Fuzzy hyper h -ideals of 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 ⊃ B . ⊃ . ( C ⊃ A ) ⊃ ( C ⊃ B )

AxiomC     A ⊃ ( B ⊃ C ) . ⊃ . B ⊃ ( A ⊃ C )

AxiomK     A ⊃ ( B ⊃ 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

Obviously, any BCK-algebra (G, * , 0) becomes a hyper BCK-algebra w.r.t the operation ι ∘ 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 ∘ ȷ = { [ 0 , ι ] , if ι < ȷ , [ 0 , ȷ ] , if ι > ȷ ≠ 0 , { ι } , if ȷ = 0 .

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}

Let ϒ be a non-empty subset of “hyper BCK-algebra” ϝ and 0 ∈ ϒ. Then ϒ is a “hyper BCK-subalgebra” (or H _BCK S) of ϝ if ι ∘ ȷ ⊆ ϒ, for any ι, ȷ ∈ ϒ, a “weak hyper BCK-ideal” (or wH _BCK I) of ϝ if ι ∘ ȷ ⊆ ϒ and ȷ ∈ ϒ imply ι ∈ ϒ, for all ι, ȷ ∈ ϝ, a “hyper BCK-ideal” (or H _BCK I) of ϝ if ι ∘ ȷ  << ϒ and ȷ ∈ ϒ imply ι ∈ ϒ, for any ι, ȷ ∈ ϝ, a “strong hyper BCK-ideal” (or sH _BCK I) 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 _BCK I) of ϝ is a sH _BCK I of ϝ.

Lemma 2.4. [11] Let ϒ be a rH _BCK I 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 _BCK I 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 _h I) of ϝ if ι ∘ (ȷ ∘ ℓ) ⊆ ϒ and ȷ ∈ ϒ imply (ι ∘ ℓ) ⊆ ϒ.

Theorem 2.7. Any H _h I (resp. wH _h I,  sH _h I,  rH _h I) of ϝ is a H _BCK I (resp. wH _BCK I,  sH _BCK I,  rH _BCK I) of ϝ.

Proof. Straightforward.□

Theorem 2.8. For any ϝ,

Any H _h I of ϝ is a wH _h I of ϝ.

Proof. (i) . Let ϒ be a H _h I 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 _h I of ϝ.

(ii) . Let ϒ be a sH _h I 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 _BCK I 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 _h Iof ϝ.

(iii) . Let ϒ be a rH _h I of ϝ. For any ι, ȷ , ℓ ∈ ϝ, let ι∘ (ȷ ∘ ℓ) ⋂ ϒ ≠ ∅ and ȷ ∈ ϒ. Being a rH _h I, ϒ is also a rH _BCK I 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 _h I 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: ∘ { 0 } { ι } { ȷ } 0 { 0 } { 0 } { 0 } ι { ι } { 0 , ι } { 0 , ι } ȷ { ȷ } { ȷ } { 0 , ȷ } Take ϒ = {0, ȷ}. Then ϒ is a wH _h I of ϝ but it is not a H _h I 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 _h I of ϝ but it is not a sH _h I 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: ∘ { 0 } { ι } { ȷ } 0 { 0 } { 0 } { 0 } ι { ι } { 0 } { 0 } ȷ { ȷ } { ι , ȷ } { 0 , ι , ȷ }

Take ϒ = {0, ι}. Then ϒ is a H _h I of ϝ but it is not a sH _h I 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 _BCK I)of ϝ if ϖ ( 0 ) ≥ ϖ ( ι ) ≥ min { inf u ∈ ι ∘ ȷ ϖ ( u ) , ϖ (ȷ)}.

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 _BCK I of ϝ is a FwH _BCK I of ϝ.

3 Fuzzy hyper h-ideals

Now, we present the notions of “fuzzy weak hyper h-ideals” (or FwH _h Is),  “fuzzy hyper h-ideals” (or FH _h Is), “fuzzy strong hyper h-ideals” (or FsH _h Is) and “fuzzy reflexive hyper h-ideals” FrH _h Is and confer some of their properties.

