Hyperfuzzy subsets and subgroupoids

1 Introduction

Hyperstructures, in particular hypergroups, has been studied by F. Marty in 1934, and hundreds of papers and several books have been written in this area. The study of fuzzy hyperstructures is an interesting research area of fuzzy sets. It can be classified into three parts: crisp hyperoperations, fuzzy hyperalgebras, fuzzy hypergroups [8]. M. Krasner introduced the notions of the hyperfield and the hyperring in 1956, and he obtained more general structures, so called a hyperring [11], and A. Connes and C. Consani [5] discussed this structures in the study of the algebraic structures of the adèle class space of a global field. R. Ameri and R. Mahjoob [4] introduced the notion of prime fuzzy hyperideal of a commutative hyperring with identity. A.D. Khalafi and B. Davvaz [6] generalized some concepts of convex analysis such as convex functions and linearfunctions on hyperstructures, and obtained useful results in convex programming. Y.Q. Yin and X.K. Huang [15] considered the relations among L-fuzzy sets, rough sets and hyperring theory, and discussed the concept of L-fuzzy hyperideals of a hyperring based on a complete residuated lattice.

The notion of the semigroup (Bin (X) , □) was introduced by H.S. Kim and J. Neggers [10]. H. Fayoumi [7] introduced the notion of the center ZBin (X) in the semigroup Bin (X) of all binary systems on a set X, and showed that if (X, •) ∈ ZBin (X), then x ¬ = y implies {x, y} = {x • y, y • x}. Moreover, she showed that a groupoid (X, •) ∈ ZBin (X) if and only if it is a locally-zero groupoid. J.S. Han et al. [9] introduced the notion of hypergroupoids (HBin (X) , □), and showed that (HBin (X) , □) is a supersemigroup of the semigroup (Bin (X) , □) via the identification x{x}. They proved that (HBin^* (X) , ⊖ , [∅]) is a BCK-algebra. S.J. Shin et al. [12] introduced the notion of abelian fuzzy subsets on a groupoid, and discussed diagonal symmetric relations, convex sets, and the fuzzy center on Bin (X). In [13] they discussed properties of a class of real-valued functions on a groupoid (X, *) and fuzzy subsets on X related to (Bin (X) , □). S.S. Ahn et al. [1] studied fuzzy upper bounds in Bin (X). J. Zhan et al. [16] generalized the left-zero semigroup by introducing the notions of a weak-zero groupoid and an (X, N)-zero groupoid. P.A. Allen et al. [3] studied several types of groupoids related to semigroups, i.e., twisted semigroups. P.J. Allen et al. [2] developed a theory of companion d-algebras, and they showed that if (X, * , 0) is a d-algebra, then (Bin (X) , ⊕ , ⋄ ₀) is also a d-algebra. S.Z. Song et al. [14] studied soft saturated values and soft dried values in BCK/BCI-algebras. Thus, it is clear that the study of groupoids (binary systems) is undergoing vigorous development at present to which this paper aims to make a further useful contribution as well.

For those who consider “universal algebra” a subject not only without boundary, but possibly also without much meaning, even when suitably restricted to binary systems and some of its natural outgrowths, i.e., fuzzy groupoids, hypergroupoids, and even fuzzy hypergroupoids, or generalizations to rings, fields, modules and vector spaces, it may be worth noticing that many well-known notions can be fitted into this context and acquire new dimensions and in this way also access to new interpretations bringing opportunities for further developments even in “classical” areas of investigation. We indicate a few examples from a much larger collection to make our point. For example, consider a random variable X and its distribution function D _X  : R → R, where D _X  (t) = Pr (X ≤ t), the probability that X ≤ t. Then, since 0 ≤ Pr (X ≤ t) ≤1, it follows that D _X  : R → [0, 1], so that D _X can be interpreted as a particular case of a fuzzy subset of R. We may also consider D _X as generating a leftoid (R, * , D _X ) in the manner described above and thus as an element of (Bin (X) , □) combine it via the □-product with other element of this semigroup in useful ways.

Given (X, • , f) where f : X → [0, 1], with x • y = f (x), then we obtain a “fuzzy leftoid”. If (X, *) is any element of Bin (X), i.e., a groupoid (binary system), then we define a composition, also denoted by □, as follows: (X, *) □ (X, • , f) has x □ y = (x * y) • (y * x) = f (x * y) ∈ [0, 1]. Thus, in this construction we obtain a new fuzzy object which, though not a fuzzy subset (leftoid) is rather interesting and close to being one. It may be considered as a fuzzy subset of X × X via X × X → * X → f [ 0 , 1 ] . If [0, 1] ⊆ X, then (X, □) = (X, • , f) □ (X, *) also makes sense where x □ y = (x • y) * (y • x) = f (x) * f (y) which need not be a fuzzy object any longer. For (R, * , D _X ) with an arbitrary groupoid (R, •), we have (R, •) □ (R, * , D _X ) yields x □ y = (x • y) * (y • x) = D _X  (x • y) ∈ [0, 1], a set of probabilistic/statiscal/algebraic structures whose properties may be studied further and perhaps be useful in applications as well.

