Fuzzy hyper h -ideals of hyper BCK-algebras

Abstract

The fuzzification of (weak, strong, reflexive) hyper h-ideals in hyper BCK-algebras is considered. It is shown that every fuzzy (weak, strong, reflexive) hyper h-ideal is a fuzzy (weak, strong, reflexive) hyper BCK-ideal. Relations among (fuzzy) weak hyper h-ideals, (fuzzy) hyper h-ideals, (fuzzy) strong hyper h-ideals and fuzzy reflexive hyper h-ideals are given. The hyper homomorphic pre-image of a fuzzy hyper h-ideal is discussed. It is proved that the product of fuzzy hyper h-ideals is also a fuzzy hyper h-ideal.

Keywords

Hyper BCK-algebras

1 Introduction

The operations of union, intersection and the set difference are the most elementary operations of set theory. The study of these operations lead to the creation of a number of branches of algebra, for instance the notion of Boolean algebra is a result of generalization of these three operations and their properties. Also the algebraic structures of distributive lattices, semi-rings, upper and lower semi-lattices are introduced on the basis of properties of intersection and union. Till 1966, different algebraic structures were discussed using the properties of intersection and union but the operation of set difference and its properties remained unexplored. Imai and Iséki [8], in 1966, considered the properties of set difference and presented the idea of a BCK-algebra. Iséki, in the same year, generalized BCK-algebras and presented the notion of BCI-algebras. BCK-algebras are inspired by BCK logic, i.e., an implicational logic based on modus ponens and the following axioms scheme:

AxiomB $A \supset B . \supset . (C \supset A) \supset (C \supset B)$

AxiomC $A \supset (B \supset C) . \supset . B \supset (A \supset C)$

AxiomK $A \supset (B \supset A)$

Similarly, BCI-algebras are inspired by BCI logic. In many cases, the connection between such algebras and their corresponding logics is much stronger. In many cases one can give a translation procedure which translates all well formed formulas and all theorems of a given logic into terms and theorems of the corresponding algebra. In some cases one can give also an inverse procedure. Nevertheless the study of algebras motivated by known logics is interesting and useful for corresponding logics, also in the case when the inverse translation procedure do not exists.

The hyper structure theory was introduced in 1934 by Marty [14] at the 8th Congress of Scandinavian Mathematicians. Such structures have applications to the following subjects: geometry, hypergraphs, binary relations, combinatorics, codes, cryptography, probability, etc. Now hyper structures are studied by many authors in various directions. For example, Rosenberg [15], in 1996, for the first time considered, in the most general meanings relations between Binary Relations and Hyperstructures. In 1994, Chvalina [3] discussed the connections of Hyperstuctures with binary relations, especially with ordered relations. In 2013, Chavlina [4, 5] et al. discussed certain proximities and pre-orderings on the transposition hypergroups of linear first-order partial differntial operators and also discussed some general actions of hypergroups and some applications. In 1996, Corsini [6] associated the Hyperstructure theory with hypergraphs. The study of hyper BCK-algebras is important with respect to their application to the theory of automaton. Such applications was firstly described by Corsini and Leoreanu [7]. In this monograph are described also many other applications of hyper structures in realworld.

Jun et al. [12] applied the hyper structures to BCK-algebras and introduced the concept of a hyper BCK-algebra which is a generalization of BCK-algebra and investigated some related properties. In this paper, we introduce the concept of fuzzification of (weak, strong, reflexive) hyper h-ideals in hyper BCK-algebras and investigate some of theirproperties.

2 Preliminaries

Consider a non-empty set ϝ equipped with a hyper operation “∘”, i.e., ∘ is a mapping from ϝ× ϝ to the family P (ϝ) of all non-empty subsets of ϝ. For two subset U and V of ϝ, the set ⋃ {u ∘ v | u ∈ U, v ∈ V} is represented as U ∘ V. We shall use ι∘ ȷ instead of ι ∘ {ȷ}, {ι} ∘ ȷ or {ι} ∘ {ȷ}.

Definition 2.1. [12] A “hyper BCK-algebra” refers to a a non-empty set ϝ with a hyperoperation “∘” and a constant 0 satisfying the axioms: ∀ ι , ȷ , ℓ ∈ ϝ,

(ι ∘ ℓ) ∘ (ȷ ∘ ℓ) << ι ∘ ȷ

(ι ∘ ȷ) ∘ ℓ = (ι ∘ ℓ) ∘ ȷ

ι ∘ H << {ι}

ι << ȷ and ȷ << ι imply ι =ȷ

where ι << ȷ is defined by 0∈ ι ∘ ȷ and for every U, V⊆ ϝ, U << V is defined by ∀ u ∈ U, ∃ v ∈ V such that u << v. The operator “<<” is termed as the “hyper order” in ϝ.

Obviously, any BCK-algebra (G, * , 0) becomes a hyper BCK-algebra w.r.t the operation $ι \circ ȷ = {ι * ȷ}$ Another interesting example of a “hyper BCK-algebra” is the set of all non-negative reals, i.e., ϝ = [0, ∞) with the operation $ι \circ ȷ = {\begin{matrix} [0, ι], & if & ι < ȷ, \\ [0, ȷ], & if & ι > ȷ \neq 0, \\ {ι}, & if & ȷ = 0 . \end{matrix}$

Proposition 2.2. [12] In any “hyper BCK-algebra” ϝ, the following axioms are valid: ∀ ι , ȷ , ℓ ∈ ϝ and for all non-empty subsets U and V of ϝ,