Definition 3.1. A fuzzy set ϖ in ϝ is

a “fuzzy weak hyper h-ideal” (or FwH _h I) of ϝ if ϖ (0) ≥ ϖ (ι) and for all t∈ ι ∘ ℓ, ϖ ( t ) ≥ min { inf u ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( u ) , ϖ ( ȷ ) }.

Theorem 3.2. Any FH _h I (resp. FwH _h I,  FsH _h I,  FrH _h I) of ϝ is a FH _BCK I (resp. FwH _BCK I,  FsH _BCK I,  FrH _BCK I) of ϝ.

proof. Straightforward.□

Theorem 3.3. For any ϝ,

Any FH _h I of ϝ is a FwH _h I of ϝ.

Proof. (i) . Let ϖ be a FH _h I of ϝ. Since any FH _h I is a FH _BCK I (by Theorem 3.2) and every FH _BCK I is a FwH _BCK I (By Theorem 2.12), therefore ϖ is also a FwH _BCK I of ϝ. Hence ϖ satisfies ϖ (0) ≥ ϖ (ι) for all ι∈ ϝ. Also being a FH _h I, ∀ ι , ȷ , ℓ ∈ ϝ and t∈ ι ∘ ℓ, ϖ satisfies: ϖ ( t ) ≥ min { inf u ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( u ) , ϖ ( ȷ ) } .

Hence ϖ is a FwH _h I of ϝ.

(ii) . Let ϖ be a FsH _h I of ϝ. Since FsH _h I is a FsH _BCK I (by Theorem 3.2.) and every FsH _BCK I is a FH _BCK I (by Theorem 2.12), therefore ϖ is also a FH _BCK I of ϝ. Hence for any ι, ȷ ∈ ϝ, if ι << ȷ then ϖ (ι) ≥ ϖ (ȷ). Also being a FsH _h I, ∀ ι , ȷ , ℓ ∈ ϝ and t∈ ι ∘ ℓ, ϖ satisfies ϖ ( t ) ≥ min { sup a ∈ ( ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) , ϖ ( ȷ ) } .

Since sup a ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) ≥ ϖ ( b ), for all b ∈ ι ∘ (ȷ ∘ ℓ), Thus, ϖ ( t ) ≥ min { sup a ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) , ϖ ( ȷ ) } ≥ min { ϖ ( b ) , ϖ ( ȷ ) }, for all b ∈ ι ∘ (ȷ ∘ ℓ) Since ϖ ( b ) ≥ inf c ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( c ) for all b ∈ ι ∘ (ȷ ∘ ℓ), Therefore, ϖ ( t ) ≥ min { ϖ ( b ) , ϖ ( ȷ ) } ≥ min { inf c ∈ ι ∘ ( ȷ ∘ ℓ )  ϖ (c) ,  ϖ (ȷ)}, i.e., ϖ (t)≥ min { inf c ∈ ι ∘ ( ȷ ∘ ℓ )  ϖ (c) ,  ϖ (ȷ)}. Hence ϖ is a FH _h I of ϝ.

(iii) . Let ϖ be a FrH _h I of ϝ. Then ϖ satisfies inf u ∈ ι ∘ ι ϖ ( u ) ≥ ϖ ( ȷ ), ∀ ι , ȷ ∈ ϝ ⇒ inf u ∈ ι ∘ ι ϖ ( u ) ≥ ϖ ( ι ), for all ι∈ ϝ. Hence the first condition for ϖ to be a FsH _h I of ϝ is satisfied. Also being a FrH _h I, for all ι, ȷ , ℓ ∈ ϝ and t∈ ι ∘ ℓ, ϖ satisfies ϖ ( t ) ≥ min { sup v ∈ ( ι ∘ ( ȷ ∘ ℓ ) ϖ ( v ) , ϖ ( ȷ ) }. Hence ϖ is a FsH _h I 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 _h I of ϝ but it is not a FH _h I of ϝ because: ι ≤ ȷ ⇒ ϖ ( ι ) = 0 < 1 = ϖ ( ȷ ) .

The next example illustrates that a FH _h I may not be FsH _h I and similarly, a FsH _h I may not be a FrH _h I.