Consider the following situation. Starting with the set of real numbers and a random variable X on R, we let Θ denote the collection of all closed intervals on R, either of the types [a, b], (- ∞ , b] , [a, ∞) or (- ∞ , ∞) = R. Now consider H _X  : Θ → D^* [0, 1], where P^* [0, 1] is the set of all non-empty subsets of [0, 1]. If we let H _X  ([a, b]) : = [D _X  (a) , D _X  (b)] , H _X  ((- ∞ , b]) : = [0, D _X  (b)] , H _X  ([a, ∞)) : = [D _X  (a) , 1] and H _X  ((- ∞ , ∞)) = H _X  (R) = [0, 1], then H _X is a hyperfuzzy subset of R generated by the random variable X. The elements of Θ are considered to be (confidence) intervals and assigned a confidence levelD _X  (b) - D _X  (a) , D _X  (b) , 1 - D _X  (a) or 1 as the “length" of the intervals H _X  (I) on [0, 1] where I ∈ Θ. In decision making using probabilistic/statistical tools, the hyperfuzzy subset H _X is of critical importance and indirectly it has been much explored in that setting. It is not unreasonable to assume that many more examples already populate the literature and that in addition many more examples may be constructed as part of a more general study of such subjects to successfully answer questions about the usefulness of such studies.

As a general comment, we note that the fuzzy subset level the theory already existing has found wide application even if certain types of fuzzy subgroupoids may have escaped contact with practical considerations as of yet. Consulting the sufficiently voluminous hyperfuzzy literature indicates that there exists a shortage in many papers of examples of even abstract cases which would be helpful to those who might be interested in what is offered in those very papers. In this paper, along with the examples developed in the commentary, we have included several others to illustrate the ideas dealt with in a manner which allows the patient and careful reader to develop a feeling/understanding of what the actual statements of theorem and propositions are attempting to convey in the hope that this will result in further exploration of the subject by an increasing number of researchers in this area, including applications thereof.

2 Preliminaries

Given a non-empty set X, we let Bin (X) denote the collection of all groupoids (X, *), where * : X × X → X is a map and where * (x, y) is written in the usual product form. Given elements (X, *) and (X, •) of Bin (X), define a product “□” on these groupoids as follows: ( X , * ) □ ( X , • ) = ( X , □ ) where x □ y = ( x * y ) • ( y * x ) for any x, y ∈ X. Using that notion, H.S. Kim and J. Neggers proved the following theorem.

Theorem 2.1. [10] (Bin (X) , □) is a semigroup, i.e., the operation “□” as defined in general is associative. Furthermore, the left- zero-semigroup is the identity for this operation.

3 Hyperfuzzy subsets

Let (X, *) be a groupoid, i.e., (X, *) ∈ Bin (X). A map μ : (X, *) → P ([0, 1]) is said to be a hyperfuzzy subset of (X, *) where P ([0, 1]) is the collection of all subsets of [0, 1]. Hence every hyperfuzzy subset μ : (X, *) → P ([0, 1]) generates at least two fuzzy subsets of (X, *). In fact, given a hyperfuzzy subset μ : (X, *) → P ([0, 1]), if we define inf μ : (X, *) → [0, 1] by (inf μ) (x) : = inf {μ (x)} and sup μ : (X, *) → [0, 1] by (sup μ) (x) : = sup {μ (x)} for all x ∈ X, then these are fuzzy subsets of (X, *).

Given a fuzzy subset μ : (X, *) → [0, 1], it generates a hyperfuzzy subset μ ^e  : (P (X) , ★) → P ([0, 1]) given by μ ^e  (A) : = {μ (a) | a ∈ A} ⊆ [0, 1] so that μ ^e  (A) ∈ P ([0, 1]) where A ∈ P (X) and A ★ B : = {a * b | a ∈ A, b ∈ B}. Such a map μ ^e is called an extended hyperfuzzy subset of P (X) by μ. Given A, B ∈ P (X), we define μ ^e  (A ★ B) : = {μ (a * b) | a ∈ A, b ∈ B} ⊆ [0, 1].

Example 3.1. Let X = {a, b, c, d} be a set with the following table.

Define a map μ : (X, *) → P ([0, 1]) by μ ( a ) : = [ 0 , 1 2 ] , μ ( b ) : = ( 1 3 , 2 3 ) , μ ( c ) : = ( 2 3 , 4 5 ] and μ (d) : = [0.2, 0.9). Then we have two fuzzy subsets inf μ and sup μ as follows: inf μ = ( a b c d 0 1 3 2 3 0.2 ) and sup μ = ( a b c d 1 2 2 3 4 5 0.9 )

Given a fuzzy subset μ : X → [0, 1] and a groupoid (X, *), we have the following associated types: for any x, y ∈ X,

μ is a fuzzy subgroupoid of (X, *), shortly an F-fuzzy subgroupoid of (X, *) if μ (x * y) ≥ min {μ (x) , μ (y)},

Example 3.2. Let X : = Z be the set of all integers and let x * y : = |x| + |y|. If we define a map μ (x) : =1 - exp ^-|x| on X, then μ (0) =1 and lim | x | → ∞ μ ( x ) = 1 . For any x, y ∈ X, we have μ (x * y) =1 - exp ^-(|x|+|y|) ≥ max {1 - exp ^-|x|, 1 - exp ^-|y|} = max {μ (x) , μ (y)}. Hence μ is a reverse-contra fuzzy subgroupoid of X.

Example 3.3. In Example 3.2, if we define μ ( x ) : = 1 1 + | x | , then μ ( x * y ) = μ ( | x | + | y | ) = 1 1 + | x | + | y | ≤ min { 1 1 + | x | , 1 1 + | y | } = min { μ ( x ) , μ ( y ) } for all x, y ∈ X. Hence μ is a reverse-fuzzy subgroupoid of X.

