Almost φ -fuzzy semi-ideals in groupoids

1. Introduction

The notion of a fuzzy subset of a set was introduced by L. A. Zadeh ([16]). His seminal paper in 1965 has opened up new insights and applications in a wide range of scientific fields. A. Rosendfeld ([14]) used the notion of a fuzzy subset to set down corner stone papers in several areas of mathematics. J. N. Mordeson and D. S. Malik ([13]) published a remarkable book, Fuzzy commutative algebra, presented a fuzzy ideal theory of commutative rings and applied the results to the solution of fuzzy intersection equations. The book included all the important work that has been done on L-subspaces of a vector space and on L-subfields of a field.

The notions of BCK-algebras and BCI-algebras were introduced by Y. Imai and K. Iséki ([8, 9]). The class of BCK-algebras is a proper subclass of the class of BCI-algebras. We refer useful textbooks for BCK-algebras and BCI-algebras to [6, 12].

For a given set X, we consider the collection Bin (X) of all binary systems (groupoids, algebras) defined on X. Given arbitrary groupoids (X, *) and (X, ∘), define an operation “ □ " as follows: x □ y : = (x * y) ∘ (y * x) for any x, y ∈ X. H. S. Kim and J. Neggers ([10]) showed that the collection Bin (X) of all groupoids defined on a set X forms a semigroup, i.e., the operation □ as defined in general is associative. Furthermore, the left-zero-semigroup is an identity for this operation. J. S. Han et al. ([4]) 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.

P. J. Allen et al. ([2]) introduced several types of groupoids related to semigroups, i.e., twisted semigroups which are related to the associative law by using the functions. J. S. Han et al. ([5]) generalized the concept of Fibonacci numbers by using the functions, and obtained several properties of Fibonacci functions. In the sequel, by using the functions we generalize the concepts of a fuzzy subalgebra, a fuzzy (semi-)ideal, (μ, φ)-extremes, (μ, φ)-special in groupoids.

S. S. Ahn et al. ([1]) discussed fuzzy upper bounds in groupoids, and obtained four different types of fuzzy subsets. S. J. Shin et al. ([15]) discussed properties of a class of real-valued functions on a set X² constructed as finite (real) linear combinations of functions denoted as [(X, *) ; μ], where (X, *) is a groupoid (binary system) and μ is a fuzzy subset of X and where [(X, *) ; μ] (x, y) : = μ (x * y) - min {μ (x), μ (y)}. Y. L. Liu et al. ([11]) extended the theory of groupoids (Bin (X), □) to the hyperfuzzy sets (HBin (X), □), and discussed hyperfuzzy subsets, convex hull of hyperfuzzy subsets and some operations of hyperfuzzy subsets. M. Farshi et al. ([3]) associated a partial g-hypergroupoid with a given g-hypergraph and analyzed the properties of this hyperstructure.

In this paper, we introduce the notions of an almost φ-fuzzy subalgebra and an almost φ-fuzzy semi-ideal in groupoids, which are generalizations of a fuzzy subalgebra and a fuzzy ideal respectively. We generalize several properties related to fuzzy algebraic structures.

2. Preliminaries

Let (X, *) ∈ Bin (X). Let S be a non-empty subset of X. Then (S, *) is called a subalgebra of X if for any x, y ∈ S, x * y ∈ S. A map μ : X → [0, 1] is called a fuzzy subalgebra of X if μ (x * y) ≥ min {μ (x), μ (y)} for any x, y ∈ X. Let (X, *) and (Y, •) be groupoids. A mapping φ : (X, *) → (Y, •) is called a groupoid homomorphism if φ (x * y) = φ (x) • φ (y) for all x, y ∈ X. A homomorphism φ : X → Y is called a groupoid epimorphism if X = Y.

A BCK-algebra ([8, 9]) is a non-empty set X with a constant 0 and a binary operation “ * " satisfying the following axioms:

x * x = 0,

We denote the BCK-algebra by (X, * , 0). A groupoid (X, * , 0) satisfying the conditions (ii) ∼ (v) is called a BCI-algebra ([8, 9]). A BCI-algebra X is said to be p-semisimple ([6]) if 0 * (0 * x) = x for all x ∈ X. The following equivalent conditions are proved:

Theorem 2.1. ([6]) Let (X, * , 0) be a BCI-algebra. Then the following conditions are equivalent:

X is p-semisimple,

3. Almost φ-fuzzy subalgebras in groupoids

Let (X, *) and (Y, •) be groupoids. A map φ : (X, *) → (Y, •) is called a (groupoid) homomorphism if f (x * y) = f (x) • f (y) for all x, y ∈ X. A groupoid homomorphism φ : X → Y is said to be an endomorphism if X = Y.

