New types of fuzzy ( m,n )-ideals in ordered semigroups

1 Introduction

Since Zadeh introduced the idea of fuzzy sets in 1965 [20], fuzzy sets and fuzzy systems theories quickly developed. In the pioneering paper of Rosenfeld [18] in 1971, the study of the fuzzy algebraic structures began. Kuroki [10] applied the theory of fuzzy sets to semigroups and fuzzy sets to ordered semigroups were first considered by Kehayopulu and Tsingelis in [6]. Jun et al. [2] introduced the concept of (α, β)-fuzzy generalized bi-ideals in ordered semigroups and provided several characterizations of regular and intra-regular ordered semigroups in terms of (∈ , ∈ ∨ q)-fuzzy generalized bi-ideals. To generalize the concept of (α, β)-fuzzy generalized bi-ideals of ordered semigroups, Khan et al. introduced the concept of (∈ , ∈ ∨ q _k )-fuzzy generalized bi-ideals in [8] of ordered semigroups. By using the concept of (∈ , ∈ ∨ q _k )-fuzzy ideals, Shabir et al. [19] characterized regular semigroups. The concept of (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy generalized bi-ideals and its (κ^*, κ)-lowe part in ordered semigroups introduced and charactized by Khan et al. [7]. Also, different notions of fuzzy set have been applied in semigroups (see for e.g., [3–5, 15–17]). For related notions of the present study, the reader is referred to [12–14].

In this paper, we establishe new types of fuzzy ideals in ordered semigroups. In fact, on ordered semigroups, the notion of (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideals is defined and some vital properties are investigated. In addition, (m, n)-regular ordered semigroups are described in terms of (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideal and their (κ^*, κ)-lower parts. The structure of paper is organized as: Section 2 summarizes some concepts and properties relating to ordered semigroups, ideals, fuzzy subsets, fuzzy subsemigroups that are required to produce our key results. The notion of (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideal of ordered semigroups is discussed in Section 3. Section 4 is dedicated to the analysis of (κ^*, κ)-lower part of (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideal. In Section 5, directions for some potential work are given. The paper finishes with a list of references.

2 Preliminaries

A semigroup (S, ·) is said to be an ordered semigroup if (S, ≤) is a partially ordered set such that a ≤ b ⇒ ac ≤ bc and ca ≤ cb, ∀a, b, c ∈ S.

For A ⊆ S, we denote (A] = {t ∈ S | t ≤ a forsome a ∈ A} . For subsets ∅ ≠ A, B ⊆ S, the product of A and B is define as AB = {ab | a ∈ A and b ∈ B}.

A subset ∅ ≠ Ω of S is called a subsemigroup if ΩΩ ⊆ Ω. Further, Ω is called a left (resp. right) ideal of S if SΩ ⊆ Ω  (resp .  ΩS ⊆ Ω) and (Ω] ⊆ Ω. It is called an ideal of S if it is both left and right ideal of S. A subsemigroup A of S is called a bi-ideal of S if ASA ⊆ A and (A] ⊆ A. A subsemigroup A of S is called an (m, n)-ideal of S [1] if A ^m SA ⁿ  ⊆ A and (A] ⊆ A, where m, n are non-negative integers. Here A⁰S = SA⁰ = S.

For (∅ ≠) A ⊆ S, the principal (m, n)-ideal [A] _(m,n) generated by A is defined [1] as follows: [ A ] ( m , n ) = ( ⋃ i = 1 m + n A i ∪ A m S A n ] .

Throughout this paper, I ( m , n ) ˆ will stand for the set of all (m, n)-ideals of S.

A mapping η : S → [0, 1] is called the fuzzy subset of S. For Ω ⊆ S, the characteristic function χ _Ω is defined as: χ Ω ( x ) = { 1 if x ∈ Ω , 0 if x ∉ Ω . For any fuzzy subsets η and ξ of S, the η ∩ ξ, η ∪ ξ and η ∘ ξ are defined as: ( η ∩ ξ ) ( x ) = min { η ( x ) , ξ ( x ) } ( η ∪ ξ ) ( x ) = max { η ( x ) , ξ ( x ) } and ( η ∘ ξ ) ( x ) = { ⋁ ( y , z ) ∈ Ω x { η ( y ) ∧ ξ ( z ) } if Ω x ≠ ∅ , 0 if Ω x = ∅ , where Ω _x  = {(r, s) ∈ S × S | x ≤ rs}. Define an order relation ⪯ on the set F ( S ) of all fuzzy subsets of S by η ⪯ ξ ⇔ η ( x ) ≤ ξ ( x ) ∀ x ∈ S . If η , ξ ∈ F ( S ) such that η ⪯ ξ, then ∀ λ ∈ F ( S ) , η ∘ λ ⪯ ξ ∘ λ and λ ∘ η ⪯ λ ∘ ξ. We denote by 1 the fuzzy subset of S defined by 1 : S → [0, 1] |x ↦ 1 (x) =1.