ι∘0 = {ι}

ι∘ ȷ << ι

0 ∘ U = {0}

U << U

U ∘ {0} = {0} implies U = {0}

U ⊆ V implies U << V

0 << ι

0 ∘ ι = {0}

0 ∘ 0 = {0}

ȷ << ℓ implies ι∘ ℓ << ι ∘ ȷ

Let ϒ be a non-empty subset of “hyper BCK-algebra” ϝ and 0 ∈ ϒ. Then ϒ is a “hyper BCK-subalgebra” (or H_BCKS) of ϝ if ι ∘ ȷ ⊆ ϒ, for any ι, ȷ ∈ ϒ, a “weak hyper BCK-ideal” (or wH_BCKI) of ϝ if ι ∘ ȷ ⊆ ϒ and ȷ ∈ ϒ imply ι ∈ ϒ, for all ι, ȷ ∈ ϝ, a “hyper BCK-ideal” (or H_BCKI) of ϝ if ι ∘ ȷ << ϒ and ȷ ∈ ϒ imply ι ∈ ϒ, for any ι, ȷ ∈ ϝ, a “strong hyper BCK-ideal” (or sH_BCKI) of ϝ if (ι ∘ ȷ) ∩ ϒ ≠ ∅ and ȷ ∈ ϒ imply ι ∈ ϒ, for all ι, ȷ ∈ ϝ. ϒ is termed as reflexive if ι ∘ ι ⊆ ϒ for any ι∈ ϝ. In the sequel, ϝ will be a “hyper BCK-algebra”.

Lemma 2.3. [11, 12] For any ϝ,

any “reflexive hyper BCK-ideal” (or rH_BCKI) of ϝ is a sH_BCKI of ϝ.

any sH_BCKI of ϝ is a H_BCKI of ϝ.

any H_BCKI of ϝ is a wH_BCKI of ϝ.

Lemma 2.4. [11] Let ϒ be a rH_BCKI of ϝ. Then (ι ∘ ȷ) ∩ I ≠ ∅ implies ι ∘ ȷ << I, ∀ ι , ȷ ∈ ϝ .

Proposition 2.5. [10] Let U be a subset of ϝ. If ϒ is a H_BCKI of H such that U << ϒ then U ⊆ ϒ.

Now, let us define the concept of (weak, strong) hyper h-ideals and elaborate it with the help of different examples.

Definition 2.6. A non-empty subset ϒ⊆ ϝ containing 0 is

a “weak hyper h-ideal” (or wH_hI) of ϝ if ι ∘ (ȷ ∘ ℓ) ⊆ ϒ and ȷ ∈ ϒ imply (ι ∘ ℓ) ⊆ ϒ.

a “hyper h-ideal” (or H_hI) of ϝ if ι ∘ (ȷ ∘ ℓ) << ϒ and ȷ ∈ ϒ imply (ι ∘ ℓ) ⊆ ϒ.

a “strong hyper h-ideal” (or sH_hI) of ϝ if (ι ∘ (ȷ ∘ ℓ)) ⋂ ϒ ≠ ∅ and ȷ ∈ ϒ imply (ι ∘ ℓ) ⊆ ϒ. for any ι, ȷ , ℓ ∈ ϝ.

Theorem 2.7. Any H_hI (resp. wH_hI, sH_hI, rH_hI) of ϝ is a H_BCKI (resp. wH_BCKI, sH_BCKI, rH_BCKI) of ϝ.

Proof. Straightforward.□

Theorem 2.8. For any ϝ,

Any H_hI of ϝ is a wH_hI of ϝ.

Any sH_hI of ϝ is a H_hI of ϝ.

Any “reflexive hyper h-ideal” (or rH_hI) of ϝ is a sH_hI of ϝ.

Proof. (i) . Let ϒ be a H_hI of ϝ. For any ι, ȷ , ℓ ∈ ϝ, let ι ∘ (ȷ ∘ ℓ) ⊆ ϒ and ȷ ∈ ϒ. Then ι ∘ (ȷ ∘ ℓ) ⊆ ϒ implies (ι ∘ ℓ) ∘ (ȷ ∘ ℓ) << ϒ (by Proposition 2.2 (v)), which along with ȷ ∈ ϒ implies ι ∘ ℓ ⊆ ϒ. Hence ϒ is a wH_hI of ϝ.

(ii) . Let ϒ be a sH_hI of ϝ. Let ι ∘ (ȷ ∘ ℓ) << ϒ and ȷ ∈ ϒ. Then for all a ∈ ι ∘ (ȷ ∘ ℓ) , ∃ b ∈ ϒ, s.t, a << b. This implies 0 ∈ a ∘ b and thus a∘ b ∩ ϒ ≠ ∅. By Theorem 3.1., ϒ is also a sH_BCKI of ϝ. Therefore a∘ b ∩ ϒ ≠ ∅ along with b ∈ ϒ implies a ∈ ϒ, that is ι ∘ (ȷ ∘ ℓ) ⊆ ϒ. Therefore (ι ∘ (ȷ ∘ ℓ)) ∩ ϒ ≠ ∅, which along with ȷ ∈ ϒ implies ι ∘ ℓ ⊆ ϒ. Hence ϒ is a H_hIof ϝ.

(iii) . Let ϒ be a rH_hI of ϝ. For any ι, ȷ , ℓ ∈ ϝ, let ι∘ (ȷ ∘ ℓ) ⋂ ϒ ≠ ∅ and ȷ ∈ ϒ. Being a rH_hI, ϒ is also a rH_BCKI of ϝ (by Theorem 3.1), therefore by Lemma 4.4, ι ∘ (ȷ ∘ ℓ) ⋂ ϒ ≠ ∅ ⇒ ι ∘ (ȷ ∘ ℓ) << ϒ, which along with ȷ ∈ ϒ implies ι ∘ ℓ ⊆ ϒ. Hence ϒ is a sH_hI of ϝ.□

The converse of above theorem isn’t valid. It can be observed by the succeeding examples.

Example 2.9. Let ϝ = {0, ι , ȷ} be a “hyper BCK-algebra” defined by the succeeding table: $\begin{matrix} \circ & {0} & {ι} & {ȷ} \\ 0 & {0} & {0} & {0} \\ ι & {ι} & {0, ι} & {0, ι} \\ ȷ & {ȷ} & {ȷ} & {0, ȷ} \end{matrix}$ Take ϒ = {0, ȷ}. Then ϒ is a wH_hI of ϝ but it is not a H_hI of ϝ because ι ∘ (ȷ ∘ ι) = {0, ι} << ϒ and ȷ ∈ ϒ but ι ∘ ι = {0, ι} ⊈ ϒ.