Definition 3.1. Let (X, *) ∈ Bin (X). A mapping μ : (X, *) → [0, 1] is called an almost φ-fuzzy subalgebra of (X, *) if, for any x, y ∈ X, μ ((x * y) * φ (x * y)) ≥ min {μ (x * φ (x)), μ (y * φ (y))}.

It is easy to see that (X, * , 0) is a BCI-algebra. Define a map φ : (X, *) → (X, *) by φ (0) = φ (3) =0 and φ (1) = φ (2) =1. It is easy to show that φ : X → X is a BCI-homomorphism. Define a map μ : X → [0, 1] by μ (3) < μ (1) = μ (2) = μ (0). It it is easy to check that μ is an almost φ-fuzzy subalgebra of X, but not a fuzzy subalgebra of (X, *), since μ (1 *2) = μ (3) < μ (1) = min {μ (1), μ (2)}.

Definition 3.3. An endomorphism of groupoids φ : (X, *) → (X, *) is said to be vanishing if x * φ (x) = x for all x ∈ X.

Proposition 3.4. Let (X, *) be a groupoid with the conditions: x * x = 0 and x * 0 = x for any x ∈ X. Define a map φ : X → X by φ (x) =0 for any x ∈ X. Then φ is vanishing.

Proof. For any x, y ∈ X, we have φ (x * y) =0 = 0 *0 = φ (x) * φ (x). Hence x * φ (x) = x * 0 = x.□

Define a map φ : X → X by φ (0) = φ (1) = φ (2) =0 and φ (3) =1. It is easy to show that φ : X → X is vanishing. Define a map ψ : X → X by ψ (0) = ψ (1) = ψ (2) =0 and ψ (3) =2. It is easy to check that ψ is also vanishing.

Given (X, *) ∈ Bin (X), we define two subsets V (X, *) and E (X, *) as follows: V (X, *) : = {φ|φ : X → X is vanishing} and E (X, *) : = {φ|φ : X → X is an endomorphism}.

Proposition 3.6. If (X, *) is a semigroup, then (V (X, *), ∘) is a subsemigroup of (E (X, *), ∘), where ∘ is a composition of mappings.

Proof. For any x ∈ X, and for any φ, ψ ∈ V (X, *), we have x * (φ ∘ ψ) (x) = (x * ψ (x)) * (φ ∘ ψ) (x) = x * (ψ (x) * (φ ∘ ψ) (x)) = x * [ψ (x) * φ (ψ (x))] = x * ψ (x) = x. Hence φ ∘ ψ ∈ (V (X, *), ∘). Similarly, we obtain ψ ∘ φ ∈ (V (X, *), ∘).□

Proposition 3.7. Let φ ∈ V (X, *). Then the class of almost φ-fuzzy subalgebras of (X, *) is equal to the class of fuzzy subalgebras of (X, *).

Proof. Straightforward. □

4. Level subsets and almost φ-fuzzy subalgebras

Definition 4.1. Let α ∈ [0, 1] and let μ be a (resp., almost φ)-fuzzy subalgebra of X. The set μ (φ, α) : = {x * φ (x) |μ (x * φ (x)) ≥ α} is called the level subset of μ (resp., with respect to φ). The set μ_* (φ, α) : = {x ∈ X|x * φ (x) ∈ μ (φ, α)} is called the pre-level subset of μ (resp., with respect to φ).

Proposition 4.2. Let φ ∈ V (X, *). Then μ (φ, α)   =  μ_*(φ, α) for any α ∈ [0, 1] and any fuzzy subset μ of X.

Proof. For any α ∈ [0, 1] and any fuzzy subset μ of X, we have μ (φ, α) = {x * φ (x) |μ (x * φ (x)) ≥ α} = {x ∈ X|x * φ (x) ∈ μ (φ, α)} = μ_* (φ, α), since φ ∈ V (X, *). □

Definition 4.3. Let φ ∈ E (X, *). Then φ is said to beweakly vanishing if μ (φ, α) = μ_* (φ, α) for any α ∈ [0, 1] and any almost φ-fuzzy subalgebra μ of X.