Let A, B ⊆ S. Then A ⊆ B ⇔ η _A  ⪯ η _B ; η _A  ∩ η _B  = η_A∩B; η _A  ∘ η _B  = η_(AB].

F ( S ) ∋ η is called a fuzzy subsemigroup [11] of S if η (ℏ ℓ) ≥ min {η (ℏ) , η (ℓ)} ∀ ℏ , ℓ ∈ S and it is called a fuzzy left (resp. right) ideal [11] of S if ℏ ≤ ℓ ⇒ η (ℏ) ≥ η (ℓ) , η (ℏ ℓ) ≥ η (ℓ) (resp. η (ℏ ℓ) ≥ η (ℏ)) ∀ ℏ , ℓ ∈ S. If η is both fuzzy left and right ideal of S, then it is called a fuzzy ideal of S. Fuzzy subset η is a strongly convex fuzzy subset ⇔ ℏ ≤ ℓ ⇒ η (ℏ) ≥ η (ℓ)   ∀   ℏ , ℓ ∈ S . A fuzzy subsemigroup η of S is called a fuzzy bi-ideal [11] of S if ℏ ≤ ℓ ⇒ η (ℏ) ≥ η (ℓ) and η (ℏ ℓ z) ≥ min {η (ℏ) , η (ı)} ∀ ℏ , ℓ , ı ∈ S.

Let a ∈ S and ω ∈ (0, 1]. An ordered fuzzy point (OFP) a _ω of S is defined by a ω ( x ) = { ω , if x ∈ ( a ] , 0 , if x ∉ ( a ] . For η ∈ F ( S ) , a _ω  ∈ η will stand for a _ω  ⊆ η. Thus a _ω  ∈ η ⇔ η (a) ≥ ω.

An OFP a _ω of S, for any κ^* ∈ (0, 1], is said to be (κ^*, q)-quasi-coincident with a fuzzy subset η of S, written as a _ω  (κ^*, q) η, if η (a) + ω > κ^* .

Let 0 ≤ κ < κ^* ≤ 1. For any OFP ℏ _ω , we say that

ℏ _ω  (κ^*, q _κ ) η if η (ℏ) + ω + κ > κ^*;

F ( S ) ∋ η is called an (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy subsemigroup of S if ℏ _ω , ℓ  _ɛ  ∈ η ⇒ (ℏ ℓ) _min{ω,ɛ} ∈ ∨ (κ^*, q _κ ) η  ∀  ω, ɛ ∈ (0, 1] , ℏ , ℓ ∈ S. It is called an (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy left (resp. right) ideal of S if ℏ≤ ℓ, ℓ _ω  ∈ η ⇒ ℏ  _ω  ∈ ∨ (κ^*, q _κ ) η, and ℏ ∈ S, ℓ _ω  ∈ η ⇒ (ℏ ℓ)  _ω  ∈ ∨ (κ^*, q _κ ) η (resp. (ℓ ℏ)  _ω  ∈ ∨ (κ^*, q _κ ) η). If it is both an (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy left and right ideal of S, then it is called an (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy ideal. An (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy subsemigroup of S is called an (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy bi-ideal if ℏ≤ ℓ, ℓ _ω  ∈ η ⇒ ℏ  _ω  ∈ ∨ (κ^*, q _κ ) η and ℏ _ω  ∈ η, ı _ɛ  ∈ η ⇒ (ℏ ℓ ı) _min{ω,ɛ} ∈ ∨ (κ^*, q _κ ) η ∀ ω, ɛ ∈ (0, 1] , ℏ , ℓ , ı ∈ S.

Theorem 2.1. [7] Let η be a fuzzy subset of S. Then η is an (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy subsemigroup of S if and only if η ( xy ) ≥ min { η ( x ) , η ( y ) , κ * - κ 2 } for all x, y ∈ S.

