Some Innovative Types of Fuzzy Ideals in AG-Groupoids

2.1 Lemma [18]

Let G be an AG-groupoid and μ a fuzzy subset of G. Then, μ is a fuzzy bi-ideal of G if and only if μ_A is a fuzzy bi-ideal of G.

Let G be an AG-groupoid and μ a fuzzy subset of G. Then, for every t∈(0, 1], the set

U(μ; t):={x|x∈G and μ(x)≥t}

is called a level set of μ with support x and value t.

2.2 Theorem [18]

Let G be an AG-groupoid and μ a fuzzy subset of G. Then, μ is a fuzzy bi-ideal of G if and only if U(μ; t)(≠∅) is a bi-ideal of G for every t∈(0, 1].

2.3 Theorem [19]

Let G be an AG-groupoid and μ a fuzzy subset of G. Then, μ is a fuzzy interior ideal of G if and only if U(μ; t)(≠∅) is an interior ideal of G for every t∈(0, 1].

Let μ be a fuzzy subset of G, then the set of the form

μ(y):={t(≠0) if y=x,0      otherwise,

is called a fuzzy point with support x and value t and is denoted by [x; t]. A fuzzy point [x; t] is said to belong to (resp. quasi-coincidence) with a fuzzy set μ, written as [x; t]∈μ (resp. [x; t]qμ) if μ(x)≥λ [resp. μ(x)+λ>1]. If [x; t]∈μ or [x; t]qμ, then we write [x; t]∈∨qμ. The symbol ∈∨q¯ means ∈∨q does not hold.

2.4 Definition

A fuzzy subset μ of G is called an (⋶, ⋶∧q̄)-fuzzy bi-ideal of G if it satisfies the following conditions:

1. (∀x, y∈G)(∀s, t∈(0, 1])([xy;s∧t]∈¯μ⇒[x;s]∈¯∨q¯μ or [y;t]∈¯∨q¯μ),

2. (∀x, a, y∈G)(∀s, t∈(0, 1])([(xa)y;s∧t]∈¯μ⇒[x;s]∈¯∨q¯μ or [y;t]∈¯∨q¯μ).

2.5 Example

Consider G={a, b, c, d, e} with the following multiplication table:

Then (G, ·) is an AG-groupoid. Define a fuzzy subset μ:G→[0, 1] as follows:

μ(x)={0.8if x=a,0.6if x∈{b, c, d, e}.

Then, by Definition 2.4, μ is an (⋶, ⋶∨q̄)-fuzzy bi-ideal of G.

2.6 Theorem

If μ is a fuzzy subset of G, then μ is an (⋶, ⋶∨q̄)-fuzzy bi-ideal of G if and only if

• (3) (∀x, y∈G)(μ(xy)∨0.5≥μ(x)∧μ(y)),

• (4) (∀x, a, y∈G)(μ((xa)y)∨0.5≥μ(x)∧μ(y)).

Proof. (1)⇒(3). If there exist x, y∈G such that μ(xy)∨0.5<t=μ(x)∧μ(y), then 0.5<t≤1, [xy; t]μ but [x; t]∈μ and [y; t]∈μ. By (1), we have [x; t]q̄μ or [y; t]q̄μ. Then, (μ(x)≥t and t+μ(x)≤1) or (μ(y)≥t and t+μ(y)≤1), which implies that t≤0.5, a contradiction.

(3)⇒(1). Let x, y∈G and s, t∈(0, 1] be such that [xy; s∧t]μ, then μ(xy)<s∧t.

(a) If μ(xy)≥μ(x)∧μ(y), then μ(x)∧μ(y)<s∧t, and consequently, μ(x)<s or μ(y)<t. It follows that [x; s] ⋶μ or [y; t]⋶μ. Thus, [x; s]⋶∨q̄μ or [y; t]⋶∨q̄μ.

(b) If μ(xy)<μ(x)∧μ(y), then by (3), we have 0.5≥μ(x)∧μ(y). Let [x; s]⋶μ or [y; t]⋶μ then s≤μ(x)≤0.5 or t≤μ(y)≤0.5. It follows that [x; s]q̄μ or [y; t]q̄μ, and [x; s]⋶∨q̄μ or [y; t]⋶∨q̄μ.

(2)⇒(4). If there exist x, a, y∈G such that μ((xa)y)∨0.5<t=μ(x) ∧ μ(y), then 0.5<t≤1, [(xa)y; t]⋶μ but [x; t] ∈μ and [y; t]∈μ. By (2), we have [x; t]q̄μ or [y; t]q̄μ. Then, (μ(x)≥t and t+μ(x)≤1) or (μ(y)≥t and t+μ(y)≤1), which implies that t≤0.5, a contradiction.

(4)⇒(2). Let x, a, y∈G and r, t∈(0, 1] be such that [(xa)y; r∧t]⋶μ, then μ((xa)y)<r∧t.

(a) If μ((xa)y)≥μ(x)∧μ(y), then μ(x)∧μ(y)<r∧t, and consequently, μ(x)<r or μ(y)<t. It follows that [x; r] ⋶μ or [y; t]⋶μ. Thus, [x; r]⋶∨q̄μ or [y; t]⋶∨q̄μ.