Corollary 4.4. If φ ∈ V (X, *), then φ is weaklyvanishing.

Proof. It follows from Proposition 4.2 and Definition 4.3.□

Definition 4.5. A groupoid (X, *) is said to medial if (x * y) * (z * w) = (x * z) * (y * w) for any x, y, z and w ∈ X.

Definition 4.6. Let μ : (X, *) → [0, 1] be a fuzzy subset of X. We say μ has the fuzzy subalgebra property with respect to φ ∈ E (X, *) if, for any α ∈ [0, 1], μ (φ, α) is a subalgebra of (X, *).

Theorem 4.7. Let (X, *) be a medial groupoid and let φ ∈ E (X, *). Then μ is an almost φ-fuzzy subalgebra of (X, *) if and only if μ has the fuzzy subalgebra property with respect to φ.

Proof. Suppose that μ is an almost φ-fuzzy subalgebra of X. Let x * φ (x), y * φ (y) ∈ μ (φ, α) for any α ∈ [0, 1]. Then μ (x * φ (x)) ≥ α and μ (y * φ (y)) ≥ α. Since X is medial and φ ∈ E (X, *), we obtain μ ((x * φ (x)) * (y * φ (y))) = μ ((x * y) * (φ (x) * φ (y))) = μ ((x * y) * φ (x * y)) ≥ min {μ (x * φ (x)), μ (y * φ (y))} ≥ α and so (x * φ (x)) * (y * φ (y)) ∈ μ (φ, α). Hence μ (φ, α) is a subalgebra of (X, *) for any α ∈ [0, 1]. Therefore μ has the fuzzy subalgebra property with respect to φ.

Assume that μ is not an almost φ-fuzzy subalgebra of (X, *). Then there exist x, y ∈ X such that μ ((x * y) * φ (x * y)) < min {μ (x * φ (x)), μ (y * φ (y))}. Let t 0 : = 1 2 ( p + q ) , where p : = μ ((x * y) * φ (x * y)) and q : = min {μ (x * φ (x)), μ (y * φ (y))}. Then x * φ (x), y * φ (y) ∈ μ (φ, t₀). Since μ (φ, t₀) is a subalgebra of (X, *), we have (x * φ (x)) * (y * φ (y)) ∈ μ (φ, t₀). Since (X, *) is medial, we obtain (x * y) * φ (x * y) = (x * y) * (φ (x) * φ (y)) = (x * φ (x)) * (y * φ (y)) ∈ μ (φ, t₀). Hence μ ((x * y) * φ (x * y)) ≥ t₀, which is a contradiction, proving the theorem. □

Definition 4.8. Let μ : (X, *) → [0, 1] be a fuzzy subset of X. Then μ has the weak fuzzy subalgebra property with respect to φ ∈ E (X, *) if for any α ∈ [0, 1], μ_* (φ, α) is a subalgebra of (X, *).

Theorem 4.9. Let (X, *) be a medial groupoid with φ ∈ E (X, *). Then μ has the weak fuzzy subalgebra property with respect to φ if and only if μ has the fuzzy subalgebra property with respect to φ.

Proof. Suppose that μ has the weak fuzzy subalgebra property with respect to φ. Let x * φ (x), y * φ (y) ∈ μ (φ, α) for any α ∈ [0, 1]. Then x, y ∈ μ_* (φ, α). By assumption, we have x * y ∈ μ_* (φ, α) and so (x * y) * φ (x * y) = (x * y) * (φ (x) * φ (y)) ∈ μ (φ, α). Since (X, *) is medial, we have (x * y) * (φ (x) * φ (y)) = (x * φ (x)) * (y * φ (y)) ∈ μ (φ, α). Therefore μ (φ, α) is a subalgebra of (X, *) for any α ∈ [0, 1]. Hence μ has the fuzzy subalgebra property with respect to φ.

Conversely, assume that μ has the fuzzy subalgebra property with respect to φ. Let x, y ∈ μ_* (φ, α) for any α ∈ [0, 1]. Then x * φ (x), y * φ (y) ∈ μ (φ, α). By assumption, we have (x * φ (x)) * (y * (φ (y)) ∈ μ (φ, α). Since (X, *) is medial, we have (x * φ (x)) * (y * φ (y)) = (x * y) * (φ (x) * φ (y)) = (x * y) * φ (x * y) ∈ μ (φ, α). Hence x * y ∈ μ_* (φ, α). Therefore μ has the weak fuzzy subalgebra property with respect to φ. □

Proposition 4.10. Let (X, *) ∈ Bin (X) and let x * x = e for all x ∈ X. If μ is an almost φ-fuzzy subalgebra of X, then μ (e) ≥ μ (x * φ (x)) for any x ∈ X, where φ ∈ E (X, *).