Now take ϒ = {0, ι}. Then ϒ is a sH_hI of ϝ but it is not a sH_hI of ϝ because ȷ ∘ ȷ = {0, ȷ} ⊈ ϒ.

Example 2.10. Let ϝ = {0, ι , ȷ} be a “hyper BCK-algebra” defined by the succeeding table: $\begin{matrix} \circ & {0} & {ι} & {ȷ} \\ 0 & {0} & {0} & {0} \\ ι & {ι} & {0} & {0} \\ ȷ & {ȷ} & {ι, ȷ} & {0, ι, ȷ} \end{matrix}$

Take ϒ = {0, ι}. Then ϒ is a H_hI of ϝ but it is not a sH_hI of ϝ because (ȷ ∘ (ι ∘0)) ∩ ϒ = {ι , ȷ} ∩ ϒ ≠ ∅ and ι ∈ ϒ but ȷ∘0 = {ȷ} ⊈ ϒ.

Definition 2.11. [9] A fuzzy set ϖ in ϝ is

a “fuzzy weak hyper BCK-ideal” (or FwH_BCKI)of ϝ if $ϖ (0) \geq ϖ (ι) \geq min {inf_{u \in ι \circ ȷ} ϖ (u),$ ϖ (ȷ)}.

a “fuzzy hyper BCK-ideal” (or FH_BCKI) of ϝif ι << ȷ implies ϖ (ι) ≥ ϖ (ȷ) and ϖ (ι) ≥ min ${inf_{u \in ι \circ ȷ} ϖ (u), ϖ (ȷ)}$ .

a “fuzzy strong hyper BCK-ideal” (or FsH_BCKI) of ϝ if $inf_{u \in ι \circ ι}$ ϖ (u) ≥ ϖ (ι) ≥ min ${sup_{v \in ι \circ ȷ} ϖ (v),$ ϖ (ȷ)}.

a “fuzzy reflexive hyper BCK-ideal” (or FrH_BCKI) of ϝ if $inf_{u \in ι \circ ι} ϖ (u) \geq ϖ (ȷ)$ and $ϖ (ι) \geq min {sup_{v \in ι \circ ȷ} ϖ (v), ϖ (ȷ)}$ . for all ι, ȷ∈ϝ.

The following result gives the relationship between different types of fuzzy hyper BCK-ideals defined above.

Theorem 2.12. [9] For any ϝ,

Any FH_BCKI of ϝ is a FwH_BCKI of ϝ.

Any FsH_BCKI of ϝ is a FH_BCKI of ϝ.

Any FrH_BCKI of ϝ is a FsH_BCKI of ϝ.

3 Fuzzy hyper h-ideals

Now, we present the notions of “fuzzy weak hyper h-ideals” (or FwH_hIs), “fuzzy hyper h-ideals” (or FH_hIs), “fuzzy strong hyper h-ideals” (or FsH_hIs) and “fuzzy reflexive hyper h-ideals” FrH_hIs and confer some of their properties.

Definition 3.1. A fuzzy set ϖ in ϝ is

a “fuzzy weak hyper h-ideal” (or FwH_hI) of ϝ if ϖ (0) ≥ ϖ (ι) and for all t∈ ι ∘ ℓ, $ϖ (t) \geq min {inf_{u \in ι \circ (ȷ \circ ℓ)} ϖ (u), ϖ (ȷ)}$ .

a “fuzzy hyper h-ideal” (or FH_hI) of ϝ if ι << ȷ implies ϖ (ι) ≥ ϖ (ȷ) and for all t∈ ι ∘ ℓ, $ϖ (t) \geq min {inf_{u \in ι \circ (ȷ \circ ℓ)} ϖ (u), ϖ (ȷ)}$ .

a “fuzzy strong hyper h-ideal” (or FsH_hI) of ϝ if $inf_{u \in ι \circ ι} ϖ (u) \geq ϖ (ι)$ and for all t∈ ι ∘ ℓ, $ϖ (t) \geq min {sup_{v \in ι \circ (ȷ \circ ℓ)} ϖ (v), ϖ (ȷ)}$ .

a “fuzzy reflexive hyper h-ideal” (or FrH_hI) of ϝ if $inf_{u \in ι \circ ι} ϖ (u) \geq ϖ (ȷ)$ and for all t∈ ι ∘ ℓ, $ϖ (t) \geq min {sup_{v \in ι \circ (ȷ \circ ℓ)} ϖ (v), ϖ (ȷ)}$ .

for any ι, ȷ , ℓ ∈ ϝ.

Theorem 3.2. Any FH_hI (resp. FwH_hI, FsH_hI, FrH_hI) of ϝ is a FH_BCKI (resp. FwH_BCKI, FsH_BCKI, FrH_BCKI) of ϝ.

proof. Straightforward.□

Theorem 3.3. For any ϝ,

Any FH_hI of ϝ is a FwH_hI of ϝ.

Any FsH_hI of H is a FH_hI of ϝ.

Any FrH_hI of H is a FsH_hI of ϝ.

Proof. (i) . Let ϖ be a FH_hI of ϝ. Since any FH_hI is a FH_BCKI (by Theorem 3.2) and every FH_BCKI is a FwH_BCKI (By Theorem 2.12), therefore ϖ is also a FwH_BCKI of ϝ. Hence ϖ satisfies ϖ (0) ≥ ϖ (ι) for all ι∈ ϝ. Also being a FH_hI, ∀ ι , ȷ , ℓ ∈ ϝ and t∈ ι ∘ ℓ, ϖ satisfies: $ϖ (t) \geq min {inf_{u \in ι \circ (ȷ \circ ℓ)} ϖ (u), ϖ (ȷ)} .$

Hence ϖ is a FwH_hI of ϝ.

