, if for all a, b, c of S we have (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) , where for any fuzzy subset μ of F (S) $(a \oplus μ) (r) = {\begin{matrix} ⋁_{t \in S} ((a \oplus t) (r) \land μ (t)), & if μ \neq 0 \\ 0, & otherwise \end{matrix}$ and $(μ \oplus a) (r) = {\begin{matrix} ⋁_{t \in S} (μ (t) \land (t \oplus a) (r)), & if μ \neq 0 \\ 0, & otherwise \end{matrix}$ for all r ∈ S .

Let μ, ν be two fuzzy subset of a fuzzy hypergroupoid (S, ⊕) , then we define (μ ⊕ ν) (t) = ⋁ _p,q∈S (μ (p) ∧ (p ⊕ q) (t) ∧ ν (q)) , for all t ∈ S . A fuzzy hypersemigroup (S, ⊕) is called a fuzzy hypergroup, if x ⊕ S = S ⊕ x = χ_S, for all x ∈ S, that is called reproducibility axiom.

Definition 2.2. [21] If (S, ⊕) be a fuzzy hypergroup, by reproducibility axiom, for every x ∈ S there exists a pair (a, b) of elements of S such that (a ⊕ b) (x) >0 .

Definition 2.3. [16] A fuzzy hyperring is a multi-valued system (R, ⊕ , ⊗) which satisfies the following axioms:

a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c, for all a, b, c ∈ R,

x ⊕ R = R ⊕ x = χ_R, for all x ∈ R,

a ⊕ b = b ⊕ a, for all a, b ∈ R,

a ⊗ (b ⊗ c) = (a ⊗ b) ⊗ c, for all a, b, c ∈ R,

The multiplication is distributive with respect to the fuzzy hyperoperation ⊕ . i.e., a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c), for all a, b, c ∈ R .

Definition 2.4. [17] Let (R, ⊕ , ⊗) be a fuzzy hyperring and (M, ⊕) be a commutative fuzzy hypergroup. M is said to be a fuzzy hypermodule over a fuzzy hyperring R, if there exists: $⊙ : R \times M \to F^{*} (M); (a, m) \mapsto a ⊙ m,$ such that for all a, b ∈ M and m₁, m₂, m ∈ M, we have

a ⊙ (m₁ ⊕ m₂) = (a ⊙ m₁) ⊕ (a ⊙ m₂)

(a ⊕ b) ⊙ m = (a ⊙ m) ⊕ (b ⊙ m) ,

(a ⊗ b) ⊙ m = a ⊙ (b ⊙ m) .

Definition 2.5. [21] Let ρ be an equivalence relation on a fuzzy hypersemigroup (S, ⊕) and let μ, ν be two fuzzy subset on (S, ⊕) . If μ (a) >0 implies there exists b ∈ S such that ν (b) >0 and aρb, and if ν (x) >0 implies there exists y ∈ S such that μ (y) >0 and xρy, then we say that $μ \bar{ρ} ν .$ Also, $μ \bar{\bar{ρ}} ν,$ if for all x ∈ S such that μ (x) >0 and for all y ∈ S such that ν (y) >0, we have xρy .

Definition 2.6. [3] Let S is a nonempty set. A fuzzy geometric space is a pair (Δ, B) such that Δ is a nonzero fuzzy subset of S and B is a nonempty family of fuzzy subsets of S such that ν ≤ Δ, for all ν ∈ B, whose elements we called fuzzy blocks.

Remark 2.7. [3] B is a covering of Δ if Δ ≤ ⋁ _ν∈Bν .