Given a fuzzy subset μ : X → [0, 1], we have the following subsets of Bin (X):

F (μ) : = {(X, *) ∈ Bin (X) | μ : anF - fuzzy subgroupoidof (X, *)},

Using the notions of this concept, we obtain the following proposition.

Proposition 3.4. Given a fuzzy subset μ : X → [0, 1], (F (μ) , □) , (CF (μ) , □) , (RF (μ) , □) , (RCF (μ) , □) are subsemigroups of (Bin (X) , □).

Proof. Straightforward. □ Let (X, *) ∈ Bin (X). A hyperfuzzy subset μ : (X, *) → P ([0, 1]) is said to be an (F, F)-hyperfuzzy subgroupoid of (X, *) if

(inf μ) (x * y) ≥ min {(inf μ) (x) , (inf μ) (y)}

Given a fuzzy subset μ : X → [0, 1], we define a subclass (F, F) (μ) of Bin (X) by ( F , F ) ( μ ) : = { ( X , * ) ∈ Bin ( X ) | μ is an ( F , F ) - hyperfuzzy subgroupoid of ( X , * ) }

Proposition 3.5. ((F, F) (μ) , □) is a subsemigroup of (Bin (X) , □).

Proof. Given groupoids (X, *) , (X, ·) ∈ (F, F) (μ), we let (X, □) : = (X, *) □ (X, ·). Then we obtain (inf μ) (x * y) ≥ min {(inf μ) (x) , (inf μ) (y)} and (inf μ) (x · y) ≥ min {(inf μ) (x) , (inf μ) (y)} for all x, y ∈ X. It follows that ( inf μ ) ( x □ y ) = ( inf μ ) ( ( x * y ) · ( y * x ) ) ≥ min { ( inf μ ) ( x * y ) , ( inf μ ) ( y * x ) } ≥ min { min { ( inf μ ) ( x ) , ( inf μ ) ( y ) } , min { ( inf μ ) ( x ) , ( inf μ ) ( y ) } } = min { ( inf μ ) ( x ) , ( inf μ ) ( y ) } .

Similarly, we obtain ( sup μ ) ( x □ y ) ≥ min { ( sup μ ) ( x ) , ( sup μ ) ( y ) } This shows that (X, □) ∈ (F, F) (μ). □

Let (X, *) ∈ Bin (X) and let μ : X → [0, 1] be a fuzzy subset of X. We denote (X, *) ∈ (A ⊓ B) (μ) if (X, *) ∈ A (inf μ) and (X, *) ∈ B (sup μ), i.e., inf μ is an A-fuzzy subgroupoid of (X, *), and sup μ is a B-fuzzy subgroupoid of (X, *) where A, B ∈ {F, CF, RF, RCF}. The following proposition can be easily proved.

Proposition 3.6. Given a fuzzy subset μ : X → [0, 1], ((A ⊓ B) (μ) , □) is a subsemigroup of (Bin (X) , □).

Theorem 3.7. Given a fuzzy subset μ : X → [0, 1], if (X, *) ∈ F (μ), then (P (X) , ★) ∈ F (inf μ ^e ).

Proof. Since (X, *) ∈ F (μ), we have μ (x * y) ≥ min {μ (x) , μ (y)} for all x, y ∈ X. For any A, B ∈ P (X), we have ( inf μ e ) ( A ★ B ) = inf { μ ( a * b ) | a ∈ A , b ∈ B } ≥ inf { min { μ ( a ) , μ ( b ) } | a ∈ A , b ∈ B } ≥ min { inf { μ ( a ) | a ∈ A } , inf { μ ( b ) | b ∈ B } } = min { ( inf μ e ) ( A ) , ( inf μ e ) ( B ) } .

Hence (P (X) , ★) ∈ F (inf μ ^e ). □

Theorem 3.8. Given a fuzzy subset μ : X → [0, 1], if (X, *) ∈ F (μ), then (P (X) , ★) ∈ F (sup μ ^e ).

Proof. Since (X, *) ∈ F (μ), we have μ (x * y) ≥ min {μ (x) , μ (y)} for all x, y ∈ X. For any A, B ∈ P (X), we have ( sup μ e ) ( A ★ B ) = sup { μ ( a * b ) | a ∈ A , b ∈ B } .

For any ∊ > 0, there exists a₀ ∈ X such that μ ( a 0 ) > sup { μ ( a ) | a ∈ A } - ∊ and there exists b₀ ∈ X such that μ ( b 0 ) > sup { μ ( b ) | b ∈ B } - ∊ .

It follows that sup { μ ( a * b ) | a ∈ A , b ∈ B } ≥ μ ( a 0 * b 0 ) ≥ min { μ ( a 0 ) , μ ( b 0 ) } ≥ min { sup { μ ( a ) | a ∈ A } - ∊ , sup { μ ( b ) | b ∈ B } - ∊ } ≥ min { ( sup μ e ) ( A ) - ∊ , ( sup μ e ) ( B ) - ∊ } = min { ( sup μ e ) ( A ) , ( sup μ e ) ( B ) } - ∊ .

This shows that (sup μ ^e ) (A ★ B) ≥ min {(sup μ ^e ) (A) , (sup μ ^e ) (B)}, proving the theorem. □

By Theorems 3.7 and 3.8, we obtain the following corollary.