(ii) . Let ϖ be a FsH_hI of ϝ. Since FsH_hI is a FsH_BCKI (by Theorem 3.2.) and every FsH_BCKI is a FH_BCKI (by Theorem 2.12), therefore ϖ is also a FH_BCKI of ϝ. Hence for any ι, ȷ ∈ ϝ, if ι << ȷ then ϖ (ι) ≥ ϖ (ȷ). Also being a FsH_hI, ∀ ι , ȷ , ℓ ∈ ϝ and t∈ ι ∘ ℓ, ϖ satisfies $ϖ (t) \geq min {sup_{a \in (ι \circ (ȷ \circ ℓ)} ϖ (a), ϖ (ȷ)} .$

Since $sup_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a) \geq ϖ (b)$ , for all b ∈ ι ∘ (ȷ ∘ ℓ), Thus, $ϖ (t) \geq min {sup_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a), ϖ (ȷ)} \geq min {ϖ (b), ϖ (ȷ)}$ , for all b ∈ ι ∘ (ȷ ∘ ℓ) Since $ϖ (b) \geq inf_{c \in ι \circ (ȷ \circ ℓ)} ϖ (c)$ for all b ∈ ι ∘ (ȷ ∘ ℓ), Therefore, $ϖ (t) \geq min {ϖ (b), ϖ (ȷ)} \geq min {inf_{c \in ι \circ (ȷ \circ ℓ)}$ ϖ (c) , ϖ (ȷ)}, i.e., ϖ (t)≥ $min {inf_{c \in ι \circ (ȷ \circ ℓ)}$ ϖ (c) , ϖ (ȷ)}. Hence ϖ is a FH_hI of ϝ.

(iii) . Let ϖ be a FrH_hI of ϝ. Then ϖ satisfies $inf_{u \in ι \circ ι} ϖ (u) \geq ϖ (ȷ)$ , ∀ ι , ȷ ∈ ϝ $\Rightarrow inf_{u \in ι \circ ι} ϖ (u) \geq ϖ (ι)$ , for all ι∈ ϝ. Hence the first condition for ϖ to be a FsH_hI of ϝ is satisfied. Also being a FrH_hI, for all ι, ȷ , ℓ ∈ ϝ and t∈ ι ∘ ℓ, ϖ satisfies $ϖ (t) \geq min {sup_{v \in (ι \circ (ȷ \circ ℓ)} ϖ (v), ϖ (ȷ)}$ . Hence ϖ is a FsH_hI of ϝ.□

The converse of the above theorem isn’t valid. Consider the “hyper BCK-algebra” ϝ = {0, ι , ȷ} delineated by the table given in Example 2. Define a fuzzy set ϖ in ϝ by: $ϖ (0) = ϖ (ȷ) = 1, ϖ (ι) = 0$

Then ϖ is a FwH_hI of ϝ but it is not a FH_hI of ϝ because: $ι \leq ȷ \Rightarrow ϖ (ι) = 0 < 1 = ϖ (ȷ) .$

The next example illustrates that a FH_hI may not be FsH_hI and similarly, a FsH_hI may not be a FrH_hI.

Example 3.4. Let ϝ = {0, ι , ȷ} be a hyper BCK-algebra defined by the succeeding table: $\begin{matrix} \circ & {0} & {ι} & {ȷ} \\ 0 & {0} & {0} & {0} \\ ι & {ι} & {0, ι} & {0, ι} \\ ȷ & {ȷ} & {ι, ȷ} & {0, ι, ȷ} \end{matrix}$

Define a fuzzy set ϖ in ϝ by: $ϖ (0) = ϖ (ι) = 0.8, ϖ (ȷ) = 0.2$

Then ϖ is a FH_hI of ϝ but it is not a FsH_hI of ϝ because: $ϖ (ȷ) = 0.2 < 0.8 = min {sup_{a \in ȷ \circ (ι \circ ȷ)} ϖ (a), ϖ (ι)}$

Again consider the hyper BCK-algebra ϝ = {0, ι , ȷ} defined by the table given in Example 3.4. Define a fuzzy set ϖ in ϝ by: $ϖ (0) = 1, ϖ (ι) = ϖ (ȷ) = \frac{1}{2}$

Then ϖ is a FsH_hI of ϝ but it is not a FrH_hI of ϝ because for ι, ȷ ∈ ȷ ∘ ȷ, $ϖ (ι) = ϖ (ȷ) = \frac{1}{2} < 1 = ϖ (0)$

Theorem 3.5. Let ϖ be a fuzzy set in ϝ. Then ϖ is a FH_hI (resp. FwH_hI, FsH_hI, FrH_hI) of ϝ if and only if for all t ∈ [0, 1], ϖ_t≠ ∅ is a H_hI (resp. wH_hI, sH_hI, rH_hI) of ϝ.

Proof. Suppose that ϖ is a FH_hI of ϝ. Since ϖ_t≠ ∅, so for any ι ∈ ϖ_t, ϖ (ι) ≥ t. Since every FH_hI is also a FwH_hI (by Theorem 3), so ϖ is also a FwH_hI of ϝ. Thus ϖ (0) ≥ ϖ (ι) ≥ t, for all ι∈ ϝ, which implies 0 ∈ ϖ_t.

Let ι ∘ (ȷ ∘ ℓ) << ϖ_t and ȷ ∈ ϖ_t, for some ι, ȷ , ℓ∈ϝ. Then for all a ∈ ι ∘ (ȷ ∘ ℓ) , ∃ b ∈ ϖ_t such that a << b. So ϖ (a) ≥ ϖ (b) ≥ t, for all a ∈ ι ∘ (ȷ ∘ ℓ). Thus $inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a) \geq t$ . Also ϖ (ȷ) ≥ t, as ȷ ∈ ϖ_t. Therefore for all v∈ ι ∘ ℓ, ϖ satisfies $\begin{matrix} ϖ (v) & \geq & min {inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a), ϖ (ȷ)} \\ \geq & min {t, t} \\ = & t \end{matrix}$ then v ∈ ϖ_t, for all v∈ ι ∘ ℓ and then ι ∘ ℓ ⊆ ϖ_t. Hence ϖ_t is H_hI of ϝ.