Proof. For any x ∈ X, we have e = φ (x) * φ (x) = φ (x * x) = φ (e). If μ is an almost φ-fuzzy subalgebra of X, then μ (e) = μ (e * e) = μ (e * φ (e)) = μ ((x * x) * φ (x * x)) ≥ min {μ (x * φ (x)), μ (x * φ (x))} = μ (x * φ (x)) for any x ∈ X. □

Corollary 4.11. Let (X, * , 0) be a BCK/BCI-algebra and let φ ∈ E (X, *). Then for any almost φ-fuzzy subalgebra of X, μ (0) ≥ μ (x * φ (x)) for any x ∈ X.

5. (μ, φ)-extremes and e-ideals

Definition 5.1. Let (X, *) ∈ Bin (X) and φ ∈ E (X, *). Let μ : (X, *) → [0, 1] be a fuzzy subset of X. Then e ∈ X is said to be a (μ, φ)-extreme if μ (e) ≥ μ (x * φ (x)) for any x ∈ X.

In Corollary 4.11, 0 is a (μ, φ)-extreme in BCK/BCI-algebras.

Definition 5.2. Let (X, *) ∈ Bin (X). A non-empty subset I of X containing e (∈ X) is said to be an e-ideal of X if it satisfies the following condition: x * y , y ∈ I implies x ∈ I .(I)

For example, every ideals in a BCK/BCI-algebra (X, * , 0) are 0-ideals of X.

Proposition 5.3. Let μ : (X, *) → [0, 1] be a fuzzy subset of X and let φ ∈ E (X, *). If μ (φ, α) is an e-ideal of X for any α ∈ [0, 1] with μ (φ, α)≠ ∅, then e is (μ, φ)-extreme, i.e., μ (e) ≥ μ (x * φ (x)) for any x ∈ X.

Proof. Assume that there exists x₀ ∈ X such that μ (e) < μ (x₀ * φ (x₀)). Let λ 0 : = 1 2 [ μ ( e ) + μ ( x 0 * φ ( x 0 ) ) ] . Then μ (e) < λ₀ < μ (x₀ * φ (x₀)), and hence x₀ * φ (x₀) ∈ μ (φ, λ₀), but e ∉ μ (φ, λ₀). Hence μ (φ, λ₀)≠ ∅ and e ∉ μ (φ, λ₀). This shows that μ (φ, λ₀) is not an e-ideal, a contradiction. □

Proposition 5.4. Let (X, *) ∈ Bin (X) and φ ∈ E (X, *). Let μ : (X, *) → [0, 1] be a fuzzy subalgebra of X. Define a set Ext (μ, φ) : = {e ∈ X|e : (μ, φ) - extreme}. Then Ext (μ, φ) is a subalgebra of (X, *) if it is not empty.

Proof. Let e₁, e₂ ∈ Ext (μ, φ). Then μ (e _i ) ≥ μ (x * φ (x)) for any x ∈ X where for any i = 1, 2. Since μ is a fuzzy subalgebra of X, we have μ (e₁ * e₂) ≥ min {μ (e₁), μ (e₂)} ≥ μ (x * φ (x)) for any x ∈ X. Hence e₁ * e₂ ∈ Ext (μ, φ), proving the proposition. □

Define a map φ : X → X by φ (0) = φ (1) =0 and φ (2) = φ (3) =2. It is easy to show that φ is a groupoid homomorphism, but φ is not vanishing, since 3 * φ (3) =3 * 2 =1. Define a map μ : X → X by μ (2) = μ (0) =0.9 and μ ( 1 ) = μ ( 3 ) = 1 2 . It is easy to check that μ is a fuzzy subalgebra of X, Ext (μ, φ) = {0, 2} and {0, 2} is a subalgebra of X.

Remark. The condition “μ is a fuzzy subalgebra of (X, *)” is very necessary in Proposition 5.4 (see Example 5.6.).