Corollary 3.9. Given a fuzzy subset μ : X → [0, 1], if (X, *) ∈ F (μ), then (P (X) , ★) ∈ (F ⊓ F) (μ ^e ).

Theorem 3.10. Given a fuzzy subset μ : X → [0, 1], if (X, *) ∈ CF (μ), then (P (X) , ★) ∈ CF (inf μ ^e ).

Proof. Since (X, *) ∈ CF (μ), we have μ (x * y) ≤ max {μ (x) , μ (y)} for all x, y ∈ X. For any A, B ∈ P (X), we have ( inf μ e ) ( A ★ B ) = inf { μ ( a * b ) | a ∈ A , b ∈ B }

For any ∊ > 0, there exists a₀ ∈ X such that μ ( a 0 ) < inf { μ ( a ) | a ∈ A } + ∊ and there exists b₀ ∈ X such that μ ( b 0 ) < inf { μ ( b ) | b ∈ B } + ∊ .

It follows that μ ( a 0 * b 0 ) ≤ max { μ ( a 0 ) μ ( b 0 ) } < max { inf { μ ( a ) | a ∈ A } + ∊ , inf { μ ( a ) | a ∈ A } + ∊ } = max { ( inf μ e ) ( A ) , ( inf μ e ) ( B ) } + ∊ .

This shows that (inf μ ^e ) (A ★ B) ≤ max {(inf μ ^e ) (A) , (inf μ ^e ) (B)}, proving the theorem. □

Theorem 3.11. Given a fuzzy subset μ : X → [0, 1], if (X, *) ∈ CF (μ), then (P (X) , ★) ∈ CF (sup μ ^e ).

Proof. Assume that (P (X) , ★) ∉ CF (sup μ ^e ). Then there exist A, B ∈ P (X) such that (sup μ ^e ) (A ★ B) > max {(sup μ ^e ) (A) , (sup μ ^e ) (B)}. It follows that there exist a₀ ∈ A and b₀ ∈ B such that μ ( a 0 * b 0 ) > max { ( sup μ e ) ( A ) , ( sup μ e ) ( B ) } Since max {μ (a₀) , μ (b₀)} ≥ μ (a₀ * b₀), we have either μ ( a 0 ) > max { ( sup μ e ) ( A ) , ( sup μ e ) ( B ) } ≥ ( sup μ e ) ( A ) or μ ( b 0 ) > max { ( sup μ e ) ( A ) , ( sup μ e ) ( B ) } ≥ ( sup μ e ) ( B ) , a contradiction. □

By Theorems 3.10 and 3.11, we obtain the following corollary.

Corollary 3.12. Given a fuzzy subset μ : X → [0, 1], if (X, *) ∈ CF (μ), then (P (X) , ★) ∈ (CF ⊓ CF) (μ ^e ).

Using the same method, we obtain the following corollary.

Corollary 3.13. Let μ : X → [0, 1] be a fuzzy subset of (X, *).

if (X, *) ∈ RF (μ), then (P (X) , ★) ∈ (RF ⊓ RF) (μ ^e ),

4 Convex hull of hyperfuzzy subsets

Let (X, *) ∈ Bin (X) and let μ : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *). Let ch (μ) : (X, *) → P ([0, 1]) be a map defined by ch (μ) (x) is a convex hull of μ (x) for all x ∈ X, i.e., ch (μ) (x) is an interval such that for all α, β ∈ μ (x), for all γ ∈ [0, 1], γα + (1 - γ) β ∈ ch (μ) (x). The following proposition can be easilyproved.

Proposition 4.1. Let μ : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *). Then

μ (x) ⊆ ch (μ) (x),

Example 4.2. Let x _N  ∈ (X, *) and N be a natural number. If μ ( x N ) : = { 1 n | n ≥ N , n ∈ N } , then μ ( x N ) = { 1 N , 1 N + 1 , ⋯ } and ch ( μ ) ( x N ) = ( 0 , 1 N ] . Hence ( sup μ ) ( x N ) = 1 N = ( sup ch ( μ ) ) ( x N ) = 1 N and (inf μ) (x _N ) = (inf ch (μ)) (x _N ) =0.

Given a hyperfuzzy subset μ : (X, *) → P ([0, 1]), we define a map l (μ) : (X, *) → [0, 1] by l (μ) (x) : = (sup μ) (x) - (inf μ) (x).

Example 4.3. Let D : R → [0, 1] be the distribution function of a random variable. If we define x * y : = 1 2 ( x + y ) for all x, y ∈ R, then x ≤ y implies x ≤ x * y ≤ y and hence D (x) ≤ D (x * y) ≤ D (y). It follows that (R, *) ∈ F (D) ∩ CF (D). If we consider the hyperfuzzy subset D ^e  : (P (R) , ★) → P ([0, 1]), then D e ( A ★ B ) = ∪ { D ( a + b 2 ) | a ∈ A , b ∈ B } for all A, B ∈ P (R). If we take A : = (-1, 1) and B : = (3, 5), then A ★ B = (1, 3) and D ^e  (A ★ B) = (D (1) , D (3)) and hence l (D) ((1, 3)) = D (3) - D (1).