Example 3.4. Let ϝ = {0, ι , ȷ} be a hyper BCK-algebra defined by the succeeding table: ∘ { 0 } { ι } { ȷ } 0 { 0 } { 0 } { 0 } ι { ι } { 0 , ι } { 0 , ι } ȷ { ȷ } { ι , ȷ } { 0 , ι , ȷ }

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

Then ϖ is a FH _h I of ϝ but it is not a FsH _h I of ϝ because: ϖ ( ȷ ) = 0.2 < 0.8 = min { sup a ∈ ȷ ∘ ( ι ∘ 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 ) = 1 2

Then ϖ is a FsH _h I of ϝ but it is not a FrH _h I of ϝ because for ι, ȷ ∈ ȷ ∘ ȷ, ϖ ( ι ) = ϖ ( ȷ ) = 1 2 < 1 = ϖ ( 0 )

Theorem 3.5. Let ϖ be a fuzzy set in ϝ. Then ϖ is a FH _h I (resp. FwH _h I,  FsH _h I,  FrH _h I) of ϝ if and only if for all t ∈ [0, 1], ϖ _t ≠ ∅ is a H _h I (resp. wH _h I,  sH _h I,  rH _h I) of ϝ.

Proof. Suppose that ϖ is a FH _h I of ϝ. Since ϖ _t ≠ ∅, so for any ι ∈ ϖ _t , ϖ (ι) ≥ t. Since every FH _h I is also a FwH _h I (by Theorem 3), so ϖ is also a FwH _h I 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 ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) ≥ t. Also ϖ (ȷ) ≥ t, as ȷ ∈ ϖ _t . Therefore for all v∈ ι ∘ ℓ, ϖ satisfies ϖ ( v ) ≥ min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) , ϖ ( ȷ ) } ≥ min { t , t } = t then v ∈ ϖ _t , for all v∈ ι ∘ ℓ and then ι ∘ ℓ ⊆ ϖ _t . Hence ϖ _t is H _h I of ϝ.

Conversely, suppose that ϖ _t ≠ ∅ is a H _h I 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 ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( c ) , ϖ ( ȷ ) }

Then ϖ (ȷ) ≥ d ⇒ ȷ ∈ ϖ _d and for all e ∈ ι ∘ (ȷ ∘ ℓ), ϖ ( e ) ≥ inf c ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( c ) ≥ 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 ) ≥ d = min { inf c ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( c ) , ϖ ( ȷ ) }

Thus ϖ is a FH _h I of ϝ.□

Theorem 3.6. If ϖ is a FH _h I (resp. FwH _h I,  FsH _h I,  FrH _h I) of ϝ then the set A = { ι ∈ ϝ ∣ ϖ ( ι ) = ϖ ( 0 ) } is a H _h I (resp. wH _h I,  sH _h I,  rH _h I) of ϝ.

Proof. Suppose that ϖ is a FH _h I 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 _h I, ϖ is also a FwH _h I 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 ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) = ϖ ( 0 ). Also ϖ (ȷ) = ϖ (0). Being a FH _h I, for all t∈ ι ∘ ℓ, ϖ satisfies ϖ ( t ) ≥ min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) , ϖ ( ȷ ) } = min { ϖ ( 0 ) , ϖ ( 0 ) } = ϖ ( 0 )

Since ϖ (0) ≥ ϖ (v), for all v∈ ϝ, therefore ϖ ( t ) = ϖ ( 0 ) , for all t ∈ ι ∘ ℓ

Thus ι ∘ ℓ ⊆ A. Hence A is a H _h I 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 ) = { t if ι ∈ A 0 if ι ∉ A for all ι∈ ϝ, where t ∈ (0, 1]. Then A is a H _h I (resp. wH _h I,  sH _h I,  rH _h I) of ϝ if and only if ϖ is a FH _h I (resp. FwH _h I,  FsH _h I,  FrH _h I) of ϝ.