Definition 2.8. [3] If ν₁, ν₂, …, ν_n are fuzzy blocks of a fuzzy geometric space (Δ, B) such that ν_i ∧ ν_i+1 > 0, for any i ∈ {1, 2, …, n - 1} , then the n-tuple (ν₁, ν₂, …, ν_n) is called a fuzzy polygonal of (Δ, B) . The concept of fuzzy polygonal allows us to define on S the following relation: x ≈ y ⇔ x = y or ∃ (ν₁, ν₂, …, ν_n) ; ν₁ (x) >0, ν_n (y) >0, where (ν₁, ν₂, …, ν_n) is a fuzzy polygonal. The relation ≈ is an equivalence and it is coincides with the transitive closure of the following relation: $x \sim y \Leftrightarrow x = y or \exists ν \in B; ν (x) > 0, ν (y) > 0,$ so ≈ is equal to ≈ = ⋃ _n≥1 ∼ ⁿ, where ∼ⁿ =∼ ∘∼ ∘ … ∘ ∼ n times. If B is a covering of Δ, the relation ∼ and ≈ is defined in the following simpler way: $x \sim y \Leftrightarrow \exists ν \in B; ν (x) > 0, ν (y) > 0,$ $x \approx y \Leftrightarrow \exists (ν_{1}, ν_{2}, \dots, ν_{n}); ν_{1} (x) > 0, ν_{n} (y) > 0 .$

Theorem 2.9. [3] A fuzzy geometric space (Δ, B) is strongly transitive, if the family B is a fuzzy cover of Δ, and the following condition is satisfied. For every pair (μ, ν) of fuzzy blocks of a fuzzy geometric space (Δ, B) and for any n ∈ N: $μ \land ν > 0, ν (x) > 0 \Rightarrow \exists η \in B, 0 < α \leq 1; (μ \lor x_{α}) \leq η .$

Theorem 2.10. [3] If (Δ, B) is a strongly transitive fuzzy geometric space, then the relation ∼ on S is transitive, thus ∼ = ≈ .

3 Strongly transitive fuzzy geometric spaces

Let (R, ⊕ , ⊗) be a fuzzy hyperring and x_ij ∈ R be elements of R . If σ ∈ S_n, then the fuzzy hypersums and hyperproducts of the elements x_ij respecting is denoted ${\sum^{˜}}_{i = 1}^{n} {\prod^{˜}}_{j = 1}^{k_{i}} x_{ij} .$ Denote by S_n is the symmetric group of all permutations of the set 1, 2, …, n in this order. Using this notations we define for every n ∈ N ∪ {0} , k_i ∈ N ∪ {0} where i = 1, 2, …, n and j = 1, …, k_i, we set: ${\tilde{B}}_{R} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) = {\sum^{˜}}_{i = 1}^{n} {\prod^{˜}}_{j = 1}^{k_{i}} x_{ij}$ We can consider the fuzzy geometric space $(Δ, {\tilde{B}}_{R})$ whose Δ is a nonzero fuzzy subset of R and fuzzy blocks are the fuzzy hypersums of hyperproducts of elements of R . Also, we can consider another fuzzy geometric space $(Δ, {\tilde{B}}_{C})$ whose Δ is a nonzero fuzzy subset of R and fuzzy blocks are the supremum of all fuzzy hypersums of hyperproducts obtained by permuting in the following possible ways: for every n ∈ N ∪ {0} , k_i ∈ N ∪ {0} and x_ij ∈ R where i = 1, 2, …, n and j = 1, …, k_i, we set: ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) = ⋁ {{\sum^{˜}}_{i = 1}^{n} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)} | σ \in S_{n}, σ_{i} \in S_{k_{i}}} .$ We can consider the fuzzy geometric space (Δ, FP_★ (R)) . By FP_★ (R) we mean the set of all fuzzy finite hypersums of hyperproducts of elements of R, that is a typical elements of FP_★ (R) is the form (x₁₁ ⊗ … ⊗ x_1k₁) ⊕ … ⊕ (x_n1 ⊗ … ⊗ x_{nk
_n}) , n ∈ N .

Notation 3.1. For x_ij ∈ R, μ ∈ FP_★ (R) we note x_ij ∈ μ ⇔ χ_{x
_ij} ≤ μ ⇔ μ (x_ij) =1 .