Let (X, *) ∈ Bin (X) and let μ : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *). Given x, y ∈ X, we define x [ μ ) y ⇔ ( sup μ ) ( x ) ≤ ( sup μ ) ( y ) , x ( μ ] y ⇔ ( inf μ ) ( x ) ≤ ( inf μ ) ( y )

We denote x [μ] y if and only if x [μ) y and x (μ] y. In particular, if μ (x) , μ (y) are singletons, then x [μ) y  ⇔  x (μ] y   ⇔  μ (x) ≤ μ (y). The following propositions can be easily proved.

Proposition 4.4. Let (X, *) ∈ Bin (X) and let μ : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *). Then, for all x, y, z ∈ X,

x [μ) x,

Proposition 4.4^′. Let (X, *) ∈ Bin (X) and let μ : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *). Then, for all x, y, z ∈ X,

x (μ] x,

Clearly, if x [μ] y, then l (μ) (x) ≤ l (μ) (y), but the converse need not be true in general. Note that, since (inf ch (μ)) (x) = (inf μ) (x) and (sup ch (μ)) (x) = (sup μ) (x), the relations corresponding are identical, i.e., x [μ) y iff x [ch (μ)) y, x (μ] y iff x (ch (μ)] y, and x [μ] y iff x [ch (μ)] y.

Given hyperfuzzy subsets μ₁, μ₂ : (X, *) → P ([0, 1]), we let μ₁ [⇒) μ₂ provided x [μ₁) y implies x [μ₂) y. Also, we let μ₁ (⇒] μ₂ provided x (μ₁] y implies x (μ₂] y, and μ₁ [⇒] μ₂ provided x [μ₁] y implies x [μ₂] y. The following proposition can be easily proved.

Proposition 4.5. Let (X, *) ∈ Bin (X) and let μ _i  : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *) (i = 1, 2). Assume μ _i  (x) , μ _i  (y) are singletons (i = 1, 2). Then

if μ₁ [⇒) μ₂, then μ₁ (x) ≤ μ₁ (y) implies μ₂ (x) ≤ μ₂ (y),

Let μ : X → P ([0, 1]) be a hyperfuzzy subset of (X, *). Given x ∈ X, we define μ ^C  (x) : = {1 - α | α ∈ μ (x)}. It follows that (μ ^C )  ^C  (x) = μ (x), i.e., (μ ^C )  ^C  = μ for all x ∈ X. We call μ ^C the complementary hyperfuzzy subset of μ.

Using the notion of the complementary hyperfuzzy subset, we prove the following propositions easily.

Proposition 4.6. Let (X, *) ∈ Bin (X) and let μ : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *). Then, for all x ∈ X, we have

(inf μ ^C ) (x) =1 - (sup μ) (x),

Proposition 4.7. Let (X, *) ∈ Bin (X) and let μ : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *). Then, for all x, y ∈ X, we have

x [μ ^C ) y ⇔ x (μ] y,

5 Some operations of hyperfuzzy subsets

Given hyperfuzzy subsets μ₁, μ₂ : (X, *) → P ([0, 1]), we define ( μ 1 ∪ μ 2 ) ( x ) : = μ 1 ( x ) ∪ μ 2 ( x ) , ( μ 1 ∩ μ 2 ) ( x ) : = μ 1 ( x ) ∩ μ 2 ( x ) , μ ′ ( x ) : = { α ∈ [ 0 , 1 ] | α ∉ μ ( x ) } .

Define a map U : (X, *) → P ([0, 1]) by U (x) = [0, 1] for all x ∈ X. Given A ∈ P ([0, 1]), x ∈ X, we define maps μ^(A,∩) (x) : = μ (x) ∩ A, μ^(A,∪) (x) : = μ (x) ∪ A, (μ₁ \ μ₂) (x) : = μ₁ (x) \ μ₂ (x), (μ₁ △ μ₂) (x) : = μ₁ (x) △ μ₂ (x) and U′ (x) : = {α | α ∉ [0, 1]}. Using this notion, we can construct a Boolean algebra on the set of all hyperfuzzy subsets of (X, *).

Theorem 5.1. Let (X, *) ∈ Bin (X). If μ _i  : (X, *) → P ([0, 1]) is a hyperfuzzy subset of (X, *) (i = 1, 2), then l ( μ 1 ∪ μ 2 ) ( x ) ≥ max { l ( μ 1 ) ( x ) , l ( μ 2 ) ( x ) } for all x ∈ X.

Proof. Given x ∈ X, we have ( inf ( μ 1 ∪ μ 2 ) ) ( x ) = inf { μ 1 ( x ) ∪ μ 2 ( x ) } ≤ inf { μ i ( x ) } = ( inf μ i ) ( x ) (i = 1, 2). It follows that ( inf ( μ 1 ∪ μ 2 ) ) ( x ) ≤ min { ( inf μ 1 ) ( x ) , ( inf μ 2 ) ( x ) }

Similarly, (sup(μ₁ ∪ μ₂)) (x) ≥ sup {μ _i  (x)} = (sup μ _i ) (x) (i = 1, 2) implies that ( sup ( μ 1 ∪ μ 2 ) ) ( x ) ≥ max { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) }

Since max {a, b} + max {c, d} ≥ max {a + c, b + d} for all a, b, c, d ∈ [0, 1], we obtain l ( μ 1 ∪ μ 2 ) ( x ) = ( sup ( μ 1 ∪ μ 2 ) ) ( x ) - ( inf ( μ 1 ∪ μ 2 ) ) ( x ) ≥ max { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } - ( inf ( μ 1 ∪ μ 2 ) ) ( x ) ≥ max { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } - min { ( inf μ 1 ) ( x ) , ( inf μ 1 ) ( x ) } ≥ max { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } + max { - ( inf μ 1 ) ( x ) , - ( inf μ 1 ) ( x ) } ≥ max { ( sup μ 1 ) ( x ) - ( inf μ 1 ) ( x ) , ( sup μ 2 ) ( x ) - ( inf μ 2 ) ( x ) } = max { l ( μ 1 ) ( x ) , l ( μ 2 ) ( x ) } , proving the theorem. □

Note that (l (μ) , ∪) looks like an RCF-hyperfuzzy subgroupoid of (X, *) in Theorem 5.1.