Proof. Suppose that A is a H _h I 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 _BCK I 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 ∈ ι ∘ ( ȷ ∘ 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 ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) , ϖ ( ȷ ) }

If ι ∘ (ȷ ∘ ℓ) notllA and ȷ ∉ A then min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) , ϖ ( ȷ ) } = 0 ≤ 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 ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ ( a ) , ϖ ( ȷ ) } = 0 ≤ ϖ ( u ) for all u∈ ι ∘ ℓ. Hence ϖ is a FH _h I 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 _h I 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: ( ∨ i ∈ I ϖ i ) ( ι ) = sup i ∈ I ϖ i ( ι ) ( ∧ i ∈ I ϖ i ) ( ι ) = inf i ∈ I ϖ i ( ι ) 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 ( ∨ i ∈ I ϖ i ) ( ι ) = sup i ∈ I ϖ i ( ι ) ≥ sup i ∈ I ϖ i ( ȷ ) = ( ∨ i ∈ I ϖ i ) ( ȷ ) and then ( ∨ i ∈ I ϖ i ) ( ι ) ≥ ( ∨ i ∈ I ϖ i ) ( ȷ )

Moreover, for any ι, ȷ , ℓ ∈ ϝ and for all t∈ ι ∘ ℓ, we have ( ∨ i ∈ I ϖ i ) ( t ) = sup i ∈ I ϖ i ( t ) ≥ sup i ∈ I [ min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ i ( a ) , ϖ i ( ȷ ) } ] = min { sup i ∈ I ( inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ i ( a ) ) , sup i ∈ I ( ϖ i ( ȷ ) ) } = min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ( sup i ∈ I ϖ i ( a ) ) , sup i ∈ I ( ϖ i ( ȷ ) ) } = min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ( ( ∨ i ∈ I ϖ i ) ( a ) ) , ( ∨ i ∈ I ϖ i ) ( ȷ ) } and then ( ∨ i ∈ I ϖ i ) ( t ) ≥ min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ( ( ∨ i ∈ I ϖ i ) ( a ) ) , ( ∨ i ∈ I ϖ i ) ( ȷ ) }

Hence ∨_i∈I ϖ _i is a FH _h I of ϝ.

Now we prove that ∧_i∈I ϖ _i is a FH _h I of ϝ. For any ι, ȷ ∈ ϝ we have, if ι << ȷ then ( ∧ i ∈ I ϖ i ) ( ι ) = inf i ∈ I ϖ i ( ι ) ≥ inf i ∈ I ϖ i ( ȷ ) = ( ∧ i ∈ I ϖ i ) ( ȷ ) and then ( ∧ i ∈ I ϖ i ) ( ι ) ≥ ( ∧ i ∈ I ϖ i ) ( ȷ ) Moreover, for any ι, ȷ , ℓ ∈ ϝ and for all t∈ ι ∘ ℓ, we have ( ∧ i ∈ I ϖ i ) ( t ) = inf i ∈ I ϖ i ( t ) ≥ inf i ∈ I [ min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ i ( a ) , ϖ i ( ȷ ) } ] = min { inf i ∈ I ( inf a ∈ ( ι ∘ ℓ ) ∘ ( ȷ ∘ ℓ ) ϖ i ( a ) ) , inf i ∈ I ( ϖ i ( ȷ ) ) } = min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ( inf i ∈ I ϖ i ( a ) ) , inf i ∈ I ( ϖ i ( ȷ ) ) } = min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ( ( ∧ i ∈ I ϖ i ) ( a ) ) , ( ∧ i ∈ I ϖ i ) ( ȷ ) } and then ( ∧ i ∈ I ϖ i ) ( t ) ≥ min { inf a ∈ ι ∘ ( ȷ ∘ ℓ ) ( ( ∧ i ∈ I ϖ i ) ( a ) ) , ( ∧ i ∈ I ϖ i ) ( ȷ ) }

Hence ∧_i∈I ϖ _i is a FH _h I 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 ) ∘ ( u 2 , v 2 ) = ( u 1 ∘ u 2 , v 1 ∘ 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 ∈ U , v ∈ V } and 0 = (0₁, 0₂), and a hyperorder “ <<” on ϝ by ( u 1 , v 1 ) < < ( u 2 , v 2 ) ⇔ 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 × 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 _h I (resp. FwH _h I,  FsH _h I,  FrH _h I) of ϝif and only if ϖ₁ and ϖ₂ are FH _h Is (resp. FwH _h Is,  FsH _h Is,  FrH _h Is) of ϝ₁ and ϝ₂ respectively.