(b) If μ((xa)y)<μ(x)∧μ(y), then by (4), we have 0.5≥μ(x)∧μ(y). Let [x; r]⋶μ or [y; t]⋶μ, then r≤μ(x)0.5 or t≤μ(y)≤0.5. It follows that [x; r]q̄μ or [y; t]q̄μ, and [x; r]⋶∨q̄μ or [y; t]⋶∨q̄μ.□

2.7 Definition

A fuzzy subset μ of G is called an (⋶, ⋶∨q̄)-fuzzy interior ideal of G if it satisfies the following conditions:

1. (∀x, y∈G)(∀s, t∈(0, 1])([xy; s∧t]∈¯μ⇒[x; s]∈¯∨q¯μ or [y; t]∈¯∨q¯μ),

2. (∀x, a, y∈G)(∀s∈(0, 1])([(xa)y; s]∈¯μ⇒[a; s]∈¯∨q¯μ.

2.8 Theorem

If μ is a fuzzy subset of G, then μ is an (⋶, ⋶∨q̄)-fuzzy interior ideal of G if and only if

• (3) (∀x, y∈G)(μ(xy)∨0.5≥μ(x)∧μ(y)),

• (4) (∀x, a, y∈G)(μ((xa)y)∨0.5≥μ(a)).

Proof. (2)⇒(4). If there exist x, a, y∈G such that μ((xa)y)∨0.5<t=μ(a), then 0.5<t≤1, [(xa)y; t]⋶μ but [a; t]∈μ. By (2), we have [a; t]q̄μ. Then, (μ(a)≥t and t+μ(a)≤1), which implies that t≤0.5, a contradiction.

(4)⇒(2). Let x, a, y∈G and r∈(0, 1] be such that [(xa)y; r]μ, then μ((xa)y)<r.

(a) If μ((xa)y)<μ(a), then μ(a)<r, and consequently, μ(a)<r. It follows that [a; r]⋶μ. Thus, [a; r]⋶∨q̄μ.

(b) If μ((xa)y)<μ(a), then by (4), we have 0.5≥μ(a). Let [a; r]⋶μ, then r≤μ(a)0.5. It follows that [a; r]q̄μ, so [a; r]⋶∨q̄μ. The remaining proof follows from Theorem 2.6. □

2.9 Definition

A fuzzy subset μ of G is called an (⋶, ⋶∨q̄)-fuzzy left (resp. right) ideal of G if it satisfies the following conditions:

(∀x, y∈G)(∀s∈(0, 1])([xy; s]∈¯μ⇒[y; s]∈¯∨q¯μ(resp. [x; s]∈¯∨q¯μ).

2.10 Theorem

If μ is a fuzzy subset of G, then μ is an (⋶, ⋶∨q̄)-fuzzy left (resp. right) ideal of G if and only if

(∀x, y∈G)(μ((xy)∨0.5≥μ(y)(resp. μ(x))).

Proof. Suppose that μ is an (⋶, ⋶∨q̄)-fuzzy left ideal. If there exist x, y∈G such that μ(xy)∨0.5<t=μ(y), then 0.5<t≤1, [xy; t]⋶μ but [y; t]∈μ. By Definition 2.9, we have [a; t]q̄μ. Then, (μ(a)≥t and t+μ(a)≤1), which implies that t≤0.5, a contradiction. Let x, y∈G and s∈(0, 1] be such that [xy; s]⋶μ, then μ(xy)<s.

(a) If μ(xy)≥μ(y), then μ(y)<s. It follows that [y; s]μ. Thus, [y; s]⋶∨q̄μ.

(b) If μ(xy)<μ(y), then by hypothesis, we have 0.5≥μ(y). Let [y; s]⋶μ, then s≤μ(y)≤0.5. It follows that [y; s]q̄μ, so [y; a]⋶∨q̄μ. In a similar way, the case for right ideal can be proved.□

2.11 Corollary

If μ is a fuzzy subset of G, then μ is an (⋶, ⋶∨q̄)-fuzzy bi-ideal of G if and only if

(∀x, y∈G)(μ((xy)∨0.5≥μ(y)∨μ(x)).

Proof. The proof follows from Theorem 2.10.□

3.1 Definition

A fuzzy subset μ of G is called an (α, β)-fuzzy AG-subgroupoid of G if it satisfies the condition

(∀x, y∈G)(∀s, t∈(γ, 1])([x; s]αμ, [x; t]αμ→[xy; s∧t]βμ).

Note that α, β∈{∈_γ, q_δ, ∈_γ∨q_δ, ∈_γ∧q_δ} and α ≠ ∈_γ∧q_δ. The case α=∈_γ∧q_δ is omitted. Because for a fuzzy subset μ such that μ(x)<δ for any x∈G. In the case [x; t]∈_γ∧q_δμ, we have μ(x)≥t and μ(x)+t>2δ. Therefore, μ(x)+μ(x)>μ(x)+t>2δ, which implies that 2μ(x)>2δ. Thus, μ(x)>δ, a contradiction. This means that {[x; t]|[x; t]∈_γ∧q_δμ}=∅.

3.2 Theorem

A fuzzy subset μ of G is an (∈_γ, ∈_γ∨q_δ)-fuzzy AG-subgroupoid of G if and only if

(∀x, y∈G)(γ, δ∈[0, 1])(μ(xy)∨γ≥μ(x)∧μ(y)∧δ).

Proof. Assume that μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy AG-subgroupoid of G. If there exist x, y∈G such that μ(xy)∨γ<μ(x)∧μ(y)∧δ. Choose t∈(γ, 1] such that μ(xy)∨γ<t≤μ(x)∧μ(y)∧δ. Then, [x; t]∈_γμ and [y; t]∈_γμ but μ(xy)<t and μ(xy)+t<2t<2δ, so [xy; t]∈γ∨qδ¯μ, a contradiction. Hence, μ(xy)∨γ≥μ(x)∧μ(y)∧δ for all x, y∈G and γ, δ∈[0, 1].