Theorem 5.2. Let (X, *) ∈ Bin (X). If μ _i  : (X, *) → P ([0, 1]) is a hyperfuzzy subset of (X, *) (i = 1, 2), then l ( μ 1 ∩ μ 2 ) ( x ) ≤ min { l ( μ 1 ) ( x ) , l ( μ 2 ) ( x ) } for all x ∈ X.

Proof. Given x ∈ X, we have ( inf ( μ 1 ∩ μ 2 ) ) ( x ) = inf { μ 1 ( x ) ∩ μ 2 ( x ) } ≥ inf { μ i ( x ) } = ( inf μ i ) ( x ) (i = 1, 2) and ( sup ( μ 1 ∩ μ 2 ) ) ( x ) = sup { μ 1 ( x ) ∩ μ 2 ( x ) } ≤ sup { μ i ( x ) } = ( sup μ i ) ( x ) (i = 1, 2). It follows that ( inf ( μ 1 ∩ μ 2 ) ) ( x ) ≥ max { ( inf μ 1 ) ( x ) , ( inf μ 2 ) ( x ) } and ( sup ( μ 1 ∩ μ 2 ) ) ( x ) ≥ min { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } Since min {a, b} + min {c, d} ≤ min {a + c, b + d} for all a, b, c, d ∈ [0, 1], we obtain l ( μ 1 ∩ μ 2 ) ( x ) = ( sup ( μ 1 ∩ μ 2 ) ) ( x ) - ( inf ( μ 1 ∩ μ 2 ) ) ( x ) ≤ min { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } - max { ( inf μ 1 ) ( x ) , ( inf μ 2 ) ( x ) } = min { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } + min { - ( inf μ 1 ) ( x ) , - ( inf μ 2 ) ( x ) } ≤ min { ( sup μ 1 ) ( x ) - ( inf μ 1 ) ( x ) , ( sup μ 2 ) ( x ) - ( inf μ 2 ) ( x ) } = min { l ( μ 1 ) ( x ) , l ( μ 2 ) ( x ) } , proving the theorem. □

Note that (l (μ) , ∩) looks like an RF-hyperfuzzy subgroupoid of (X, *) in Theorem 5.2.

We may take (sup μ^(∅,∩)) (x) = (inf μ^(∅,∩)) (x) =0 for all x ∈ X, and hence we may assume that l (μ^(∅,∩)) (x) =0 for all x ∈ X.

Proposition 5.3. Let (X, *) ∈ Bin (X). If μ : (X, *) → P ([0, 1]) is a hyperfuzzy subset of (X, *), then

l (μ ∪ μ′) (x)  ≥  max {l (μ) (x) , l (μ′) (x)} for all x ∈ X,

Proof. (i) Since (μ ∪ μ′) (x) = μ (x)  ∪ μ′ (x) = μ (x) ∪ {α | α ∉ μ (x)} = [0, 1] for all x ∈ X, we have l ( μ ∪ μ ′ ) ( x ) = ( sup ( μ ∪ μ ′ ) ) ( x ) - ( inf ( μ ∪ μ ′ ) ) ( x ) = 1 ≥ max { l ( μ ) ( x ) , l ( μ ′ ) ( x ) } (ii) Since (μ ∩ μ′) (x) = μ (x) ∩ μ′ (x) = ∅ = μ^(∅,∩) (x), we have l (μ ∩ μ′) (x) = (sup μ^(∅,∩)) (x) - (inf μ^(∅,∩)) (x) =0. □

Proposition 5.4. Let (X, *) ∈ Bin (X). If μ _i  : (X, *) → P ([0, 1]) is a hyperfuzzy subset of (X, *) (i = 1, 2), then l ( μ 1 \ μ 2 ) ( x ) ≤ l ( μ 1 ) ( x ) for all x ∈ X.

Proof. Since (μ₁ \ μ₂) (x) = μ₁ (x) \ μ₂ (x) ⊆ μ₁ (x), we have (sup(μ₁ \ μ₂)) (x) ≤ (sup μ₁) (x) and (inf(μ₁ \ μ₂)) (x) ≥ (inf μ₁) (x). It follows that l (μ₁ \ μ₂) (x) ≤ l (μ₁) (x), proving the proposition. □

Theorem 5.5. Let (X, *) ∈ Bin (X). If μ : (X, *) → P ([0, 1]) is a hyperfuzzy subset of (X, *), then ( RCF , RF ) ( μ ) ⊆ RCF ( l ( μ ) ) for all x ∈ X.