Lemma 3.2. Let (R, ⊕ , ⊗) be a fuzzy hyperring. Then

${\tilde{B}}_{R} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \leq {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) .$

Moreover, we have ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) = ⋁ {{\tilde{B}}_{R} ([x_{σ (1) σ_{σ (1)} (1)}^{σ (1) σ_{σ (1)} (k_{σ (1)})}], \dots, [x_{σ (n) σ_{σ (n)} (1)}^{σ (n) σ_{σ (n)} (k_{σ (n)})}] | σ \in S_{n}, σ_{i} \in S_{k_{i}}} .$

Lemma 3.3. Let (R, ⊕ , ⊗) be a fuzzy hyperring. Then for every y ∈ R we have

${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \oplus y = {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}], [y]) .$

$y \oplus {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) = {\tilde{B}}_{C} ([y], [x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) .$

${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \otimes y = {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}, y], \dots, [x_{n 1}^{{nk}_{n}}, y]) .$

$y \otimes {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) = {\tilde{B}}_{C} ([y, x_{11}^{1 k_{1}}], \dots, [y, x_{n 1}^{{nk}_{n}}]) .$

${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \oplus {\tilde{B}}_{C} ([y_{11}^{1 k_{1}}], \dots, [y_{m 1}^{{ml}_{m}}]) = {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}], [y_{11}^{1 k_{1}}], \dots, [y_{m 1}^{{ml}_{m}}]) .$

${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \otimes {\tilde{B}}_{C} ([y_{11}^{1 k_{1}}], \dots, [y_{m 1}^{{ml}_{m}}]) = {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}, y_{11}^{1 l_{1}}], \dots, [x_{11}^{1 k_{1}}, y_{m 1}^{{ml}_{m}}], \dots, [x_{n 1}^{{nk}_{n}}, y_{11}^{1 l_{1}}], \dots, [x_{n 1}^{{nk}_{n}}, y_{m 1}^{{ml}_{m}}]) .$

Proof. It is straightforward.

Lemma 3.4. Lemma 3.3 is true for the fuzzy geometric space $(Δ, {\tilde{B}}_{R}) .$

Lemma 3.5. Let (R, ⊕ , ⊗) be a fuzzy hyperring. Then for every σ ∈ S_n and σ_i ∈ S_{k
_i} we have ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) = {\tilde{B}}_{C} ([x_{σ (1) σ_{σ (1)} (1)}^{σ (1) σ_{σ (1)} (k_{σ (1)})}], \dots, [x_{σ (n) σ_{σ (n)} (1)}^{σ (n) σ_{σ (n)} (k_{σ (n)})}])$ Moreover, if (R, ⊕ , ⊗) is a commutative fuzzy hyperring then two fuzzy geometric spaces $(Δ, {\tilde{B}}_{R})$ and $(Δ, {\tilde{B}}_{C})$ are equal.

Proof. It obtain from definition of fuzzy geometric spaces $(Δ, {\tilde{B}}_{R})$ and $(Δ, {\tilde{B}}_{C}) .$

Lemma 3.6. Let (R, ⊕ , ⊗) be a fuzzy hyperring. Then

If x_rs ∈ a ⊗ b then ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \leq {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{r 1}^{r (s - 1)}, a, b, x_{r (s + 1)}^{{rk}_{r}}], \dots, [x_{n 1}^{{nk}_{n}}]) .$

If x_rs ∈ a ⊕ b then ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \leq {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{(r - 1) 1}^{(r - 1) k_{r - 1}}], [x_{r 1}^{r (s - 1)}, a, x_{r (s + 1)}^{{rk}_{r}}], [x_{r 1}^{r (s - 1)}, b, x_{r (s + 1)}^{{rk}_{r}}], [x_{(r + 1) 1}^{(r + 1) k_{r + 1}}], \dots, [x_{n 1}^{{nk}_{n}}]) .$