Conversely, suppose that ϖ_t≠ ∅ is a H_hI of ϝ for all t ∈ [0, 1]. Let ι << ȷ for some ι, ȷ ∈ ϝ and put ϖ (ȷ) = t. Then ȷ ∈ ϖ_t. So ι << ȷ ∈ ϖ_t ⇒ ι << ϖ_t. Being a hyper h-ideal, ϖ_t is also a hyper BCK-ideal of ϝ (by Theorem 2.7) therefore by Proposition 2.2, ι ∈ ϖ_t. Hence ϖ (ι) ≥ t = ϖ (ȷ). That is ι << ȷ ⇒ ϖ (ι) ≥ ϖ (ȷ), for all ι, ȷ ∈ ϝ. Moreover for any ι, ȷ , ℓ ∈ ϝ, let $d = min {inf_{c \in ι \circ (ȷ \circ ℓ)} ϖ (c), ϖ (ȷ)}$

Then ϖ (ȷ) ≥ d ⇒ ȷ ∈ ϖ_d and for all e ∈ ι ∘ (ȷ ∘ ℓ), $ϖ (e) \geq inf_{c \in ι \circ (ȷ \circ ℓ)} ϖ (c) \geq d$ , which implies e ∈ ϖ_d. Thus ι ∘ (ȷ ∘ ℓ) ⊆ ϖ_d. By Proposition 2.2(v), ι ∘ (ȷ ∘ ℓ) ⊆ ϖ_d ⇒ ι ∘ (ȷ ∘ ℓ) << ϖ_d, which along with ȷ ∈ ϖ_d implies ι ∘ ℓ ⊆ ϖ_d. Hence for all u∈ ι ∘ ℓ, we get $ϖ (u) \geq d = min {inf_{c \in ι \circ (ȷ \circ ℓ)} ϖ (c), ϖ (ȷ)}$

Thus ϖ is a FH_hI of ϝ.□

Theorem 3.6. If ϖ is a FH_hI (resp. FwH_hI, FsH_hI, FrH_hI) of ϝ then the set $A = {ι \in ϝ ∣ ϖ (ι) = ϖ (0)}$ is a H_hI (resp. wH_hI, sH_hI, rH_hI) of ϝ.

Proof. Suppose that ϖ is a FH_hI of ϝ. Clearly 0 ∈ A. Let ι ∘ (ȷ ∘ ℓ) << A and ȷ ∈ A for some ι, ȷ , ℓ ∈ ϝ. Then for all a ∈ ι ∘ (ȷ ∘ ℓ) , ∃ b ∈ A such that a << b. Therefore ϖ (a) ≥ ϖ (b) = ϖ (0). But being a FH_hI, ϖ is also a FwH_hI of ϝ (by Theorem 3(i)), so ϖ satisfies ϖ (0) ≥ ϖ (v), for all v∈ ϝ. This implies ϖ (0) ≥ ϖ (a), for all a ∈ ι ∘ (ȷ ∘ ℓ). Therefore ϖ (a) = ϖ (0), for all a ∈ ι ∘ (ȷ ∘ ℓ), that is, $inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a) = ϖ (0)$ . Also ϖ (ȷ) = ϖ (0). Being a FH_hI, for all t∈ ι ∘ ℓ, ϖ satisfies $\begin{matrix} ϖ (t) & \geq & min {inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a), ϖ (ȷ)} \\ = & min {ϖ (0), ϖ (0)} \\ = & ϖ (0) \end{matrix}$

Since ϖ (0) ≥ ϖ (v), for all v∈ ϝ, therefore $ϖ (t) = ϖ (0), for all t \in ι \circ ℓ$

Thus ι ∘ ℓ ⊆ A. Hence A is a H_hI of ϝ.□

The transfer principle for fuzzy sets described in [13] suggest the following theorem.

Theorem 3.7. For any subset A of a hyper BCK-algebra ϝ, let ϖ be a fuzzy set in ϝ defined by: $ϖ (ι) = {\begin{matrix} t & if ι \in A \\ 0 & if ι \notin A \end{matrix}$ for all ι∈ ϝ, where t ∈ (0, 1]. Then A is a H_hI (resp. wH_hI, sH_hI, rH_hI) of ϝ if and only if ϖ is a FH_hI (resp. FwH_hI, FsH_hI, FrH_hI) of ϝ.

Proof. Suppose that A is a H_hI of ϝ. Let ι << ȷ for some ι, ȷ ∈ ϝ and put ϖ (ȷ) = t. Then ȷ ∈ ϖ_t. So ι << ȷ ∈ ϖ_t ⇒ ι << ϖ_t. Being a hyper h-ideal, ϖ_t is also a H_BCKI of ϝ (by Theorem 2.7) therefore by Proposition 2.2, ι ∈ ϖ_t. Hence ϖ (ι) ≥ t = ϖ (ȷ). That is ι << ȷ ⇒ ϖ (ι) ≥ ϖ (ȷ), for all ι, ȷ ∈ ϝ.

Moreover for any ι, ȷ , ℓ ∈ ϝ, if ι ∘ (ȷ ∘ ℓ) << A and ȷ ∈ A then ι ∘ ℓ ⊆ A. Since A is a hyper h-ideal of ϝ, so by Proposition 2.2, ι ∘ (ȷ ∘ ℓ) ⊆ A. Thus ϖ (a) = t, for all a ∈ ι ∘ (ȷ ∘ ℓ) which implies $inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a) = t$ . Also ϖ (ȷ) = t. Since ι ∘ ℓ ⊆ A, for all u∈ ι ∘ ℓ, we have $ϖ (u) = t = min {inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a), ϖ (ȷ)}$

If ι ∘ (ȷ ∘ ℓ) notllA and ȷ ∉ A then $min {inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a), ϖ (ȷ)} = 0 \leq ϖ (u)$ for all u∈ ι ∘ ℓ. If ι ∘ (ȷ ∘ ℓ) notllA and ȷ ∈ A (OR) If ι ∘ (ȷ ∘ ℓ) << A and ȷ ∉ A. Then in both of these cases we have $min {inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ (a), ϖ (ȷ)} = 0 \leq ϖ (u)$ for all u∈ ι ∘ ℓ. Hence ϖ is a FH_hI of ϝ.