Proof. If (X, *) ∈ (RCF, RF) (μ), then ( sup μ ) ( x * y ) ≥ max { ( sup μ ) ( x ) , ( sup μ ) ( y ) } and ( inf μ ) ( x * y ) ≤ max { ( inf μ ) ( x ) , ( inf μ ) ( y ) } for all x, y ∈ X. It follows that l ( μ ) ( x * y ) = ( sup μ ) ( x * y ) - ( inf μ ) ( x * y ) ≥ ( sup μ ) ( x ) - ( inf μ ) ( x ) = l ( μ ) ( x )

Similarly, we obtain l (μ) (x * y) ≥ l (μ) (y). Hence l ( μ ) ( x * y ) ≥ max { l ( μ ) ( x ) , l ( μ ) ( y ) } , which shows that (X, *) ∈ RCF (l (μ)). □

By the similar method, we obtain the following theorem, and we omit its proof.

Theorem 5.6. Let (X, *) ∈ Bin (X). If μ : (X, *) → P ([0, 1]) is a hyperfuzzy subset of (X, *), then ( RF , RCF ) ( μ ) ⊆ RF ( l ( μ ) ) for all x ∈ X.

Let (X, *) ∈ Bin (X). Let μ _i  : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *) (i = 1, 2). We define a new map μ₁ ∨ μ₂ : (X, *) → P ([0, 1]) by ( μ 1 ∨ μ 2 ) ( x ) : = { α ∈ [ 0 , 1 ] | α = max { u , v } , u ∈ μ 1 ( x ) , v ∈ μ 2 ( x ) } for all x ∈ X.

Example 5.7. Let (X, *) ∈ Bin (X) and let x₀ ∈ X. If we define μ 1 ( x 0 ) : = { 0 , 1 2 , 4 5 } , μ 2 ( x 0 ) : = { 0 , 2 3 } for some x₀ ∈ (X, *), then (μ₁ ∨ μ₂) (x₀) = { max { 0 , 0 } , max { 0 , 2 3 } , max { 1 2 , 0 } , max { 1 2 , 2 3 } , max { 4 5 , 0 } , max { 4 5 , 2 3 } } = { 0 , 1 2 , 2 3 , 4 5 } . Hence ( sup ( μ 1 ∨ μ 2 ) ) ( x 0 ) = 4 5 , ( inf ( μ 1 ∨ μ 2 ) ) ( x 0 ) = 0 .

Proposition 5.8. Let (X, *) ∈ Bin (X). If μ _i  : (X, *) → P ([0, 1]) are hyperfuzzy subsets of (X, *) (i = 1, 2), then ( sup ( μ 1 ∨ μ 2 ) ) ( x ) = max { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } for all x ∈ X.

Proof. Given x ∈ X, if we let α : = (sup μ₁) (x) , β : = (sup μ₂) (x), then ( sup ( μ 1 ∨ μ 2 ) ) ( x ) = sup { a ∈ [ 0 , 1 ] | a = max { u , v } , u ∈ μ 1 ( x ) , v ∈ μ 2 ( x ) } = sup { max { u , v } | u ∈ μ 1 ( x ) , v ∈ μ 2 ( x ) } = sup { max { α , v | v ∈ μ 2 ( x ) } , max { α , β } , max { u , v | u ∈ μ 1 ( x ) , v ∈ μ 2 ( x ) } , max { u , β | u ∈ μ 1 ( x ) } } = max { α , β } = max { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } , which proves the proposition. □

Proposition 5.9. Let (X, *) ∈ Bin (X). If μ _i  : (X, *) → P ([0, 1]) are hyperfuzzy subsets of (X, *) (i = 1, 2), then ( inf ( μ 1 ∨ μ 2 ) ) ( x ) = max { ( inf μ 1 ) ( x ) , ( inf μ 2 ) ( x ) } for all x ∈ X.

Proof. Given x ∈ X, if we let α : = (inf μ₁) (x) , β : = (inf μ₂) (x), then the proof is similar to Proposition 5.9, and we omit it. □

Theorem 5.10. Let (X, *) ∈ Bin (X). If μ _i  : (X, *) → P ([0, 1]) are hyperfuzzy subsets of (X, *) (i = 1, 2), then ( l ( μ 1 ∨ μ 2 ) ) ( x ) ≥ min { ( l ( μ 1 ) ) ( x ) , ( l ( μ 2 ) ) ( x ) } for all x ∈ X.

Proof. Since max {a, b} - max {c, d} ≥ min {a - c, b - d}, by Propositions 5.8 and 5.9, we obtain ( l ( μ 1 ∨ μ 2 ) ) ( x ) = ( sup ( μ 1 ∨ μ 2 ) ) ( x ) - ( inf ( μ 1 ∨ μ 2 ) ) ( x ) = max { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } - max { ( inf μ 1 ) ( x ) , ( inf μ 2 ) ( x ) } ≥ min { ( sup μ 1 ) ( x ) - ( inf μ 1 ) ( x ) , ( sup μ 2 ) ( x ) - ( inf μ 2 ) ( x ) } = min { ( l ( μ 1 ) ) ( x ) , ( l ( μ 2 ) ) ( x ) } , which proves the theorem. □

Note that max {a, b} - max {c, d} ≥ max {a - c, b - d} does not hold in general.