If $x_{rs} \in B = {\tilde{B}}_{C} ([y_{11}^{1 l_{1}}], \dots, [y_{m 1}^{{ml}_{m}}]),$ then ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \leq {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{r 1}^{r (s - 1)}, B, x_{r (s + 1)}^{{rk}_{r}}], \dots, [x_{n 1}^{{nk}_{n}}])$ $= {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{(r - 1) 1}^{(r - 1) k_{r - 1}}], [.], [x_{(r + 1) 1}^{(r + 1) k_{r + 1}}], \dots, [x_{n 1}^{{nk}_{n}}]),$ where $[.] = [x_{r 1}^{r (s - 1)}, y_{11}^{1 l_{1}}, x_{r (s + 1)}^{{rk}_{r}}], \dots, [x_{r 1}^{r (s - 1)}, y_{m 1}^{{ml}_{m}}, x_{r (s + 1)}^{{rk}_{r}}] .$

Proof. (1) Let x_rs ∈ a ⊗ b and ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) (y) > 0 .$ Then there exists σ ∈ S_n, σ_r ∈ S_{k
_r}, such that $({\sum^{˜}}_{i = 1}^{n} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)}) (y) > 0,$ if σ (u) = r and σ_r (v) = s, then we have x_{σ(u)σ_σ(u)(v)} = x_rs, such that (a ⊗ b) (x_{σ(u)σ_σ(u)(v)}) =1, we have ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) (y) = = ({\sum^{˜}}_{i = 1}^{n} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)}) (y)$ $= (({\sum^{˜}}_{i = 1}^{u - 1} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)}) \oplus ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)} \otimes x_{rs} \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)}) \oplus ({\sum^{˜}}_{i = u + 1}^{n} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)})) (y),$ we set: ${\sum^{˜}}_{i = 1}^{u - 1} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)} = μ,$ ${\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)} \otimes x_{rs} \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)} = ν,$ and ${\sum^{˜}}_{i = u + 1}^{n} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)} = η .$ Then ${\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) (y) = ⋁ (μ \oplus ν \oplus η) (y)$ = ⋁ {⋁ _p∈R [(μ ⊕ ν) (p) ∧ (p ⊕ η) (y)]} = ⋁ {⋁ _p∈R [(⋁ _q∈Rν (q) ∧ (μ ⊕ q) (p)) ∧ (p ⊕ η) (y)]} $= ⋁ {⋁_{p \in R} [(⋁_{q \in R} ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)} \otimes x_{rs} \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)}) (q) \land (μ \oplus q) (p)) \land (p \oplus η) (y)]}$ $= ⋁ {⋁_{p \in R} [(⋁_{q \in R} (⋁_{z \in R} ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)} \otimes x_{rs}) (z) \land (z \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)})) (q) \land (μ \oplus q) (p)) \land (p \oplus η) (y)]}$ $= ⋁ {⋁_{p \in R} [(⋁_{q \in R} (⋁_{z \in R} (⋁_{t \in R} ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)}) (t) \land (t \otimes x_{rs}) (z) \land (z \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)})) (q) \land (μ \oplus q) (p)) \land (p \oplus η) (y)]}$ $\leq ⋁ {⋁_{p \in R} [(⋁_{q \in R} (⋁_{z \in R} (⋁_{t \in R} ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)}) (t) \land (t \otimes x_{rs}) (z) \land (a \otimes b) (x_{rs}) \land (z \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)})) (q) \land (μ \oplus q) (p)) \land (p \oplus η) (y))]}$ $\leq ⋁ {⋁_{p \in R} [(⋁_{q \in R} (⋁_{z \in R} (⋁_{t \in R} ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)}) (t) \land$ $⋁_{w \in R} [(t \otimes w) (z) \land (a \otimes b) (w)] \land (z \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)})) (q) \land (μ \oplus q) (p)) \land (p \oplus η) (y))]}$ $\leq ⋁ {⋁_{p \in R} [(⋁_{q \in R} (⋁_{z \in R} (⋁_{t \in R} ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)}) (t) \land$ $(t \otimes (a \otimes b)) (z) \land (z \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)})) (q) \land (μ \oplus q) (p)) \land (p \oplus η) (y))]}$ $= ⋁ {⋁_{p \in R} [(⋁_{q \in R} (⋁_{z \in R} ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)} \otimes (a \otimes b)) (z) \land (z \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)})) (q) \land (μ \oplus q) (p)) \land (p \oplus η) (y)]}$ $= ⋁ {⋁_{p \in R} [(⋁_{q \in R} ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)} \otimes a \otimes b \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)}) (q) \land (μ \oplus q) (p)) \land (p \oplus η) (y)]}$ $= ⋁ {⋁_{p \in R} [(μ \oplus ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)} \otimes a \otimes b \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)})) (p)] \land (p \oplus η) (y)]}$ $= ⋁ (({\sum^{˜}}_{i = 1}^{u - 1} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)}) \oplus ({\prod^{˜}}_{j = 1}^{v - 1} x_{r σ_{σ (r)} (j)} \otimes a \otimes b \otimes {\prod^{˜}}_{j = v + 1}^{k_{u}} x_{r σ_{σ (r)} (j)}) \oplus ({\sum^{˜}}_{i = u + 1}^{n} {\prod^{˜}}_{j = 1}^{k_{σ (i)}} x_{σ (i) σ_{σ (i)} (j)})) (y),$ $= {\tilde{B}}_{C} ([x_{11}^{1 k_{1}}], \dots, [x_{r 1}^{r (s - 1)}, a, b, x_{r (s + 1)}^{{rk}_{r}}], \dots, [x_{n 1}^{{nk}_{n}}])$ and the proof is complete. The proof of (2) is similar to (1) and (3) obtains from (1) and (2) .