Conversely, assume that μ(xy)∨γ≥μ(x)∧μ(y)∧δ for all x, y∈G and γ, δ[0, 1]. Let [x; s]∈_γμ and [y; t]∈_γμ for some s, t∈(γ, 1], then μ(x)≥s>γ and μ(y)≥t>γ and by hypothesis,

μ(xy)∨γ≥μ(x)∧μ(y)∧δ≥s∧t∧δ={s∧t if s∧t≤δδ   if δ<s∧t.

That is, μ(xy)∨γ≥s∧t but s∧t>γ, therefore μ(xy)≥s∧t>γ and hence [xy; s∧t]∈_γμ or μ(xy)∨γ≥δ but γ<δ, therefore μ(xy)≥δ, thus μ(xy)+s∧t≥δ+s∧t>2δ, that is [xy; s∧t]q_δμ and hence [xy; s∧t]∈_γ∨q_δμ. Consequently, μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy AG-subgroupoid of G. □

3.3 Definition

A fuzzy subset μ of G is called an (α, β)-fuzzy bi-ideal of G if it satisfies the following conditions:

1. (∀x, y∈G)(∀s, t∈(γ, 1])([x; s]αμ, [x; t]αμ→[xy; s∧t]βμ),

2. (∀x, a, y∈G)(∀s, t∈(γ, 1])([x; s]αμ, [y; t]αμ→[(xa)y; s∧t]βμ).

3.4 Example

Let G be an AG-groupoid as shown in Example 2.5. Then {a}, {a, c, d, e} and G are bi-ideals of G. Define a fuzzy subset μ:S→[0, 1] as follows:

μ(x)={0.8if x=a,0.6if x=b,0.2if x=c,0.3if x=d,0.4if x=e.

Then, μ is an (∈_0.1, ∈_0.1∨q_0.2)-fuzzy bi-ideal of G.

3.5 Theorem

Let μ be a fuzzy subset of G. Then, the following conditions are equivalent:

1. μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.

2. μ satisfies the following conditions:

   • (2.1) (∀x, y∈G)(∀γ, δ∈[0, 1])(μ(xy)∨γ≥μ(x)∧μ(y)∧δ),

   • (2.2) (∀x, a, y∈G)(∀γ, δ∈[0, 1))(μ((xa)y)∨γ≥μ(x)∧μ(y)∧δ).

Proof. Let μ be an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G and consider (μ(xy)∨γ<μ(x)∧μ(y)∧δ) for some x, y∈G and γ, δ∈[0, 1], then (μ(xy)∨γ<t≤μ(x)∧μ(y)∧δ) for some t∈(γ, 1]. From this, we observe that [x; t]∈_γμ and [y; t]∈_γμ but μ(xy)<t and μ(xy)+t<2t<2δ, so [xy; t]∈γ∨qδ¯μ, a contradiction. Hence, μ(xy)∨γ≥μ(x)∧μ(y)∧δ for all x, y∈G and γ, δ∈[0, 1]. Next, μ((xa)y)∨γ<μ(x)∧μ(y)∧δ for some x, a, y∈G and γ, δ∈[0, 1]; then there exist some t∈(γ, 1] such that μ((xa)y)∨γ<t≤μ(x)∧μ(y)∧δ. This shows that [x; t]∈_γμ and [y; t]∈_γμ but μ((xa)y)<t and μ((xa)y)+t<2t<2δ, so [(xa)y; λ]∈γ ∨qδ¯μ, a contradiction. Therefore, μ((xa)y)∨γ≥μ(x)∧μ(y)∧δ for all x, a, y∈G and γ, δ∈[0, 1].

Conversely, let μ satisfy conditions (2.1) and (2.2) and consider [x; s]∈_γμ, [y; t]∈_γμ for some x, y∈G and s, t∈(γ, 1]. Then, by (2.1):

μ(xy)∨γ≥μ(x)∧μ(y)∧δ≥s∧t∧δ={s∧tif s∧t≤δδif δ<s∧t.

This shows that [xy; s∧t]∈_γ∨q_δμ, and by (2.2),

μ((xa)y)∨γ≥μ(x)∧μ(y)∧δ≥s∧t∧δ={s∧tif s∧t≤δδif δ<s∧t.

It follows from Theorem 3.2 that [(xa)y; s∧t]∈_γ∨q_δμ. Consequently, μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.□

3.6 Proposition

Let 2δ=1+γ and A be a non-empty subset of G. Then, A is an AG-subgroupoid of G if and only if the fuzzy subset μ of G defined as

μ(x)={≥δ if x∈A,γ    if x∈G\A,

is an (α,∈_γ∨q_δ)-fuzzy AG-subgroupoid of G.

Proof. The proof follows from Ref. [20].□

3.7 Theorem

Let μ be a fuzzy subset of G. Then, the following conditions are equivalent:

1. μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy AG-subgroupoid of G.

2. U(μ; t)(≠∅) is an AG-subgroupoid of G for all t∈(γ, δ].

Proof. Suppose that μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy AG-subgroupoid of G and x, y∈U(μ; t) for some t∈(γ, δ]. Then, μ(x)≥t and μ(y)≥t and by hypothesis

μ(xy)∨γ≥μ(x)∧μ(y)∧δ≥t∧t∧δ=t(as t∈(γ, δ]).

Hence, μ(xy)≥t>γ; therefore, μ(xy)∈U(μ; t). Thus, U(μ; t) is an AG-subgroupoid of G.