Theorem 2.2. [7] Let η be a fuzzy subset of S. Then η is an (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy bi-ideal of S if and only if

x ≤ y ⇒ η ( x ) ≥ min { η ( y ) , κ * - κ 2 } , and

3 (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideals

Definition 3.1. A fuzzy subset η of S is called an (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideal of S if

x¹ ≤ z¹, z ω 1 1 ∈ η ⇒ x ω 1 1 ∈ ∨ ( κ * , q κ ) η ,

for all ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n  ∈ (0, 1] and x¹, x², …, x ^m , y, z¹, z², …, z ⁿ  ∈ S

Throughout the paper, the set of all (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideals and fuzzy bi-ideals of S will be denoted by F ( m , n ) ˆ and F B ˆ .

Lemma 3.3. Let η ∈ F ( S ) . Then η ∈ F B ˆ ⇒ η ∈ F ( m , n ) ˆ ∀ m , n ∈ ℤ + .

Proof. Let η ∈ F B ˆ . Then, first two conditions of (∈ , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideal hold. Suppose that x ω 1 1 , x ω 2 2 , ⋯ , x ω m m ∈ η , z ɛ 1 1 , z ɛ 2 2 , ⋯ , z ɛ n n ∈ η for any x¹, x², …, x ^m , y, z¹, z², …, z ⁿ  ∈ S and ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n  ∈ (0, 1]. By Theorem 2.2, we have η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) ≥ min { η ( x 1 ) , η ( x 2 ) , … , η ( x m ) , η ( z 1 ) , η ( z 2 ) , … , η ( z n ) , κ * - κ 2 } .

Since x ω 1 1 , x ω 2 2 , … , x ω m m ∈ η and z ɛ 1 1 , z ɛ 2 2 , … , z ɛ n n ∈ η , we have η (x¹) ≥ ω₁, η (x²) ≥ ω₂, …, η (x ^m ) ≥ ω _m and η (z¹) ≥ ɛ₁, η (z²) ≥ ɛ₂, …, η (z ⁿ ) ≥ ɛ _n . So η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) ≥ min  {η (x¹) , η (x²) , … , η (x ^m ) , η (z¹) , η (z²) , … , η ( z n ) , κ * - κ 2 } ≥ min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n , κ * - κ 2 } . If min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } ≤ κ * - κ 2 , then η ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) ≥ min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } . So, (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ η. If min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } > κ * - κ 2 , then η ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) ≥ κ * - κ 2 . So η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) + min {ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ n } > κ * - κ 2 + κ * - κ 2 = κ * - κ , which implies that (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} (κ^*, q _κ ) η. So (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ ∨ (κ^*, q _κ ) η. Hence η ∈ F ( m , n ) ˆ . □

Remark 3.4. In general, η ∈ F ( m , n ) ˆ ≱ η ∈ F B ˆ .

Example 3.5. Let S be an ordered semigroup of Example 3.2. Define a fuzzy subset η of S as follows: η ( x ) = { 0.3 if x ∈ { ℏ , ℓ } , 0 if x ∈ { ı , 𝚥 } . Take κ^* = 0.7 and κ = 0.1. Then η ∈ F ( m , n ) ˆ ∀ m, n ≥ 2, but η ∉ F B ˆ because 0 = η ( ı ) = η ( ℓ 𝚥 ℓ ) ≱ min { η ( ℓ ) , η ( ℓ ) , κ * - κ 2 } = 0.3 .

Theorem 3.6. Let η ∈ F ( S ) . Then η ∈ F ( m , n ) ˆ ⇔

x 1 ≤ z 1 ⇒ η ( x 1 ) ≥ min { η ( z 1 ) , κ * - κ 2 } ,

Proof. (⇒) Let x¹, z¹ ∈ S such that x¹ ≤ z¹. Now on contrary suppose that η ( x 1 ) < min { η ( z 1 ) , κ * - κ 2 } for some x¹, z¹ ∈ S. So ∃ ω ∈ (0, 1] such that η ( x 1 ) < ω ≤ min { η ( z 1 ) , κ * - κ 2 } . Then z ω 1 ∈ η but x ω 1 ∈ ∨ ( κ * , q κ ) ¯ η , which is a contradiction. Hence η ( x 1 ) ≥ min { η ( z 1 ) , κ * - κ 2 } . Thus (1) holds.