Example 5.6. Consider a set X = {0, 1, 2, 3} as in Example 5.5. Define a map μ : X → [0, 1] by μ (0) =0.1, μ (1) =0.5 and μ (2) = μ (3) =0.8. It is easy to show that Ext (μ, φ) = {2, 3}. But Ext (μ, φ) is not a subalgebra of (X, *), since 2 * 3 =1 ∉ Ext (μ, φ). Moreover, μ is not a fuzzy subalgebra of X, since μ (2 *3) = μ (1) =0.5 < 0.8 = min {μ (2), μ (3)}. Since μ ((1 * 3) * φ (1 *3)) = μ (2 * φ (2)) = μ (0) =0.1 < 0.5 = μ (1) = min {μ (1 * φ (1)), μ (3 * φ (3))}, μ is not an almost φ-fuzzy subalgebra of X.

Proposition 5.7. Let (X, *) ∈ Bin (X) and φ ∈ E (X, *). If μ : (X, *) → [0, 1] is a constant map, then Ext (μ, φ) = X.

Proof. Straightforward.□

Definition 6.1. Let (X, *) ∈ Bin (X) and let φ ∈ E (X, *). A fuzzy subset μ : (X, *) → [0, 1] is called an almost φ-fuzzy semi-ideal of X if μ (x * φ (x)) ≥ min {μ ((x * y) * φ (x * y)), μ (y * φ (y))} for any x, y ∈ X.

Proposition 6.2. Let (X, *) ∈ Bin (X) with 0 * x = 0 for any x ∈ X. Let μ : (X, *) → [0, 1] be a fuzzy subset of (X, *) such that μ (0) ≥ μ (x) for all x ∈ X. If μ is an almost φ-fuzzy semi-ideal of X where φ ∈ E (X, *), then x * y = 0 implies μ (x * φ (x)) ≥ μ (y * φ (y)) for any x, y ∈ X.

Proof. Assume that x * y = 0 for any x, y ∈ X. Then (x * y) * φ (x * y) =0 * φ (x * y) =0. Since μ is an almost φ-fuzzy semi-ideal of X, we have μ (x * φ (x)) ≥ min {μ ((x * y) * φ (x * y)), μ (y * φ (y))} = min {μ (0), μ (y * φ (y))} = μ (y * φ (y)).□

Example 6.3. (1) Consider a BCI-algebra (X, * , 0), and maps φ, μ as in Example 3.2. It is easy to see that μ is an almost φ-fuzzy semi-ideal.

Then it is easy to see that (X, * , 0) is a BCI-algebra. Define a map φ : (X, *) → (X, *) defined by φ (0) = φ (3) =3, φ (1) =1 and φ (2) =2. It is easy to show that φ : X → X is a BCI-homomorphism. Define a map μ : X → [0, 1] by μ (3) < μ (1) = μ (2) = μ (0). It it is easy to see that μ is an almost φ-fuzzy semi-ideal of X. But μ is not an almost φ-fuzzy subalgebra of X, since μ ((2 * 1) * φ (2 *1)) = μ (3 * φ (3)) = μ (3) < μ (1) = min {μ (2) , μ (1)} = min {μ (2 * φ (2)) , μ (1 * φ (1)}}.

Then it is easy to see that (X, * , 0) is a BCI-algebra. Define a map φ : (X, *) → (X, *) defined by φ (0) = φ (2) =0 and φ (1) = φ (3) =1. It is easy to show that φ : X → X is a BCI-homomorphism. Define a map μ : X → [0, 1] by μ (3) < μ (1) = μ (2) = μ (0). It is easy to see that μ is both an almost φ-fuzzy subalgebra of X and an almost φ-fuzzy semi-ideal of X.

Define a map φ : (X, *) → (X, *) defined by φ (0) = φ (1) =0, φ (2) =2 and φ (3) =3. It is easy to show that φ : X → X is a BCK-homomorphism. Define a map μ : X → [0, 1] by μ (1) < μ (0) = μ (2) = μ (3). It is easy to see that μ is an almost φ-fuzzy subalgebra of X, but it is not an almost φ-fuzzy semi-ideal of X, since μ (1 * φ (1)) = μ (1) < μ (0) = min {μ ((1 * 2) * φ (1 *2)) , μ (2 * φ (2))}.