Conversely, assume that U(μ; t)(≠∅) is an AG-subgroupoid of G for all t∈(γ, δ]. If there exist x, y∈G such that μ(xy)∨γ<μ(x)∧μ(y)∧δ, then μ(xy)∨γ<t≤μ(x)∧μ(y)∧δ for some t∈(γ, 1]. From this, we see that x, y∈U(μ; t) but xy∉U(μ; t), a contradiction. Hence, μ(xy)∨γ≥μ(x)∧μ(y)∧δ for all x, y∈G and γ, δ∈[0, 1].□

The proof of Proposition 3.8 is straightforward and is omitted.

3.8 Proposition

Let 2δ=1+γ and A be a non-empty subset of G. Then, A is a bi-ideal of G if and only if the fuzzy subset μ of G defined as

μ(x)={≥δ if x∈A,γ    if x∈G\A,

is an (α, ∈_γ ∨q_δ)-fuzzy bi-ideal of G.

Proof. The proof follows from Ref. [20].□

3.9 Theorem

Let μ be a fuzzy subset of G. Then, the following conditions are equivalent:

1. μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.

2. U(μ; t)(≠∅) is a bi-ideal of G for all t∈(γ, δ].

Proof. Suppose that μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G and let x, a, y∈G be such that x, y∈U(μ; t) for some t∈(γ, δ]. Then, μ(x)≥t and μ(y)≥t and by hypothesis

μ((xa)y)∨γ≥μ(x)∧μ(y)∧δ)≥t∧t∧δ=t (as t∈(γ, δ]).

Hence, μ((xa)y)≥t>γ. This shows that (xa)y∈U(μ; t) and hence U(μ; t)(≠∅) is a bi-ideal of G for all x, a, y∈G and t∈(γ, δ].

Conversely, assume that U(μ; t)(≠∅) is a bi-ideal of G for all t∈(γ, δ]. If there exist x, a, y∈G such that μ((xa)y)∨γ<μ(x)∧μ(y)∧δ, then choose t∈(γ, δ] such that μ((xa)y)∨γ<t≤μ(x)∧μ(y)∧δ. From here, we have x, y∈U(μ; t) but (xa)y∉U(μ; t), a contradiction. Hence, μ((xa)y)∨γ≥μ(x)∧μ(y)∧δ for all x, a, y∈G and t∈(γ, δ]. The remaining proof is a consequence of Theorem 3.7. Therefore, μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.□

3.10 Lemma

A non-empty subset A of G is a bi-ideal if and only if the characteristic function μ_A of A is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.

Proof. The proof is straightforward.□

3.11 Definition

A fuzzy subset μ of G is called an (∈_γ, ∈_γ∨q_δ)-fuzzy left (resp. right) ideal of G if the following condition holds:

(∀x, y∈S)(∀γ, δ∈[0, 1])([y; t]∈γμ→[xy; t]∈γ ∨qδμ (resp. [yx; t]∈γ ∨qδμ).

3.12 Theorem

Let 2δ=1+γ and A be a non-empty subset of G. Then, A is a left (resp. right) ideal of G if and only if the fuzzy subset μ of G defined as

μ(x)={≥δ if x∈A,γ    if x∈G\A,

is an (α, ∈_γ ∨q_δ)-fuzzy left (resp. right) ideal of G.

Proof. To prove this, we consider the following three cases:

Case 1. Consider A is a left ideal of G. Let x, y∈G and t∈(γ, 1] be such that [y; t]∈_γμ, then y∈A. As A is a left ideal of G, therefore xy∈A, that is μ(xy)≥δ. If t≤δ, then μ(xy)≥t>γ and so [xy; t]∈_γμ. If t>δ, then μ(xy)+t>δ+t>2δ and hence [xy; t]q_δμ. Thus, [xy; t]∈_γ∨q_δμ. This shows that μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy left ideal of G.

Case 2. Let x, y∈G and t∈(γ, 1] be such that [y; t]q_δμ, then y∈A. Therefore, xy∈A, as A is a left ideal of G, that is μ(xy)≥δ. If t≤δ, then μ(xy)≥t>γ and so [xy; t]∈_γμ. If t>δ, then μ(xy)+t>δ+t>2δ, and hence [xy; t]q_δμ. Thus, [xy; t]∈_γ∨q_δμ. This shows that μ is a (q_δ, ∈_γ∨q_δ)-fuzzy left ideal of G.

Case 3. Let x, y∈G and t∈(γ, 1] be such that [y; t]∈_γ∨q_δμ, follows from case 1 and case 2.□

3.13 Proposition

Every (∈_γ, ∈_γ)-fuzzy bi-ideal of G is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.

Proof. It is straightforward, as for [x; t]∈_γμ, we have [x; t]∈_γ∨q_δμ for all x∈G and t∈(γ, 1].□

The converse of Proposition 3.13 is not true in general, as shown in the following example.

3.14 Example

Consider an AG-groupoid G and a fuzzy subset μ as defined in Example 2.5. Then, μ is an (∈_0.1, ∈_0.1∨q_0.2)-fuzzy bi-ideal of G but μ is not an (∈_0.1, ∈_0.1)-fuzzy bi-ideal of G, as [d; 0.28]∈_0.1μ [e; 0.38]∈_0.1μ and but [(dc)e; 0.28∧0.38]=[c; 0.28]∈¯0.1μ.

3.15 Remark

A fuzzy subset μ of an AG-groupoid G is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G if and only if it satisfies conditions (2.1), and (2.2) of Theorem 3.5.

3.16 Remark

Every fuzzy bi-ideal of an AG-groupoid G is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G. However, the converse is not true in general.