Next, suppose to the contrary that η ( x 1 z 1 ) < min { η ( x 1 ) , η ( z 1 ) , κ * - κ 2 } for some x¹, z¹ ∈ S. So ∃ ω ∈ (0, 1] such that η ( x 1 z 1 ) < ω ≤ min { η ( x 1 ) , η ( z 1 ) , κ * - κ 2 } . Then x ω 1 ∈ η and z ω 1 ∈ η , but ( x 1 z 1 ) ω ∈ ∨ ( κ * , q κ ) ¯ η , which is a contradiction. Thus η ( x 1 z 1 ) ≥ min { η ( x 1 ) , η ( z 1 ) , κ * - κ 2 } which proves (2).

Finally, to show that (3) holds, suppose to the contrary that η ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) < min { η ( x 1 ) , η ( x 2 ) , … , η ( x m ) , η ( z 1 ) , η ( z 2 ) , … , η ( z n ) , κ * - κ 2 } for some x¹, x², …, x ^m , y, z¹, z², …, z ⁿ  ∈ S. So ∃ ω ∈ (0, 1] such that η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) < ω and ω ≤ min { η ( x 1 ) , η ( x 2 ) , … , η ( x m ) , η ( z 1 ) , η ( z 2 ) , … , η ( z n ) , κ * - κ 2 } . Then x ω 1 1 , x ω 2 2 , ⋯ , x ω m m ∈ η and z ɛ 1 1 , z ɛ 2 2 , ⋯ , z ɛ n n ∈ η , but ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) ω ∈ ∨ ( κ * , q κ ) ¯ η , which is a contradiction. Hence η (x¹x²⋯ x ^m yz¹z² ⋯ z ⁿ ) ≥ min {η (x¹) , η (x²) , …, η (x ^m ) , η (z¹) , η (z²) , … , η ( z n ) , κ * - κ 2 } . Thus (3) holds.

(⇐) Assume that the conditions (1), (2) and (3) hold. By (1), we have η ( x 1 ) ≥ min { η ( z 1 ) , κ * - κ 2 } for any x¹, z¹ ∈ S with x¹ ≤ z¹. Let z ω 1 ∈ η (ω ∈ (0, 1]). Then η (z¹) ≥ ω. Thus η ( x 1 ) ≥ min { ω , κ * - κ 2 } . Now, if ω ≤ κ * - κ 2 , then η (x¹) ≥ ω. So x ω 1 ∈ η . Again, if ω > κ * - κ 2 , then η ( x 1 ) ≥ κ * - κ 2 . So η ( x 1 ) + ω > κ * - κ 2 + κ * - κ 2 = κ * - κ , which implies that x ω 1 ( κ * , q κ ) η . Hence x ω 1 ∈ ∨ ( κ * , q κ ) η .

Next, assume that x ω 1 1 ∈ η and z ɛ 1 1 ∈ η (ω₁, ɛ₁ ∈ (0, 1]). Then η (x¹) ≥ ω₁ and η (z¹) ≥ ɛ₁. Now, as η ( x 1 z 1 ) ≥ min { η ( x 1 ) , η ( z 1 ) , κ * - κ 2 } , we have η ( x 1 z 1 ) ≥ min { ω 1 , ɛ 1 , κ * - κ 2 } . If min { ω 1 , ɛ 1 } ≤ κ * - κ 2 , then η (x¹z¹) ≥ min {ω, ɛ} implies that (x¹z¹) _min{ω1,ɛ1} ∈ η . In the other case when min { ω 1 , ɛ 1 } > κ * - κ 2 , we have η ( x 1 z 1 ) ≥ κ * - κ 2 . So η ( x 1 z 1 ) + min { ω 1 , ɛ 1 } > κ * - κ 2 + κ * - κ 2 = κ * - κ implies (x¹z¹) _min{ω,ɛ} (κ^*, q _κ ) η. Thus (x¹z¹) _min{ω,ɛ} ∈ ∨ (κ^*, q _κ ) η.