Theorem 6.4. Let (X, *) ∈ Bin (X) and (X, *) be medial. Then μ is an almost φ-fuzzy semi-ideal of (X, *) if and only if μ (φ, α) satisfies the condition (I) for any α ∈ [0, 1] with μ (φ, α)≠ ∅.

Proof. Assume that μ (φ, α)≠ ∅. Let (x * φ (x)) * (y * φ (y)) , y * φ (y) ∈ μ (φ, α). Then μ ((x * φ (x)) * (y * φ (y))) ≥ α and μ (y * φ (y)) ≥ α. Since μ is an almost φ-fuzzy semi-ideal of X and (X, *) is medial, we have μ ( x * φ ( x ) ) ≥ min { μ ( ( x * y ) * φ ( x * y ) ) , μ ( y * φ ( y ) ) } = min { μ ( ( x * y ) * ( φ ( x ) * φ ( y ) ) ) , μ ( y * φ ( y ) ) } = min { μ ( ( x * φ ( x ) ) * ( y * φ ( y ) ) ) , μ ( y * φ ( y ) ) } ≥ α .

Hence x * φ (x) ∈ μ (φ, α).

Conversely, assume that there exist x₀, y₀ ∈ X such that μ (x₀ * φ (x₀)) < min {μ ((x₀ * y₀) * φ (x₀ * y₀)) , μ (y₀ * φ (y₀))}. Let κ₀ : = min {μ ((x₀ * y₀) * φ (x₀ * y₀)) , μ (y₀ * φ (y₀))} and let λ 0 : = 1 2 [ μ ( x 0 * φ ( x 0 ) ) + κ 0 ] . Then μ (x₀ * φ (x₀)) < λ₀ < κ₀ ≤ 1 and so μ (y₀ * φ (y₀)) ≥ κ₀ > λ₀. Hence y₀ * φ (y₀) ∈ μ (φ, λ₀). Since (X, *) is medial, we have (x₀ * φ (x₀)) * (y₀ * φ (y₀)) = (x₀ * y₀) * (φ (x₀) * φ (y₀)) = (x₀ * y₀) * φ (x₀ * y₀) ∈ μ (φ, λ₀). By assumption, x₀ * φ (x₀) ∈ μ (φ, λ₀). Therefore μ (x₀ * φ (x₀)) ≥ λ₀, a contradiction. This proves the theorem. □

Corollary 6.5. Let (X, * , 0) be a p-semisimple BCI-algebra. If μ is an almost φ-fuzzy semi-simple ideal of X, then μ (φ, α) is a 0-ideal of X for any α ∈ [0, 1] with μ (φ, α)≠ ∅.

Proof. It follows immediately from Theorem 6.4 and Theorem 2.1.□

Proposition 6.6. Let (X, *) be a medial groupoid. If μ is an almost φ-fuzzy semi-ideal of (X, *), then μ_* (φ, α) satisfies the condition (I) for any α ∈ [0, 1] with μ_* (φ, α)≠ ∅.

Proof. Let x * y, y ∈ μ_* (φ, α). Then (x * y) * φ(x * y) , y * φ (y) ∈ μ (φ, α). Since (X, *) is medial. we have (x* φ (x)) * (y * φ (y)) = (x * y) * (φ (x) *φ (y)) = (x * y) * φ (x * y) ∈ μ (φ, α). Hence μ ((x*φ (x)) * (y * φ (y))) ≥ α and μ (y * φ (y)) ≥ α. Since μ is an almost φ-fuzzy semi-ideal of X, we obtain μ (x * φ (x)) ≥ min {μ ((x * y) * φ (x * y)) , μ (y * φ (y))} = {μ ((x * φ (x)) * (y * φ (y))) , μ (y * φ (y))} ≥ α. Hence x * φ (x) ∈ μ (φ, α). Therefore x ∈ μ_* (φ, α), proving the proposition.□

Proposition 6.7. Let (X, *) be a medial groupoid and let μ be an almost φ-fuzzy semi-ideal of (X, *). Assume that μ (e * φ (e)) ≥ μ (x * (φ (x)) for all x ∈ X. Then μ_* (φ, α) is an e-ideal of X for any α ∈ [0, 1] with μ_* (φ, α)≠ ∅.