3.17 Example

Consider the AG-groupoid as given in Example 2.5 and define a fuzzy subset μ as follows:

μ(a)=0.8, μ(b)=0.6, μ(c)=0.2, μ(d)=0.3, μ(e)=0.4.

Then, μ is an (∈_0.1, ∈_0.1∨q_0.2)-fuzzy bi-ideal of G. However,

1. μ is not an (∈_0.1, q_0.2)-fuzzy bi-ideal of G, as [d; 0.18]∈_0.1μ and [e; 0.20]∈_0.1μ but

   [(dc)e; 0.18∧0.20]=[c; 0.18]q¯0.2μ.

2. μ is not an (q_0.2, ∈_0.1)-fuzzy bi-ideal of G, as [c; 0.28]∈_0.1μ and [d; 0.38]∈_0.1μ but

   [(cd);  0.28∧0.38]=[c; 0.28]∈¯0.1μ.

3. μ is not an (∈_0.1∨q_0.2, q_0.2)-fuzzy bi-ideal of G, as [e; 0.18]∈_0.1∨q_0.2μ and [d; 0.28]∈_0.1∨q_0.2μ but

   [(ec); 0.18∧0.20]=[c; 0.18]q¯0.2μ.

Thus, μ is not an (α, β)-fuzzy bi-ideal of G, for every α, β∈{∈_0.1, q_0.2, ∈_0.1∨q_0.2}.

3.18 Proposition

If {μ_i}_i∈I is a family of (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideals of an AG-groupoid G, then ∩i∈Iμi is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.

Proof. Let {μ_i}_i∈I be a family of (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideals of G. Let x, y∈G. Then

(∩i∈I μi)(xy)∨γ=limi∈Iμi(xy)∨γ≥limi∈I(μi(x)∧μi(y)∧δ)=(limi∈I(μi(x)∧δ)∧limi∈I(μi(y)∧δ))=(∩limi∈Iμi)(x)∧(∩limi∈Iμi)(y)∧δ.

Let x, a, y∈G. Then

(∩limi∈Iμi)((xa)y)∨γ=limi∈Iμi((xa)y)∨γ≥limi∈I(μi(x)∧μi(y)∧δ)=(limi∈I(μi(x)∧δ)∧limi∈I(μi(y)∧δ))=(∩limi∈Iμi)(x)∧(∩limi∈Iμi)(y)∧δ.

Thus ∩limi∈Iμi is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.□

3.19 Lemma

The intersection of any family (∈_γ, ∈_γ∨q_δ)-fuzzy left (resp. right) ideals of an AG-groupoid G is an (∈_γ, ∈_γ∨q_δ)-fuzzy left (resp. right) ideal of G.

Proof. We consider the case for a left ideal. The case for a right ideal can be proved similarly. Let {μ_i}_i∈I be a family of (∈_γ, ∈_γ∨q_δ)-fuzzy left ideals of G and x, y∈G. Then, (lim_i∈Iμ_i)(xy)=lim_i∈I (μ_i(xy)). As each μ_i is an (∈_γ, ∈_γ∨q_δ)-fuzzy left ideal of G, we have μ_i(xy)∨γ≥μ_i(y)∧δ for all i∈I. Thus

(limi∈Iμi)(xy)∨γ=limi∈I(μi(xy)∨γ)≥limi∈I(μi(y)∧δ)=(limi∈Iμi(y))∧δ=(limi∈Iμi)(y)∧δ.

Hence, lim_i∈Iμ_i is an (∈_γ, ∈_γ∨q_δ)-fuzzy left ideal of G.□

3.20 Lemma

The union of any family (∈_γ, ∈_γ∨q_δ)-fuzzy left (resp. right) ideals of an AG-groupoid G is an (∈_γ, ∈_γ∨q_δ)-fuzzy left (resp. right) ideal of G.

For a detailed study of (α, β)-fuzzy ideals, the readers are referred to Refs. [4, 5, 6, 7, 8, 9, 34, 35] to properly understand the characteristics and properties of this type of ideal theory.

4.1 Definition

A fuzzy subset μ of an AG-groupoid G is called an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G if it satisfies the following conditions:

1. (∀x, y∈G)(∀s, t∈(γ, 1])([x; s]∈_γμ, [y; t]∈_γμ→[xy; s∧t]∈_γ∨q_δμ),

2. (∀x, a, y∈G)(∀s∈(γ, 1])([a; s]∈_γμ→[(xa)y; s]∈_γ∨q_δμ).

4.2 Theorem

Let 2δ=1+γ and A be a non-empty subset of G. Then, A is an interior ideal of G if and only if the fuzzy subset μ of G defined as

μ(x)={≥δ if x∈A,γ    if x∈G\A,

is an (α, ∈_γ∨q_δ)-fuzzy interior ideal of G.

Proof. We consider the following three cases:

• Case 1: Consider A is an interior ideal of G. Let x, a, y∈G and [a; t]∈_γμ for some t∈(γ, 1], then a∈A. As A is an interior ideal of G, therefore xay∈A, that is μ(xay)≥δ. If t≤δ, then μ(xay)≥t and so [xay; t]∈_γμ. If t>δ, then μ(xay)+t>δ+t>2δ and hence [xay; t]q_δμ. Thus, [xay; t]∈_γ∨q_δμ. This shows that μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.

• Case 2: Let x, a, y∈G and t∈(γ, 1] such that [a; t]q_δμ, then a∈A. As A is an interior ideal of G, therefore xay∈A and hence μ(xay)≥δ. If t≤δ, then μ(xay)≥t and so [xay; t]∈_γμ. If t>δ, then μ(xay)+t>δ+t>2δ and hence [xay; t]q_δμ. Thus, [xay; t]∈_γ∨q_δμ. This shows that μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy left ideal of G.