Finally, take any x¹, x², …, x ^m , y, z¹, z², …, z ⁿ  ∈ S and ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n in (0, 1] such that x ω 1 1 , x ω 2 2 , ⋯ , x ω m m ∈ η , z ɛ 1 1 , z ɛ 2 2 , ⋯ , z ɛ n n ∈ η . By (3), η (x¹x² ⋯ x ^m yz¹z² ⋯ z n ) ≥ min { η ( x 1 ) , η ( x 2 ) , … , η ( x m ) , η ( z 1 ) , η ( z 2 ) , … , η ( z n ) , κ * - κ 2 } . As x ω 1 1 , x ω 2 2 , … , x ω m m ∈ η and z ɛ 1 1 , z ɛ 2 2 , ⋯ , z ɛ n n ∈ η for all ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n  ∈ (0, 1], we have η (x¹) ≥ω₁, η (x²) ≥ ω₂, …, η (x ^m ) ≥ ω _m and η (z¹) ≥ ɛ₁, η (z²) ≥ ɛ₂, …, η (z ⁿ ) ≥ ɛ _n . So η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) ≥ min { η ( x 1 ) , η ( x 2 ) , … , η ( x m ) , η ( z 1 ) , η ( z 2 ) , … , η ( z n ) , κ * - κ 2 } ≥ min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n , κ * - κ 2 } . If min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } ≤ κ * - κ 2 , then η (x¹x² ⋯ x ^m yz¹z², ⋯ z ⁿ ) ≥ min {ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n } implies that ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } ∈ η . Again, if min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } > κ * - κ 2 , then η ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) ≥ κ * - κ 2 . So η ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) + min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } > κ * - κ 2 + κ * - κ 2 = κ * - κ , which implies that (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} (κ^*, q _κ ) η . Therefore (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ ∨ (κ^*, q _κ ) η. Hence η ∈ F ( m , n ) ˆ .□

Definition 3.7. S is said to be (m, n)-regular if ∀ a ∈ S  ∃  x ∈ S such that a ≤ a ^m xa ⁿ .

Lemma 3.8 If S is (m, n)-regular, then η ∈ F ( m , n ) ˆ ⇒ η ∈ F B ˆ .

Proof. Suppose that η ∈ F ( m , n ) ˆ and x, a, y ∈ S . Since S is (m, n)-regular, there exist p, q ∈ S such that xay ≤ x ^m px ⁿ ay ^m qy ⁿ . Now η (xay) = η (x ^m px ⁿ ay ^m qy ⁿ ) = η (x ^m  (px ⁿ ay ^m q) y ⁿ ) ≥ min { η ( x ) , η ( x ) , … , η ( x ) , η ( y ) , η ( y ) , … , η ( y ) , κ * - κ 2 } = min { η ( x ) , η ( y ) , κ * - κ 2 } .

Hence by Theorem 2.2, η ∈ F B ˆ .□

Lemma 3.9 Let { η i ∈ F ( m , n ) ˆ | i ∈ I } . Then ⋂ i ∈ I η i ∈ F ( m , n ) ˆ .

Proof. Take any x, y ∈ S such that x ≤ y and y _ω  ∈ ⋂ _i∈Iη _i . Then y _ω  ∈ η _i for each i ∈ I. As each η i ∈ F ( m , n ) ˆ , x _ω  ∈ η _i for each i ∈ I. Thus x _ω  ∈ ⋂ _i∈Iη _i .

Next take any x, y ∈ S and ω, ɛ ∈ (0, 1] such that x _ω , y _ɛ  ∈ ⋂ _i∈Iη _i . Then x _ω , y _ɛ  ∈ η _i for each i ∈ I. So η _i  (x) ≥ ω,  η _i  (y) ≥ ɛ. Thus, we have ⋂_i∈Iη _i  (xy) = ⋀ _i∈Iη _i  (xy) ≥ ⋀ _i∈I min {η _i  (x) , η _i  (y)} ≥ min {ω, ɛ} . So (xy) _min{ω,ɛ} ∈ ⋂ _i∈Iη _i  (xy).

Finally, take any x ω 1 1 , x ω 2 2 , ⋯ , x ω m m ∈ ⋂ i ∈ I η i and z ɛ 1 1 , z ɛ 2 2 , ⋯ , z ɛ n n ∈ ⋂ i ∈ I η i for each x₁, x₂, ⋯ , x _m , z, y₁, y₂, ⋯ , y _n  ∈ S and ω₁, ω₂, ⋯ , ω _m , ɛ₁, ɛ₂, ⋯ , ɛ _n  ∈ (0, 1]. Therefore x ω 1 1 , x ω 2 2 , ⋯ , x ω m m ∈ η i and z ɛ 1 1 , z ɛ 2 2 , ⋯ , z ɛ n n ∈ η i for i ∈ I. As each η i ∈ F ( m , n ) ˆ , so (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ ∨ (κ^*, q _κ ) η _i   ∀  i ∈ I. Thus (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ η _i or η _i  (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) + min {ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n } + κ ≥ κ^*. If (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,} ɛ₂, …, ɛ _n } ∈ η _i , then

⋂_i∈Iη _i  (x₁x₂ ⋯ x _m zy₁y₂ ⋯ y _n ) = ⋀ _i∈Iη _i  (x₁x₂ ⋯ x _m zy₁y₂ ⋯ y _n ) ≥ ⋀ _i∈I min {ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n } = min {ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n } .