Conversely suppose that ϖ is a fuzzy hyper h-ideal of ϝ. Then by Theorem 3.5, for all t ∈ (0, 1], ϖ_t = A is a H_hI of ϝ.□

For a family {ϖ_i ∣ i ∈ I} of fuzzy sets in a non-empty set X, define the join ∨_i∈I ϖ_i and meet ∧_i∈I ϖ_i as follows: $\begin{matrix} (\lor_{i \in I} ϖ_{i}) (ι) & = & sup_{i \in I} ϖ_{i} (ι) \\ (\land_{i \in I} ϖ_{i}) (ι) & = & inf_{i \in I} ϖ_{i} (ι) \end{matrix}$ for all ι ∈ X, where I is any indexing set.

Theorem 3.8. The family of fuzzy (weak, strong, reflexive) hyper h-ideals of ϝ is a completely distributive lattice with respect to join and meet.

Proof. Let {ϖ_i ∣ i ∈ I} be a family of fuzzy hyper h-ideals of ϝ. Since [0, 1] is a completely distributive lattice with respect to the usual ordering in [0, 1], it is sufficient to show that ∨_i∈I ϖ_i and ∧_i∈I ϖ_i are fuzzy hyper h-ideals of ϝ. For any ι, ȷ ∈ ϝ, if ι << ȷthen $\begin{matrix} (\lor_{i \in I} ϖ_{i}) (ι) & = & sup_{i \in I} ϖ_{i} (ι) \\ \geq & sup_{i \in I} ϖ_{i} (ȷ) \\ = & (\lor_{i \in I} ϖ_{i}) (ȷ) \end{matrix}$ and then $(\lor_{i \in I} ϖ_{i}) (ι) \geq (\lor_{i \in I} ϖ_{i}) (ȷ)$

Moreover, for any ι, ȷ , ℓ ∈ ϝ and for all t∈ ι ∘ ℓ, we have $\begin{matrix} (\lor_{i \in I} ϖ_{i}) (t) \\ = & sup_{i \in I} ϖ_{i} (t) \\ \geq & sup_{i \in I} [min {inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ_{i} (a), ϖ_{i} (ȷ)}] \\ = & min {sup_{i \in I} (inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ_{i} (a)), sup_{i \in I} (ϖ_{i} (ȷ))} \\ = & min {inf_{a \in ι \circ (ȷ \circ ℓ)} (sup_{i \in I} ϖ_{i} (a)), sup_{i \in I} (ϖ_{i} (ȷ))} \\ = & min {inf_{a \in ι \circ (ȷ \circ ℓ)} ((\lor_{i \in I} ϖ_{i}) (a)), (\lor_{i \in I} ϖ_{i}) (ȷ)} \end{matrix}$ and then $\begin{matrix} (\lor_{i \in I} ϖ_{i}) (t) \geq \\ min {inf_{a \in ι \circ (ȷ \circ ℓ)} ((\lor_{i \in I} ϖ_{i}) (a)), (\lor_{i \in I} ϖ_{i}) (ȷ)} \end{matrix}$

Hence ∨_i∈I ϖ_i is a FH_hI of ϝ.

Now we prove that ∧_i∈I ϖ_i is a FH_hI of ϝ. For any ι, ȷ ∈ ϝ we have, if ι << ȷ then $\begin{matrix} (\land_{i \in I} ϖ_{i}) (ι) & = & inf_{i \in I} ϖ_{i} (ι) \\ \geq & inf_{i \in I} ϖ_{i} (ȷ) \\ = & (\land_{i \in I} ϖ_{i}) (ȷ) \end{matrix}$ and then $(\land_{i \in I} ϖ_{i}) (ι) \geq (\land_{i \in I} ϖ_{i}) (ȷ)$ Moreover, for any ι, ȷ , ℓ ∈ ϝ and for all t∈ ι ∘ ℓ, we have $\begin{matrix} (\land_{i \in I} ϖ_{i}) (t) \\ = & inf_{i \in I} ϖ_{i} (t) \\ \geq & inf_{i \in I} [min {inf_{a \in ι \circ (ȷ \circ ℓ)} ϖ_{i} (a), ϖ_{i} (ȷ)}] \\ = & min {inf_{i \in I} (inf_{a \in (ι \circ ℓ) \circ (ȷ \circ ℓ)} ϖ_{i} (a)), inf_{i \in I} (ϖ_{i} (ȷ))} \\ = & min {inf_{a \in ι \circ (ȷ \circ ℓ)} (inf_{i \in I} ϖ_{i} (a)), inf_{i \in I} (ϖ_{i} (ȷ))} \\ = & min {inf_{a \in ι \circ (ȷ \circ ℓ)} ((\land_{i \in I} ϖ_{i}) (a)), (\land_{i \in I} ϖ_{i}) (ȷ)} \end{matrix}$ and then $\begin{matrix} (\land_{i \in I} ϖ_{i}) (t) \geq \\ min {inf_{a \in ι \circ (ȷ \circ ℓ)} ((\land_{i \in I} ϖ_{i}) (a)), (\land_{i \in I} ϖ_{i}) (ȷ)} \end{matrix}$

Hence ∧_i∈I ϖ_i is a FH_hI of ϝ.

Thus the family of fuzzy hyper h-ideals of ϝ is a completely distributive lattice with respect to join and meet.□

4 Product of fuzzy hyper h-ideals

In this section, we discuss the product of different types of fuzzy hyper h-ideals. By making use of transfer principle for fuzzy sets it will be proved that the product of two fuzzy hyper h-ideals is also a fuzzy hyper h-ideal.