• Case 3: Let x, a, y∈G and t∈(γ, 1] be such that [a; t]∈_γ∨q_δμ follows from case 1 and case 2. The remaining proof follows directly from Proposition 3.8.□

4.3 Lemma

A non-empty subset A of G is an interior ideal if and only if the characteristic function μ_A of A is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.

Proof. The proof follows from Lemma 3.10.□

4.4 Theorem

Suppose that μ is a fuzzy subset of G. Then, the following conditions are equivalent:

1. μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.

2. μ satisfies the following conditions:

   • (2.1) (∀x, y∈G)(∀γ, δ∈[0, 1])(μ(xy)∨γ≥μ(x)∧μ(y)∧δ),

   • (2.2) (∀x, a, y∈G)(∀γ, δ∈[0, 1))(μ((xa)y)∨γ≥μ(a)∧δ).

Proof. Let μ be an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G. Suppose that (μ(xy)∨γ<μ(x)∧μ(y)∧δ) for some x, y∈G and γ, δ∈[0, 1], then (μ(xy)∨γ<t≤μ(x)∧μ(y)∧δ) for some t∈(γ, 1]. From this, we observe that [x; t]∈_γμ and [y; t]∈_γμ but μ(xy)<t and μ(xy)+t<2t≤2δ, so [xy; t]∈γ∨qδ¯μ, a contradiction. Hence, μ(xy)∨γ≥μ(x)∧μ(y)∧δ for all x, y∈G and γ, δ∈[0, 1]. Next, μ((xa)y)∨γ<μ(a)∧δ for some x, a, y∈G and γ, δ∈[0, 1], then there exist some t∈(γ, 1] such that μ((xa)y)∨γ<t≤μ(a)∧δ. This shows that [a; t]∈_γμ but μ((xa)y)≥t and μ((xa)y)+t<2t≤2δ, so [(xa)y; t]∈γ∨qδ¯μ, a contradiction. This implies μ((xa)y)∨γ≥μ(a)∧δ for all x, a, y∈G and γ, δ∈[0, 1].

Conversely, let μ satisfy conditions (2.1) and (2.2) and consider [x; s]∈_γμ, [y; t]∈_γμ for some x, y∈G and s, t∈(γ, 1]. Then, by (2.1)

μ(xy)∨γ≥μ(x)∧μ(y)∧δ≥s∧t∧δ={s∧tif s∧t≤δδif δ<s∧t.

This shows that [xy; s∧t]∈_γ∨q_δμ. Next, we let x, a, y∈G such that [a; s]∈_γμ and by (2.2)

μ((xa)y)∨γ≥μ(a)∧δ≥s∧δ={s if s≤δδ if δ<s.

From this, we can say that [xy; s∧t]∈_γ∨q_δμ. Consequently, μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.□

4.5 Theorem

Let μ be a fuzzy subset of G. Then, the following conditions are equivalent:

1. μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.

2. U(μ; t)(≠∅) is an interior ideal of G for all t∈(γ, δ].

Proof. Suppose that μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G and let x, a, y∈G be such that a∈U(μ; t) for some t∈(γ, δ]. Then, μ(a)≥t and by hypothesis

μ((xa)y)∨γ≥μ(a)∧δ)≥t∧δ=t(as t∈(γ, δ]).

Hence, μ((xa)y)≥t>γ. This shows that (xa)y∈U(μ; t), and hence U(μ; t)(≠∅) is an interior of G for all x, a, y∈G and t∈(γ, δ].

Conversely, assume that U(μ; t)(≠∅) is an interior of G for all t∈(γ, δ]. If there exist x, a, y∈G such that μ((xa)y)∨γ<μ(a)∧δ, then choose t∈(γ, δ] such that μ((xa)y)∨γ<t≤μ(a)∧δ. From here, we have a∈U(μ; t) but (xa)y∉U(μ; t), a contradiction. Hence, μ((xa)y)∨γ≥μ(a)∧δ for all x, a, y∈G and t∈(γ, δ]. The remaining proof is a consequence of Theorem 3.9. Therefore, μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.□

4.6 Proposition

Every (∈_γ, ∈_γ)-fuzzy interior ideal of G is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.

Proof. The proof follows from Proposition 3.13.□

4.7 Remark

A fuzzy subset μ of an AG-groupoid G is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G if and only if it satisfies conditions (2.1) and (2.2) of Theorem 4.4.

4.8 Proposition

Every (∈_γ∨q_δ, ∈_γ∨q_δ)-fuzzy interior ideal of G is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.

Proof. Let μ be an (∈_γ∨q_δ, ∈_γ∨q_δ)-fuzzy interior ideal of G, then consider x, y∈G and s, t∈(γ, 1] such that [x; s]∈_γ∨q_δμ, [y; t]∈_γ∨q_δμ. Therefore, [x; s]∈_γ∨qδμ, [y; t]∈_γ∨q_δμ implies [x; s]∈_γμ or [x; s]q_δμ and [y; t]∈_γμ or [y; t]q_δμ, which leads to [x; s]∈_γμ, [y; t]∈_γμ. Hence, by Definition 4.1, it implies that [xy; s∧t]∈_γ∨q_δμ). Also, suppose x, a, y∈G and s∈(γ, 1] such that [a; s]∈_γ∨q_δμ. Therefore, [a; s]∈_γ∨q_δμ, implies [a; s]∈_γμ or [a; s]q_δμ, which leads to [x; s]∈_γμ. Therefore, Definition 4.1 implies that [xay; s]∈_γ∨q_δμ). Thus, μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G.