Therefore (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ ⋂ _i∈Iη _i , which implies that (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ ∨ (κ^*, q _κ ) ⋂ _i∈Iη _i . Similarly, if η _i  (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) + min {ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n } + κ ≥ κ^*, then (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ ∨ (κ^*, q _κ ) ⋂ _i∈Iη _i  . Hence ⋂ i ∈ I η i ∈ F ( m , n ) ˆ . □

Remark 3.10. Let { η i ∈ F ( m , n ) ˆ | i ∈ I } . Then ⋃ i ∈ I η i ∉ F ( m , n ) ˆ in general. The following example validates the above claim:

Then S is an ordered semigroup. Define η 1 , η 2 ∈ F ( S ) by η 1 ( ℏ ) = 0.4 , η 1 ( ı ) = 0.4 , η 1 ( 𝚥 ) = 0 , η 1 ( ℓ ) = 0 ; η 2 ( ℏ ) = 0.4 , η 2 ( ı ) = 0 , η 2 ( 𝚥 ) = 0.4 , η 2 ( ℓ ) = 0 . Then η 1 , η 2 ∈ F ( m , n ) ˆ , but η 1 ∪ η 2 ∉ F ( m , n ) ˆ because 0 = η 1 ( ℓ ) ∨ η 1 ( ℓ ) = ( η 1 ∪ η 2 ) ( ℓ ) = ( η 1 ∪ η 2 ) ( ı 𝚥 ) < min { ( η 1 ∪ η 2 ) ( ı ) , ( η 1 ∪ η 2 ) ( 𝚥 ) , κ * - κ 2 } .

Definition 3.12. [9] Let η ∈ F ( S ) . Then the set [ η ] ω = { x ∈ S | x ω ∈ ∨ ( κ * , q κ ) η } , where ω ∈ (0, 1] , is called an (∈ ∨ (κ^*, q _κ ))-level subset of η.

Theorem 3.13. Let η be any strongly convex fuzzy subset of S. Then η ∈ F ( m , n ) ˆ ⇔ the (∈ ∨ (κ^*, q _κ ))-level subset [ η ] ω ∈ I ( m , n ) ˆ , ∀ ω ∈ (0, 1].

Proof. (⇒) Take any x ∈ S and y ∈ [η]  _ω such that x ≤ y. As y ∈ [η]  _ω , we have y _ω  ∈ ∨ (κ^*, q _κ ) η; that is η (y) ≥ ω or η (y) + ω + κ > κ^*. As η is strongly convex, η (x) ≥ η (y) ≥ ω or η (x) ≥ η (y) ≥ κ^* - ω - κ; that is x _ω  ∈ ∨ (κ^*, q _κ ) η. So x ∈ [η]  _ω .

Next, take any x¹, x², …, x ^m , z¹, z², …, z ⁿ  ∈ [η]  _ω for ω ∈ (0, 1]. Then x ω i ∈ ∨ ( κ * , q κ ) η and z ω j ∈ ∨ ( κ * , q κ ) η ; that is η (x ⁱ ) ≥ ω or η (x ⁱ ) + ω + κ > κ^* and η (z ^j ) ≥ ω or η (z ^j ) + ω + κ > κ^* for each i ∈ {1, 2, …, m} and j ∈ {1, 2, …, n}. Since η ∈ F ( m , n ) ˆ , by Theorem 3.6, η ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) ≥ min { η ( x 1 ) , η ( x 2 ) , … , η ( x m ) , η ( z 1 ) , η ( z 2 ) , … , η ( z n ) , κ * - κ 2 } .