Proof. Let α ∈ [0, 1] be such that μ_* (φ, α)≠ ∅. Then there exists x ∈ X such that x * φ (x) ∈ μ (φ, α). Hence μ (x * φ (x)) ≥ α. By assumption, we have μ (e * φ (e)) ≥ μ (x * φ (x)) ≥ α. Therefore e ∈ μ_* (φ, α). By Proposition 6.6, μ_* (φ, α) is an e-ideal of X.□

7. (μ, φ)-special elements

Let (X, *) ∈ Bin (X) and let μ : (X, *) → [0, 1] be fuzzy subset of (X, *). An element e (∈ X) is said to be (μ, φ)-special if

e * φ (e) = e,

Note that every (μ, φ)-special element is a (μ, φ)-extreme element.

Proposition 7.1. Let (X, *) be a medial groupoid and let e be a (μ, φ)-special element. If μ is an almost φ-fuzzy semi-ideal of X, then μ (φ, α) and μ_* (φ, α) are e-ideals of X for any α ∈ [0, 1] with μ (φ, α)≠ ∅.

Proof. Let α ∈ [0, 1] be such that μ (φ, α)≠ ∅. Then there exists x ∈ X such that x * φ (x) ∈ μ (φ, α). It follows that μ (x * φ (x)) ≥ α. Since e is (μ, φ)-special, we have μ (e * φ (e)) ≥ μ (x * φ (x)) ≥ α for all x ∈ X. Hence e = e * φ (e) ∈ μ (φ, α). By Theorem 6.4, μ (α, φ) is an e-ideal of X. Also by Proposition 6.7, μ_* (φ, α) is an e-ideal of X. □

Proposition 7.2. Let (X, *) be a medial groupoid and let μ be a fuzzy subalgebra of X. If e₁, e₂ are (μ, φ)-special elements, then e₁ * e₂ is also (μ, φ)-special element.

Proof. Let e₁, e₂ be (μ, φ)-special elements and μ be an almost φ-fuzzy subalgebra of (X, *). Since (X, *) is medial, we have e₁ * e₂ = (e₁ * φ (e₁)) * (e₂ * φ (e₂)) = (e₁ * e₂) * (φ (e₁) * φ(e₂)) = (e₁ * e₂) * φ (e₁ * e₂). Since μ is a fuzzy subalgebra of X, we have μ (e₁ * e₂) ≥ min {μ(e₁) , μ (e₂)} = min {μ (e₁ * φ (e₁)) , μ (e₂ * φ (e₂))} ≥ min {μ (x * φ (x)) , μ (x * φ (x))} = μ (x * φ (x)) for any x ∈ X. Hence μ ((e₁ * e₂) * φ (e₁ * e₂)) = μ (e₁ * e₂) ≥ μ (x * φ (x)) for all x ∈ X. Therefore e₁ * e₂ is a (μ, φ)-special element.□

Proposition 7.3. Let (X, *) be a medial groupoid and let φ ∈ E (X, *). If μ is an almost φ-fuzzy semi-ideal of (X, *) and if e is a (μ, φ)-special element, then the set A : = {x * φ (x) |μ (x * φ (x)) = μ (e)} is an e-ideal of X.

Proof. Let μ be an almost φ-fuzzy semi-ideal of (X, *) and let e be a (μ, φ)-special element. Then μ (e) ≥ μ (x * φ (x)) for all x ∈ X. If we let λ : = μ (e), then A = μ (φ, λ). In fact, if x * φ (x) ∈ A, then μ (x * φ (x)) = μ (e) = λ and hence x * φ (x) ∈ μ (φ, λ). Hence A ⊆ μ (φ, λ). If q ∈ μ (φ, λ), then there exists x₀ ∈ X such that q = x₀ * φ (x₀) and μ (q) ≥ λ = μ (e). Now, since e is (μ, φ)-special, we have μ (e) ≥ μ (x * φ (x)) for all x ∈ X. Hence μ (e) ≥ μ (x₀ * φ (x₀)) = μ (q). Therefore μ (e) = μ (q) = μ (x₀ * φ (x₀)) and so q = x₀ * φ (x₀) ∈ A. Thus μ (φ, λ) ⊆ A. By Proposition 7.1, A is an e-ideal of X. □

8. Some diagrams

Let (X, *), (Y, ⊛) be groupoids and let ψ : (X, *) → (Y, ⊛) be a groupoid epimorphism. Assume that following diagram commutes: OPEN IN VIEWER

i.e., φ ˆ ∘ ψ = ψ ∘ φ . We say φ ˆ can be lifted to φ via ψ and φ can be lowered to φ ˆ via ψ.