Note that, every (∈_γ, ∈_γ∨q_δ)-fuzzy ideal of G is an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal. Similarly, every (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal is an (∈_γ, ∈_γ∨q_δ)-fuzzy generalized bi-ideal of G. In the following propositions, we provide the conditions under which the converses of the above statements are true.□

4.9 Proposition

If G is an intra-regular AG-groupoid, then every (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal is an (∈_γ, ∈_γ∨q_δ)-fuzzy ideal of G.

Proof. Let μ be an (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideal of G. Let a, b∈G. As G is intra-regular, there exists x, y∈G such that a=(xa²)y. Thus,

μ(ab)∨γ=μ(((xa2)y)b)∨γ=μ((by)(xa2))∨γ=μ((by)(x(aa)))∨γ=μ((by)(a(xa))∨γ=μ((by)a(xa)))∨γ≥μ(a)∧δ.

Similarly, we can show that μ(ab)∨γ≥μ(b)∧δ, and hence μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy ideal of G.□

4.10 Proposition

If G is an intra-regular AG-groupoid, then every (∈_γ, ∈_γ∨q_δ)-fuzzy generalized bi-ideal is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.

Proof. Let μ be an (∈_γ, ∈_γ∨q_δ)-fuzzy generalized bi-ideal of G. Let a, b∈G. As G is intra-regular, there exists x, y∈G such that a=(xa²)y. Thus,

μ(ab)∨γ=μ(((xa2)y)b)∨γ=μ(((xa2)(ey))b)∨γ=μ(((ye)(a2x)b)∨γ=μ((a2((ye)x))b)∨γ=μ(((aa)((ye)x))b)∨γ=μ(((x(ye))(aa))b)∨γ=μ((a((x(ye)a))b)∨γ≥μ(a)∧μ(b)∧δ.

Thus, μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy AG-subgroupoid of G and consequently, μ is an (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideal of G.

In Propositions 4.9 and 4.10, if we take G as regular, then the concepts of (∈_γ, ∈_γ∨q_δ)-fuzzy ideals, (∈_γ, ∈_γ∨q_δ)-fuzzy interior ideals, (∈_γ, ∈_γ∨q_δ)-fuzzy generalized bi-ideals, and (∈_γ, ∈_γ∨q_δ)-fuzzy bi-ideals coincide.□

5.1 Definition

A fuzzy subset μ of G is called an (β̄, ᾱ)-fuzzy bi-ideal of G if it satisfies the following conditions:

1. (∀x, y∈G)(∀s, t∈(γ, 1])([xy; s∧t]β¯μ→[x; s]α¯μ or [y; t]α¯μ),

2. (∀x, a, y∈G)(∀s, t∈(γ, 1])([(xa)y; s∧t]β¯μ→[x;  s]α¯μ or [y;  t]α¯μ).

5.2 Theorem

If μ is a fuzzy subset μ of G, then the following conditions are equivalent:

1. μ is an (⋶_γ, ⋶_γ∨q̄_δ)-fuzzy bi-ideal of G.

2. μ satisfies the following conditions:

   1. (∀x, y∈G)(μ(xy)∨δ≥μ(x)∧μ(y)),

   2. (∀x, a, z∈G)(μ((xa)z)∨δ≥μ(x)∧μ(z)).

Proof. Let μ be an (⋶_γ, ⋶_γ∨q̄_δ)-fuzzy bi-ideal of G. If there exists x, y∈G such that μ(xy)∨δ<μ(x)∧μ(y), then μ(xy)∨δ<t≤μ(x)∧μ(y) for some t∈(γ, 1], we see that [xy; t]⋶_γμ but [x; t]∈_γμ and [y; t]∈_γμ, a contradiction, and hence μ(xy)∨δ≥μ(x)∧μ(y) for all x, y∈G. Next, we suppose that there exist a, b, c∈G such that μ((ab)c)∨δ<μ(a)∧μ(c), then μ((ab)c)∨δ<s≤μ(a)∧μ(c) for some s∈(γ, 1] shows that [(ab)c; t]⋶_γμ but [a; t]∈_γμ and [c; t]∈_γμ, a contradiction; therefore, μ((xa)z)∨δ≥μ(x)∧μ(z) for all x, a, z∈G.

Conversely, suppose μ satisfies conditions (i) and (ii). On the contrary, assume that for x, y∈G, [xy; s∧t]⋶_γμ such that [x; s]∈¯γ∨q¯δ¯μ and [y; t]∈¯γ∨q¯δ¯μ. Hence, μ(xy)<s∧t, μ(x)≥s, μ(x)+s>2δ, μ(y)≥t, and μ(y)+t>2δ. We claim that μ(x)>δ and μ(y)>δ. This is because if μ(x)>δ and μ(y)>δ, then s≤μ(x)<δ implies that s<δ, and similarly t<δ. Thus, μ(x)+s≤δ+s≤δ+δ=2δ, which leads to [x; s]q̄_δμ. Hence, [x; s]⋶_γ∨q̄_δμ, a contradiction to our assumption. Therefore, μ(x)>δ and μ(y)>δ. Hence

μ(x)∧μ(y)>(s∧t)∨δ>μ(xy)∨δ.