Definition 4.1. [2] Let (ϝ ₁, ∘ ₁, 0₁) and (ϝ ₂, ∘ ₂, 0₂) are hyper BCK-algebras and ϝ = ϝ ₁ × ϝ ₂. We define a hyperoperation “∘” on ϝ by $(u_{1}, v_{1}) \circ (u_{2}, v_{2}) = (u_{1} \circ u_{2}, v_{1} \circ v_{2}),$ for all (u₁, v₁) , (u₂, v₂)∈ ϝ, where for U ⊆ ϝ ₁ and V ⊆ ϝ ₂ by (U, V) we mean $(U, V) = {(u, v) : u \in U, v \in V}$ and 0 = (0₁, 0₂), and a hyperorder “ <<” on ϝ by $(u_{1}, v_{1}) < < (u_{2}, v_{2}) \Leftrightarrow u_{1} < < u_{2} and v_{1} < < v_{2}$

Thus (ϝ , ∘ , 0) is a “hyper BCK-algebra”.

Let ϖ and ζ be fuzzy sets in hyper BCK-algebras ϝ₁ and ϝ₂ respectively. Then ϖ × ζ, the product of ϖ and ζ of ϝ = ϝ ₁ × ϝ ₂ is delineated as $(ϖ \times ζ) ((ι, ȷ)) = min {ϖ (ι), ζ (ȷ)}$

In the sequel, ϝ₁ and ϝ₂ will bee hyper BCK-algebras and ϝ = ϝ ₁ × ϝ ₂.

Definition 4.2. Let ϖ be a fuzzy set in ϝ. Then fuzzy sets ϖ₁ and ϖ₂ on ϝ₁ and ϝ₂ respectively, are defined as $ϖ_{1} (ι) = ϖ ((ι, 0)), ϖ_{2} (ȷ) = ϖ ((0, ȷ))$

Theorem 4.3. Let ϖ be a fuzzy set in ϝ. Then ϖis a FH_hI (resp. FwH_hI, FsH_hI, FrH_hI) of ϝif and only if ϖ₁ and ϖ₂ are FH_hIs (resp. FwH_hIs, FsH_hIs, FrH_hIs) of ϝ₁ and ϝ₂ respectively.

Proof. Let ϖ be a FH_hI of ϝ and let ι₁ << ι ₂ for some ι₁, ι ₂ ∈ ϝ ₁. Then (ι ₁, 0) << (ι ₂, 0) which implies ϖ ((ι ₁, 0)) = ϖ₁ (ι ₁) ≥ ϖ ((ι ₂, 0)) = ϖ₁ (ι ₂), that is, ϖ₁ (ι ₁) ≥ ϖ₁ (ι ₂).

Moreover for any ι₁, ȷ ₁, ℓ ₁ ∈ ϝ ₁, let $t = min {inf_{a \in ι_{1} \circ (ȷ_{1} \circ ℓ_{1})} ϖ_{1} (a), ϖ_{1} (ȷ_{1})}$

Then for all b ∈ ι ₁ ∘ (ȷ ₁ ∘ ℓ ₁), $ϖ_{1} (b) \geq inf_{a \in ι_{1} \circ (ȷ_{1} \circ ℓ_{1})} ϖ_{1} (a) \geq t and ϖ_{1} (ȷ_{1}) \geq t$ hence $ϖ ((b, 0)) \geq t and ϖ ((ȷ_{1}, 0)) \geq t$ for all (b, 0) ∈ (ι ₁, 0) ∘ ((ȷ ₁, 0) ∘ (ℓ ₁, 0)), $(b, 0) \in ϖ_{t} and (ȷ_{1}, 0) \in ϖ_{t}$ for all (b, 0) ∈ (ι ₁, 0) ∘ ((ȷ ₁, 0) ∘ (ℓ ₁, 0)), $(ι_{1}, 0) \circ ((ȷ_{1}, 0) \circ (ℓ_{1}, 0)) \subseteq ϖ_{t} and (ȷ_{1}, 0) \in ϖ_{t}$

By Transfer principle for fuzzy sets, ϖ_t≠ ∅ is a H_hI of ϝ and so is a wH_hI of ϝ (by Theorem 3.8). Thus $(ι_{1}, 0) \circ ((ȷ_{1}, 0) \circ (ℓ_{1}, 0)) \subseteq ϖ_{t} and (ȷ_{1}, 0) \in ϖ_{t}$ imply (ι ₁, 0) ∘ (ℓ ₁, 0) ⊆ ϖ_t. Therefore, ϖ ((s, 0))≥t, for all (s, 0) ∈ (ι ₁, 0) ∘ (ℓ ₁, 0) = (ι ₁ ∘ ℓ ₁, 0). Hence $ϖ_{1} (s) \geq t = min {inf_{a \in ι_{1} \circ (ȷ_{1} \circ ℓ_{1})} ϖ_{1} (a), ϖ_{1} (ȷ_{1})}$ for all s ∈ ι ₁ ∘ ℓ ₁. Hence ϖ₁ is a FH_hI of ϝ₁.

Similarly, we can prove that ϖ₂ is a FH_hI of ϝ₂.

Conversely, let ϖ₁ and ϖ₂ be FH_hIs of ϝ₁ and ϝ₂ respectively. For any (ι , u) , (ȷ , v) ∈ ϝ, where ι, ȷ ∈ ϝ ₁ and u, v ∈ ϝ ₂, let (ι , u) << (ȷ , v), since

(ι , u) << (ȷ , v) ⇔ ι << ȷ and u << v ⇒ϖ₁ (ι) ≥ ϖ₁ (ȷ) and ϖ₂ (u) ≥ ϖ₂ (v) ⇒ min {ϖ₁ (ι) , ϖ₂ (u)} ≥ min {ϖ₁ (ȷ) , ϖ₂ (v)} ⇒ (ϖ₁ × ϖ₂) ((ι , u)) ≥ (ϖ₁ × ϖ₂) ((ȷ , v)) ⇒ϖ ((ι , u)) ≥ ϖ ((ȷ , v))

Thus (ι , u) << (ȷ , v) ⇒ ϖ ((ι , u)) ≥ ϖ ((ȷ , v)).