Lemma 3.7. Lemma 3.6, is true for the fuzzy geometric space $(Δ, {\tilde{B}}_{R}) .$

Theorem 3.8. If (R, ⊕ , ⊗) be a fuzzy hyperfield then

The fuzzy geometric space $(Δ, {\tilde{B}}_{R})$ is a strongly transitive fuzzy geometric space.

The fuzzy geometric space $(Δ, {\tilde{B}}_{C})$ is a strongly transitive fuzzy geometric space.

Proof. Let $B_{1} = {\tilde{B}}_{R} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]), B_{2} = {\tilde{B}}_{R} ([y_{11}^{1 l_{1}}], \dots, [y_{m 1}^{{ml}_{m}}])$ be two fuzzy block of ${\tilde{B}}_{R}$ such that $B_{1} \land B_{2} > 0, B_{2} (x) > 0 .$

Let for b ∈ R, we have B₁ (b) >0 and B₂ (b) >0 . Since R is a fuzzy hyperfield, thus there exist $u_{1}^{n} \in R$ and v ∈ R such that (u_i ⊗ x) (x_{ik
_i}) =1 and (b ⊗ v) (x) =1 . Then we have x_{ik
_i} ∈ u_i ⊗ x, x ∈ b ⊗ v . By Lemma 3.6 we have $B_{1} = {\tilde{B}}_{R} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \leq {\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1)}, u_{1}, x], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, x])$ $\leq {\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1)}, u_{1}, b, v], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, b, v])$ $\leq {\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1)}, u_{1}, B_{2}, v], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, B_{2}, v]) .$ Moreover, since (b ⊗ v) (x) =1, B₁ (b) >0, then $⋁_{b} [{\tilde{B}}_{R} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) (b) \land (b \otimes v) (x)] > 0,$ that is equal to $({\tilde{B}}_{R} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \otimes v) (x) > 0,$ then by Lemma 3.6, $({\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1}, u_{1}, x], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, x]) \otimes v) (x) > 0 .$