Thus, a contradiction to (i). This is due to our wrong supposition. Therefore, for x, y∈G such that [xy; s∧t]⋶_γμ, then it implies that [x; s]⋶_γ∨q̄_δμ or [y; t]⋶_γ∨q̄_δμ. Similarly, for x, a, y∈G, [(xa)y; s∧t]⋶_γμ such that [x; s]∈¯γ∨q¯δ¯μ and [y; t]∈¯γ∨q¯δ¯μ. Therefore, μ((xa)y)<s∧t, μ(x)≥s, μ(x)+s>2δ, μ(y)≥t and μ(y)+t>2δ. We claim that μ(x)>δ and μ(y)>δ. This is because if μ(x)>δ and μ(x)>δ, then s≤μ(x)<δ implies that s<δ, and similarly t<δ. Hence, μ(x)+s≤δ+s≤δ+δ=2δ implies that [x; s]q̄_δμ. Hence, [x; s]⋶_γ∨q̄_δμ, a contradiction. Therefore, μ(x)>δ and μ(y)>δ. Hence

μ(x)∧μ(y)>(s∧t)∨δ>μ((xa)y)∨δ.

Thus, a contradiction to (ii). Therefore, for x, y∈G such that [(xa)y; s∧t]_γμ, then it implies that [x; s]⋶_γ∨q̄_δμ or [y; t]⋶_γ∨q̄_δμ. Hence, μ is an (⋶_γ, ⋶_γ∨q̄_δ)-fuzzy bi-ideal of G.□

5.3 Definition

A fuzzy subset μ of G is called an (β̄, ᾱ)-fuzzy interior ideal of G if it satisfies the following conditions:

(∀x, y∈G)(∀s, t∈(γ, 1])([xy; s∧t]β¯μ→[x; s]α¯μ or [y; t]α¯μ),

(∀x, a, y∈G)(∀s∈(γ, 1])([(xa)y; s]β¯μ→[a; s]α¯μ).

5.4 Theorem

If μ is a fuzzy subset μ of G, then the following conditions are equivalent:

1. μ is an (⋶_γ, ⋶_γ∨q̄_δ)-fuzzy interior ideal of G.

2. μ satisfies the following conditions:

   1. (∀x, y∈G)(μ(xy)∨δ≥μ(x)∧μ(y)),

   2. (∀x, a, z∈G)(μ((xa)z)∨δ≥μ(a)).

Proof. It is an immediate consequence of Theorem 5.2.□

5.5 Lemma

If μ is a fuzzy subset of G, then the following are equivalent:

1. U(μ; t) is a bi-ideal of G for all t∈( δ, 1].

2. μ is an (⋶_γ, ⋶_γ∨q̄_δ)-fuzzy bi-ideal.

Proof. Assume that U(μ; t) is a bi-ideal of G for all t∈(δ, 1]. If there exist x, y∈G such that μ(xy)∨δ<μ(x)∧μ(y)=t₁, then t₁∈(δ, 1], x, y∈U(μ; t₁). However, μ(xy)<t₁ implies xy∉U(μ; t₁), a contradiction. Hence, μ(xy)∨δ≥μ(x)∧μ(y) for all x, y∈G.

If there exist x, y, z∈G such that μ((xy)z)∨δ<μ(x)∧μ(z)=t₂, then t₂∈(δ, 1], x, z∈U(μ; t₂). However, μ((xy)z)<t₂ implies (xy)z∉U(μ; t₂), a contradiction. Hence, μ((xy)z)∨δ≥μ(x)∧μ(z) for all x, y, z∈G.

Conversely, suppose that for x, y∈U(μ; t). By Theorem 5.2, we get

μ(xy)∨δ≥μ(x)∧μ(y)≥t,

and so μ(xy)≥t. It follows that xy∈U(μ; t). Let x, z∈U(μ; t), then μ(x)≥t and μ(z)≥t. By Theorem 5.2, we get

μ((xy)z)∨δ≥μ(x)∧μ(z)≥t,

and so μ((xy)z)≥t. It follows that (xy)z∈U(μ; t). Thus, U(μ; t) is a bi-ideal of G for all t∈(δ, 1].□

5.6 Lemma

If μ is a fuzzy subset of G, then the following are equivalent:

1. U(μ, t) is an interior ideal of G for all t∈(δ, 1].

2. μ is an (⋶_γ, ⋶_γ∨q̄_δ)-fuzzy interior ideal.

Proof. This follows directly from Lemma 5.5.□

Next, we discuss fuzzy bi-ideals with thresholds and fuzzy interior ideals with thresholds.

5.7 Definition

Let γ, δ∈[0, 1] and γ<δ, then a fuzzy subset μ of G is called a fuzzy bi-ideal with thresholds (γ, δ] of G if it satisfies the following conditions:

1. (∀x, y∈G)(μ(xy)∨γ≥μ(x)∧μ(y)∧δ),

2. (∀x, y, z∈G)(μ((xy)z)∨γ≥μ(x)∧μ(z)∧δ).

5.8 Theorem

A fuzzy subset μ of G is a fuzzy bi-ideal with thresholds (γ, δ] of G if and only if U(μ; t)(≠∅) is a bi-ideal of G for all γ<t≤δ.

5.9 Definition

Let γ, δ∈(0, 1] and γ<δ, then a fuzzy subset μ of G is called a fuzzy interior ideal with thresholds (γ, δ] of G if it satisfies the following conditions:

1. (∀x, y∈G)(μ(xy)∨γ≥μ(x)∧μ(y)∧δ),

2. (∀x, y, z∈G)(μ((xy)z)∨γ≥μ(a)∧δ).

5.10 Theorem

A fuzzy subset μ of G is a fuzzy interior ideal with thresholds (γ, δ] of G if and only if U(μ; t)(≠∅) is an interior ideal of G for all γ<t≤δ.