Moreover for any (ι , u) , (ȷ , v) , (ℓ , w) ∈ ϝ, where for all ι, ȷ , ℓ ∈ ϝ ₁ and u, v, w ∈ ϝ ₂ and for all (a, b) ∈ (ι , u) ∘ (ℓ , w) = (ι ∘ ℓ , u ∘ w), $\begin{matrix} ϖ ((a, b)) \\ = & (ϖ_{1} \times ϖ_{2}) ((a, b)) \\ = & min {ϖ_{1} (a), ϖ_{2} (b)} \\ \geq & min [min {inf_{c \in ι \circ (ȷ \circ ℓ)} ϖ_{1} (c), ϖ_{1} (ȷ)}, \\ min {inf_{d \in u \circ (v \circ w)} ϖ_{2} (d), ϖ_{2} (v)}] \\ = & min [min {inf_{c \in ι \circ (ȷ \circ ℓ)} ϖ_{1} (c), inf_{d \in u \circ (v \circ w)} ϖ_{2} (d)}, \\ min {ϖ_{1} (ȷ), ϖ_{2} (v)}] \\ = & min [inf_{c \in ι \circ (ȷ \circ ℓ), d \in u \circ (v \circ w)} {min {ϖ_{1} (c), ϖ_{2} (d)}}, \\ min {ϖ_{1} (ȷ), ϖ_{2} (v)}] \\ = & min {inf_{(c, d) \in (ι \circ (ȷ \circ ℓ), u \circ (v \circ w))} (ϖ_{1} \times ϖ_{2}) min d)), \\ (ϖ_{1} \times ϖ_{2}) ((ȷ, v))} \\ = & min {inf_{(c, d) \in (ι \circ (ȷ \circ ℓ), u \circ (v \circ w))} ϖ ((c, d)), ϖ ((ȷ, v))} \end{matrix}$ and then $\begin{matrix} ϖ ((a, b)) \geq \\ min {inf_{(c, d) \in (ι, u) \circ ((ȷ, v) \circ (ℓ, w))} ϖ ((c, d)), ϖ ((ȷ, v))} \end{matrix}$

Hence, ϖ is a FH_hI of H.□

5 Conclusion

By extending the idea of hyper BCK-algebras introduced by Jun et al. [12], we have applied the hyperstructures to h-ideals in BCK-algebras. We have been able to explore a number of new results by characterizing hyper h-ideals with respect to various aspects and by describing their connections. We extended the study of hyper h-ideals by involving the concepts of weak, strong and reflexive hyper BCK-ideals and also by considering the ideas of hyper homomorphism and the product of these ideals.

It has been proved that the family of fuzzy hyper h-ideals is a completely distributive lattice with respect to join and meet. Using transfer principle for fuzzy sets, we proved that a fuzzy set ϖ is a fuzzy hyper h-ideal if and only if ϖ_t≠ ∅ is a hyper h-ideal. Lastly we discussed the product of fuzzy hyper h-ideals and proved that the product of any two fuzzy hyper h-ideals is again a fuzzy hyper h-ideal. This work can further be extended by applying the idea of intuitionistic fuzzy sets presented by Atanassov [1] to hyper h-ideals introduced here and the discussion of the properties of the intuitionistic fuzzy hyper h-ideals to be presented in this case by using the ideas of level subsets, strongest fuzzy relations, hyperhomomorphism and the products.

References

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Set Syst20 (1986), 87–96.

Borzooei

R.A.

, Hasankhani

, Zahedi

M.M.

and Jun

Y.B.

, On hyper K-algebras, Math Japonica52(1) (2000), 113–121.

Chvalina

, Commutative hypergroups in the sense of Marty and ordered sets, Summer School on General Algebra and Ordered Sets, Proc Internat Conf Olomouc (Czech Rep), 1994, pp. 19–30.

Chvalina

and Mayerova

S.H.

, On certain proximities and preorderings on the transposition hypergroups of linear first-order partial differential operators, Analele Stiintifice ale Uni. Ovidius Constanta, Ser Matematica21(1) (2013), 59–82.

Chvalina

, Mayerova

S.H.

and Nezhad

A.D.

, General actions of hypergroups and some applications, Analele Stiintifice ale Uni Ovidius Constanta, Ser Matematica21(1) (2013), 59–82.

Corsini

, Hypergraphs and hypergroups, Algebra Universalis36 (1996), 548–555.

Corsini

and Leoreanu

, Applications of hyperstructure theory, Advances in Mathematics, Vol. 5, Kluwer Academic Publishers, 2003.

Imai

and Iseki

, On axioms of propositional calculi, XIV Proc, Jpn Acad42 (1966), 19–22.

Jun

Y.B.

and Xin

X.L.

, Fuzzy hyper BCK-ideals of hyper BCKalgebras, Sci Math Jpn53(2) (2001), 415–422.

10.

Jun

Y.B.

and Xin

X.L.

, Scalar elements and hyperatoms of hyper BCK-algebras, Sci Math Jpn2(3) (1999), 303–309.

11.

Jun

Y.B.

, Xin

X.L.

, Zahedi

M.M.

and Roh

E.H.

, Strong hyper BCK-ideals of hyper BCK-algebras, Math Jpn51(3) (2000), 493–498.

12.

Jun

Y.B.

, Zahedi

M.M.

, Xin

X.L.

and Borzooei

R.A.

, On hyper BCK-algebras, Ital J Pure Appl Math8 (2000), 127–136.

13.

Kondo

and Dudek

W.A.

, On the transfer principle in fuzzy theory, Mathware Soft Comput12 (2005), 41–55.

14.

Marty

, Sur une generalization de la notion de groupe, 8th Congress Math Scandinaves, Stockholm, 1934, pp. 45–49.

15.

Rosenberg

I.G.

, An algebraic approach to hyperalgebras, Proc 26th Int Symp Multiple-Valued Logic, Santiago de Compostela, IEEE, 1996.