Case (1). Let η (x^ı) ≥ ω, η (z^𝚥) ≥ ω for each ı ∈ {1, 2, . . , m} , 𝚥 ∈ {1, 2, …, n}. If ω > κ * - κ 2 , then η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) ≥ min { η ( x 1 ) , η ( x 2 ) , … , η ( x m ) , η ( z 1 ) , η ( z 2 ) , … , η ( z n ) , κ * - κ 2 } ≥ min { ω , ω , . . . , ω , ω , ω , … , ω , κ * - κ 2 } = κ * - κ 2 , and, thus, (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ )  _ω  (κ^*, q _κ ) η. If ω ≤ κ * - κ 2 , then η (xy) ≥ min {η (x¹) , η (x²) , …, η (x ^m ) , η (z¹) , η (z²) , …, η ( z n ) , κ * - κ 2 } ≥ min { ω , ω , . . , ω , ω , ω , … , ω , κ * - κ 2 } = ω and, so, (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ )  _ω  ∈ η. Hence (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ )  _ω  ∈ ∨ (κ^*, q _κ ) η.

Case (2). Let η (x^ı) ≥ ω for each ı ∈ {1, 2, . . , m}, η (z^𝚥) ≥ ω for each 𝚥 ∈ {1, 2, . . , n - 1}, and η (z ⁿ ) + ω + κ > κ^*. If ω > κ * - κ 2 , then

η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) ≥ min {η (x¹) , η (x²) , …, η (x ^m ) , η (z¹) , η (z²) , … η ( z n ) , κ * - κ 2 } ≥ { ω , ω , … , ω , ω , ω , … ω , η ( z n ) , κ * - κ 2 } ≥ { ω , η ( z n ) , κ * - κ 2 } > min { ω , ( κ * - ω - κ ) , κ * - κ 2 } = κ^* - ω - κ ; i.e., η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) + ω + κ > κ^*, and, so, (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ )  _ω  (κ^*, q _κ ) η. Again, if ω ≤ κ * - κ 2 , then

η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) ≥ min  {η (x¹) , η (x²) , …, η (x ^m ) , η (z¹) , η (z²) , …, η ( z n ) , κ * - κ 2 } ≥ { ω , ω , … , ω , ω , ω , … ω , η ( z n ) , κ * - κ 2 } ≥ { ω , η ( z n ) , κ * - κ 2 } > min { ω , ( κ * - ω - κ ) , κ * - κ 2 } = ω and, thus, (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ )  _ω  ∈ η. Hence (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ )  _ω  ∈ ∨ (κ^*, q _κ ) η.

Case (3). Let η (x^ı) ≥ ω for each ı ∈ {1, 2, …, m}, η (z^𝚥) ≥ ω for each 𝚥 ∈ {1, 2, …, n - 2}, and η (z ^κ ) + ω + κ > κ^* for κ ∈ {n - 1, n}. In this case, the proof is similar to the proof of case (2).

Continuing in this way, we may show that (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ )  _ω  ∈ ∨ (κ^*, q _κ ) η.

Thus in each case, we have (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ )  _ω  ∈ ∨ (κ^*, q _κ ) η, and so x¹x² ⋯ x ^m yz¹ z² ⋯ z ⁿ  ∈ [η]  _ω . Hence [ η ] ω ∈ I ( m , n ) ˆ . (⇐) Let x¹, z¹ ∈ S such that x¹ ≤ z¹. Suppose to the contrary that η ( x 1 ) < min { η ( z 1 ) , κ * - κ 2 } . Then η ( x 1 ) < ω ≤ min { η ( z 1 ) , κ * - κ 2 } , for some ω ∈ ( 0 , κ * - κ 2 ] , implies that z¹ ∈ [η]  _ω , but x¹ ∉ [η]  _ω , a contradiction. Thus η ( x 1 ) ≥ min { η ( z 1 ) , κ * - κ 2 } .

Next, we assume that if, η ( x 1 z 1 ) < min { η ( x 1 ) , η ( z 1 ) , κ * - κ 2 } for some x¹, y¹ ∈ S. Then ∃ ω ∈ ( 0 , κ * - κ 2 ] such that η ( x 1 z 1 ) < ω ≤ min { η ( x 1 ) , η ( z 1 ) , κ * - κ 2 } . Now x¹, z¹ ∈ [η]  _ω which implies that x¹z¹ ∈ [η]  _ω . Then η (x¹z¹) ≥ ω or η (x¹z¹) + ω + κ > κ^* which is not possible. Therefore η ( x 1 z 1 ) ≥ min { η ( x 1 ) , η ( z 1 ) , κ * - κ 2 } for all x¹, z¹ ∈ S. Using the similar argument, we get η ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) ≥ min { η ( x 1 ) , η ( x 2 ) , … , η ( x m ) , η ( z 1 ) , η ( z 2 ) , … , η ( z n ) , κ * - κ 2 } for all x¹, x², …, x ^m , y, z¹, z², …, z ⁿ  ∈ S. Hence by Theorem 3.6, η ∈ F ( m , n ) ˆ .□