Proposition 8.1. Let (X, *), (Y, •) be groupoids such that the following diagram commutes: OPEN IN VIEWER

where ψ is a groupoid epimorphism and φ , φ ˆ are groupoid homomorphisms. Assume that ν : (Y, •) → [0, 1] is an almost φ ˆ -fuzzy semi-ideal of (Y, •). If we define μ : = ν ∘ ψ, then μ is an almost φ-fuzzy semi-ideal of (X, *).

Proof. Since ν is an almost φ ˆ -fuzzy semi-ideal of Y, we have μ ( x * φ ( x ) ) = ( ν ∘ ψ ) ( x * φ ( x ) ) = ν ( ψ ( x ) • ψ ( φ ( x ) ) ) = ν ( ψ ( x ) • ( ψ ∘ φ ) ( x ) ) = ν ( ψ ( x ) • ( φ ˆ ∘ ψ ) ( x ) ) = ν ( ψ ( x ) • φ ˆ ( ψ ( x ) ) ) ≥ min { ν ( ψ ( y ) • φ ˆ ( ψ ( y ) ) ) , ν ( ( ψ ( x ) • ψ ( y ) ) • φ ˆ ( ψ ( x ) • ψ ( y ) ) ) } for any x, y ∈ X. Now, for any y ∈ X, we obtain ν ( ψ ( y ) • φ ˆ ( ψ ( y ) ) ) = ν ( ψ ( y ) • ( φ ˆ ∘ ψ ) ( y ) ) = ν ( ψ ( y ) • (ψ ∘ φ) (y)) = ν (ψ (y) • ψ (φ (y))) = ν (ψ (y * φ (y)))= μ (y * φ (y)). Similarly, we get ν ( ( ψ ( x ) • ψ ( y ) ) • φ ˆ ( ψ ( x ) • ψ ( y ) ) ) = μ ( ( x * y ) * φ ( x * y ) ) . Therefore μ (x * φ (x)) ≥ min {μ ((x * y) * φ (x * y)) , μ (y * φ(y))} for any x, y ∈ X. Thus μ is an almost φ-fuzzy semi-ideal of (X, *).□

Proposition 8.2. Let (X, *), (Y, •) be groupoids such that the following diagram commutes: OPEN IN VIEWER

where ψ is a groupoid epimorphism and φ , φ ˆ are groupoid homomorphisms. Assume that μ is an almost φ-fuzzy semi-ideal of (X, *) and ν : (Y, •) → [0, 1] is a map such that ν ∘ ψ = μ. Then ν is an almost φ ˆ -fuzzy semi-ideal of (Y, •).

Proof. Let y₁, y₂ ∈ Y. Since ψ is an epimorphism, there exist x₁, x₂ ∈ X such that ψ (x _i ) = y _i for i = 1, 2. Hence ν ( y 1 • φ ˆ ( y 1 ) ) = ν ( ψ ( x 1 ) • φ ˆ ( ψ ( x 1 ) ) ) = ν ( ψ ( x 1 ) • ( φ ˆ ∘ ψ ) ( x 1 ) ) = ν ( ψ ( x 1 ) • ( ψ ∘ φ ) ( x 1 ) ) = ν ( ψ ( x 1 ) • ψ ( φ ( x 1 ) ) ) = ν ( ψ ( x 1 * φ ( x 1 ) ) ) = μ ( x 1 * φ ( x 1 ) ) .

Similarly, we obtain ν ( ( y 1 • y 2 ) • φ ˆ ( y 1 • y 2 ) ) = μ ( ( x 1 * x 2 ) * φ ( x 1 * x 2 ) ) and ν ( y 2 • φ ˆ ( y 2 ) ) = μ ( x 2 * φ ( x 2 ) ) . Since μ is an almost φ-fuzzy semi-ideal of (X, *), we obtain ν ( y 1 • φ ˆ ( y 1 ) ) = μ ( x 1 * φ ( x 1 ) ) ≥ min { μ ( ( x 1 * x 2 ) * φ ( x 1 * x 2 ) ) , μ ( x 2 * φ ( x 2 ) ) } = min { ν ( ( y 1 • y 2 ) • φ ˆ ( y 1 • y 2 ) ) , ν ( y 2 • φ ˆ ( y 2 ) ) } .

Hence ν is an almost φ ˆ -fuzzy semi-ideal of(Y, •).□