So by Lemma 3.3, $0 < {\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1}, u_{1}, x], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, x]) \otimes v = {\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1}, u_{1}, x, v], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, x, v]),$ since $B_{2} (x) = {\tilde{B}}_{R} ([y_{11}^{1 l_{1}}], \dots, [y_{m 1}^{{ml}_{m}}]) (x) > 0,$ and by Lemma 3.6, we have $0 < {\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1}, u_{1}, x, v], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, x, v]) (x)$ $\leq {\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1}, u_{1}, B_{2}, v], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, B_{2}, v]) (x)$ therefore there exists 0 < α ≤ 1 such that ${\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1}, u_{1}, B_{2}, v], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, B_{2}, v]) (x)$ ≥α . By the concept of fuzzy point x_α, we have ${\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1}, u_{1}, B_{2}, v], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, B_{2}, v]) \geq x_{α} .$ Therefore ${\tilde{B}}_{R} ([x_{11}^{1 k_{1}}], \dots, [x_{n 1}^{{nk}_{n}}]) \lor x_{α} \leq {\tilde{B}}_{R} ([x_{11}^{1 (k_{1} - 1}, u_{1}, B_{2}, v], \dots, [x_{n 1}^{n (k_{n} - 1)}, u_{n}, B_{2}, v]) .$ and the fuzzy geometric space $\tilde{G} = (Δ, {\tilde{B}}_{R})$ is strongly transitive. In a similar way we obtain (2) .

Definition 3.9. Let (R, ⊕ , ⊗) be a fuzzy hyperring. We define the relation Γ as follows:

$x Γ y \Leftrightarrow \exists n \in N, k_{i} \in N, \exists (x_{i 1}, \dots, x_{{ik}_{i}}) \in R^{k_{i}}, 1 \leq i \leq n; {\sum^{˜}}_{i = 1}^{n} ({\prod^{˜}}_{j = 1}^{k_{i}} x_{ij}) (x) > 0, {\sum^{˜}}_{i = 1}^{n} ({\prod^{˜}}_{j = 1}^{k_{i}} x_{ij}) (y) > 0 .$

Definition 3.10. Let (R, ⊕ , ⊗) be a fuzzy hyperring. We define the relation α as follows: xαy⇔ ∃ n ∈ N, k_i ∈ N, ∃ σ ∈ S_n, ∃ (x_i1, …, x_{ik
_i}) ∈ R^{k
_i}, ∃ σ_i ∈ S_{k
_i}1 ≤ i ≤ n ; ${\sum^{˜}}_{i = 1}^{n} ({\prod^{˜}}_{j = 1}^{k_{i}} x_{ij}) (x) > 0, {\sum^{˜}}_{i = 1}^{n} (A_{i}) (y) > 0,$

where $A_{i} = {\prod^{˜}}_{j = 1}^{k_{i}} x_{i σ_{i} (j)} .$

The relation α and Γ are reflexive and symmetric. We take Γ^∗, α^∗ be the transitive closure of Γ, α . Then Γ^∗, α^∗ is an equivalence relation on R .

Theorem 3.11. Let (R, ⊕ , ⊗) be a hyperfield. Then

Γ = Γ^∗ .

α = α^∗ .

Proof. (2) Let (R, ⊕ , ⊗) be a fuzzy hyperfield. Then the relation ∼ defined on fuzzy geometric space $(Δ, {\tilde{B}}_{C})$ coincides with the relation α on the fuzzy hyperfield (R, ⊕ , ⊗) . Also the relation ≈ defined on the fuzzy geometric space $(Δ, {\tilde{B}}_{C})$ coincides the relation α^∗ on the fuzzy hyperfield (R, ⊕ , ⊗) . Now, if (R, ⊕ , ⊗) is a hyperfield then the fuzzy geometric space $(Δ, {\tilde{B}}_{C})$ is strongly transitive by Theorem 2.13, we have $α = \sim = \approx = α^{*} .$ The proof of (1) is similar.

Conclusion

The study of hyperoperations was initiated by Marty in [18] and continued by others (see [2 , 20]). The above discussion shows that fuzzy geometric space can be done for fuzzy hyperrings, which have recently appeared in the previous paper (for more see [1]). This paper provides useful condition for doing new research in the field of fuzzy geometric space associated to fuzzy hypermodules and fundamental relation for fuzzy hyperstructures.

Footnotes

Acknowledgements

The first author partially has been supported by the “Research Center in Algebraic Hyperstructures and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran” and “Algebraic Hyperstructure Excellence, Tarbiat Modares University, Tehran, Iran”.

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