Theorem 3.14. Let B ∈ I ( m , n ) ˆ and let η be defined by η ( b ) = { κ * - κ 2 if b ∈ B , 0 if b ∉ B . Then

η is an ((k^*, q) , ∈ ∨ (κ^*, q _κ ))-fuzzy (m,n)-ideal of S; and

Proof. (1). Let x¹, z¹ ∈ S with x¹ ≤ z¹ and let ω ∈ (0, 1] be such that z ω 1 ( k * , q ) η . Then z¹ ∈ B, η (z¹) + ω > k^*. Since B ∈ I ( m , n ) ˆ and x¹ ≤ z¹ ∈ B, we have x¹ ∈ B. Thus η ( x 1 ) ≥ κ * - κ 2 . If ω ≤ κ * - κ 2 , then η (x¹) ≥ ω. So x ω 1 ∈ η . If ω > κ * - κ 2 , then η ( x 1 ) + ω > κ * - κ 2 + κ * - κ 2 = κ * - κ . Therefore x ω 1 ( κ * , q κ ) η .

Next, we assume that x¹, z¹ ∈ S and ω₁, ɛ₁ ∈ (0, 1] be such that x ω 1 1 ( k * , q ) η and z ɛ 1 1 ( k * , q ) η . Then x¹, z¹ ∈ B, η (x¹) + ω₁ > k^* and η (z¹) + ɛ₁ > k^*. As B ∈ I ( m , n ) ˆ , we have x¹z¹ ∈ B. Thus η ( x 1 z 1 ) ≥ κ * - κ 2 . If min { ω 1 , ɛ 1 } ≤ κ * - κ 2 , then η (x¹z¹) ≥ min {ω₁, ɛ₁} and, so, (x¹z¹) _min{ω1,ɛ1} ∈ η. Again, if min { ω 1 , ɛ 1 } > κ * - κ 2 , then

η ( x 1 z 1 ) + min { ω 1 , ɛ 1 } > κ * - κ 2 + κ * - κ 2 = κ * - κ .

So (x¹z¹) _min{ω1,ɛ1} (κ^*, q _κ ) η. Therefore (x¹z¹) _min{ω1,ɛ1} ∈ ∨ (κ^*, q _κ ) η .

Finally, assume that x¹, x², …, x ^m , y, z¹, z², …, z ⁿ  ∈ S and ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n  ∈ (0, 1] be such that x ω 1 1 ( k * , q ) η , x ω 2 2 ( k * , q ) η , … , x ω m m ( k * , q ) η and z ɛ 1 1 ( k * , q ) η , z ɛ 2 2 ( k * , q ) η , … , z ɛ n n ( k * , q ) η . Then x¹, x², …, x ^m , y, z¹, z², …, z ⁿ  ∈ B, η (x¹) + ω₁ > k^*, η (x²) + ω₂ > k^*, …, η (x ^m ) + ω _m  > k^* and η (z¹) + ɛ₁ > k^*, η (z²) + ɛ₂ > k^*, …, η (z ⁿ ) + ɛ _n  > k^*. As B is an (m, n)-ideal, x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ  ∈ B. Thus η ( x 1 x 2 ⋯ x m y z 1 z 2 ⋯ z n ) ≥ κ * - κ 2 . If min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } ≤ κ * - κ 2 , then η (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) ≥ min {ω₁, ω₂, …, ω _m , ɛ₁, ɛ₂, …, ɛ _n } , and, so, (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ η. Again, if min {ω₁, ω₂, …, ω _m , ɛ 1 , ɛ 2 , … , ɛ n } > κ * - κ 2 , then η ( x 1 x 2 ⋯ x m yz 1 z 2 ⋯ z n ) + min { ω 1 , ω 2 , … , ω m , ɛ 1 , ɛ 2 , … , ɛ n } > κ * - κ 2 + κ * - κ 2 = κ * - κ . Thus (x¹x² ⋯ x ^m yz¹z² ⋯ z ⁿ ) _{min{ω1,ω2,…,ω m ,ɛ1,ɛ2,…,ɛ n }} ∈ ∨ (κ^*, q _κ ) η. Hence η is a ((k^*, q) , ∈ ∨ (κ^*, q _κ ))-fuzzy (m, n)-ideal of S.