Proof. Let ϖ be a FH _h I 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 ∈ ι 1 ∘ ( ȷ 1 ∘ ℓ 1 ) ϖ 1 ( a ) , ϖ 1 ( ȷ 1 ) }

Then for all b ∈ ι ₁ ∘ (ȷ ₁ ∘ ℓ ₁), ϖ 1 ( b ) ≥ inf a ∈ ι 1 ∘ ( ȷ 1 ∘ ℓ 1 ) ϖ 1 ( a ) ≥ t and ϖ 1 ( ȷ 1 ) ≥ t hence ϖ ( ( b , 0 ) ) ≥ t and ϖ ( ( ȷ 1 , 0 ) ) ≥ t for all (b, 0) ∈ (ι ₁, 0) ∘ ((ȷ ₁, 0) ∘ (ℓ ₁, 0)), ( b , 0 ) ∈ ϖ t and ( ȷ 1 , 0 ) ∈ ϖ t for all (b, 0) ∈ (ι ₁, 0) ∘ ((ȷ ₁, 0) ∘ (ℓ ₁, 0)), ( ι 1 , 0 ) ∘ ( ( ȷ 1 , 0 ) ∘ ( ℓ 1 , 0 ) ) ⊆ ϖ t and ( ȷ 1 , 0 ) ∈ ϖ t

By Transfer principle for fuzzy sets, ϖ _t ≠ ∅ is a H _h I of ϝ and so is a wH _h I of ϝ (by Theorem 3.8). Thus ( ι 1 , 0 ) ∘ ( ( ȷ 1 , 0 ) ∘ ( ℓ 1 , 0 ) ) ⊆ ϖ t and ( ȷ 1 , 0 ) ∈ 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 ) ≥ t = min { inf a ∈ ι 1 ∘ ( ȷ 1 ∘ ℓ 1 ) ϖ 1 ( a ) , ϖ 1 ( ȷ 1 ) } for all s ∈ ι ₁ ∘ ℓ ₁. Hence ϖ₁ is a FH _h I of ϝ₁.

Similarly, we can prove that ϖ₂ is a FH _h I of ϝ₂.

Conversely, let ϖ₁ and ϖ₂ be FH _h Is 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), ϖ ( ( a , b ) ) = ( ϖ 1 × ϖ 2 ) ( ( a , b ) ) = min { ϖ 1 ( a ) , ϖ 2 ( b ) } ≥ min [ min { inf c ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ 1 ( c ) , ϖ 1 ( ȷ ) } , min { inf d ∈ u ∘ ( v ∘ w ) ϖ 2 ( d ) , ϖ 2 ( v ) } ] = min [ min { inf c ∈ ι ∘ ( ȷ ∘ ℓ ) ϖ 1 ( c ) , inf d ∈ u ∘ ( v ∘ w ) ϖ 2 ( d ) } , min { ϖ 1 ( ȷ ) , ϖ 2 ( v ) } ] = min [ inf c ∈ ι ∘ ( ȷ ∘ ℓ ) , d ∈ u ∘ ( v ∘ w ) { min { ϖ 1 ( c ) , ϖ 2 ( d ) } } , min { ϖ 1 ( ȷ ) , ϖ 2 ( v ) } ] = min { inf ( c , d ) ∈ ( ι ∘ ( ȷ ∘ ℓ ) , u ∘ ( v ∘ w ) ) ( ϖ 1 × ϖ 2 ) min d ) ) , ( ϖ 1 × ϖ 2 ) ( ( ȷ , v ) ) } = min { inf ( c , d ) ∈ ( ι ∘ ( ȷ ∘ ℓ ) , u ∘ ( v ∘ w ) ) ϖ ( ( c , d ) ) , ϖ ( ( ȷ , v ) ) } and then ϖ ( ( a , b ) ) ≥ min { inf ( c , d ) ∈ ( ι , u ) ∘ ( ( ȷ , v ) ∘ ( ℓ , w ) ) ϖ ( ( c , d ) ) , ϖ ( ( ȷ , v ) ) }

Hence, ϖ is a FH _h I 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.