Example 5.11. Let (X, *) ∈ Bin (X) and let x₀ ∈ X. If we define μ 1 ( x 0 ) : = [ 1 3 , 2 3 ] , μ 2 ( x 0 ) : = 90.4 , 0.9 ] for some x₀ ∈ (X, *), then (μ₁ ∨ μ₂) (x₀) = (0.4, 0.9]. It shows that (inf(μ₁ ∨ μ₂)) (x₀) =0.4 and (sup(μ₁ ∨ μ₂)) (x₀) =0.9. Hence (l (μ₁ ∨ μ₂)) (x₀) =0.9 - 0.4 = 0.5 and ( l ( μ 1 ) ) ( x 0 ) = 2 3 - 1 3 = 1 3 , (l (μ₂)) (x₀) =0.9 - 0.4 = 0.5. Hence (l (μ₁ ∨ μ₂)) (x₀) ≥ min {(l (μ₁)) (x₀) , (l (μ₂)) (x₀).

Let (X, *) ∈ Bin (X). Let μ _i  : (X, *) → P ([0, 1]) be a hyperfuzzy subset of (X, *) (i = 1, 2). We define a new map μ₁ ∧ μ₂ : (X, *) → P ([0, 1]) by ( μ 1 ∧ μ 2 ) ( x ) : = { α ∈ [ 0 , 1 ] | α = min { u , v } , u ∈ μ 1 ( x ) , v ∈ μ 2 ( x ) } for all x ∈ X.

Example 5.12. In Example 5.7, it is easy to see that ( μ 1 ∧ μ 2 ) ( x 0 ) = { 0 , 1 2 , 2 3 } , and hence ( sup ( μ 1 ∧ μ 2 ) ) ( x 0 ) = 2 3 , ( inf ( μ 1 ∧ μ 2 ) ) ( x 0 ) = 0 .

Proposition 5.13. Let (X, *) ∈ Bin (X). If μ _i  : (X, *) → P ([0, 1]) are hyperfuzzy subsets of (X, *) (i = 1, 2), then ( i ) ( sup ( μ 1 ∧ μ 2 ) ) ( x ) = min { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } , ( ii ) ( inf ( μ 1 ∧ μ 2 ) ) ( x ) = min { ( inf μ 1 ) ( x ) , ( inf μ 2 ) ( x ) } , for all x ∈ X.

Proof. The proof is similar to the proofs of Propositions 5.8 and 5.9. □

Theorem 5.14. Let (X, *) ∈ Bin (X). If μ _i  : (X, *) → P ([0, 1]) are hyperfuzzy subsets of (X, *) (i = 1, 2), then ( l ( μ 1 ∧ μ 2 ) ) ( x ) ≤ max { ( l ( μ 1 ) ) ( x ) , ( l ( μ 2 ) ) ( x ) } for all x ∈ X.

Proof. Since min {a, b} - min {c, d} ≤ max {a - c, b - d}, by Proposition 5.13, we obtain ( l ( μ 1 ∧ μ 2 ) ) ( x ) = ( sup ( μ 1 ∧ μ 2 ) ) ( x ) - ( inf ( μ 1 ∧ μ 2 ) ) ( x ) = min { ( sup μ 1 ) ( x ) , ( sup μ 2 ) ( x ) } - min { ( inf μ 1 ) ( x ) , ( inf μ 2 ) ( x ) } ≤ max { ( sup μ 1 ) ( x ) - ( inf μ 1 ) ( x ) , ( sup μ 2 ) ( x ) - ( inf μ 2 ) ( x ) } = max { ( l ( μ 1 ) ) ( x ) , ( l ( μ 2 ) ) ( x ) } , which proves the theorem. □

6 Conclusion

In the theory developed above we have noted that extending a fuzzy subset in a natural way and employing the usual definition of a fuzzy (sub)groupoid μ (x * y) ≥ min {μ (x) , μ (y)}, μ : X → [0, 1], and extending it in the natural way of extending both concepts, allows us to discuss the notion of hyperfuzzy (sub)groupoids also as a natural extension of the basic concepts underlying the results obtained in this paper. In particular, in the semigroup (Bin (X) , □), we are able to identify new subsemigroups of binary systems (groupoids) from natural structures identified in the corresponding semigroups of hyperfuzzy groupoids, showing that the flow of generalizations is a natural one. Being a large collection of objects than Bin (X), there are introduced new parameters on the collection of hyperfuzzy groupoids which collapse to trivial forms on ordinary groupoids in Bin (X). This is also a good sign that a more extensive theory, such as proposed in this paper, is available to the researcher.

7 Future works

As observed in the conclusion above, further investigations into the theoretical aspects of a theory of hyperfuzzy (sub)groupoids of groupoids on set X is certainly indicated by results obtained in the paper above in this direction already. Thus, it is logical to continue with further research in this area. The introduction of fuzzy concepts and thus of hyperfuzzy concepts, as is done for groupoids on sets X in the case of this paper, and the reasons for doing so in general, indicate that one should look for practical results and conclusions where possible. This might be done indirectly, e.g., by converting “graph theory” into a branch of the theory of groupoids, i.e., selective groupoids and thus obtain new graph-theoretic results, and other similar approaches and more direct approach to other classes of problems and their models as well. For example, certain parameters, such as the independence number of a graph for example, can be translated into a similar parameter for groupoids (or an interesting subclass thereof and naturally expand to related concepts for hyperfuzzy groupoids which may then be directly interpreted. It is among our plans for future work to include this approach in several different settings including the situation just mentioned. The gate is wide-open and the opportunities available here look inviting enough for future projects to succeed in this area.