In this paper, we introduce new types of fuzzy (m, n)-ideals in ordered semigroups. In fact, the notion of (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals of the ordered semigroups is introduced. Further, we present the characterzations of this notion in different ways. Then the (κ*, κ)-lower part of the (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals is defined and its associated properties are investigated. After that, (m, n)-regular ordered semigroups are characterized in terms of its (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals and their (κ*, κ)-lower parts.
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 (∈ , ∈ ∨ qk)-fuzzy generalized bi-ideals in [8] of ordered semigroups. By using the concept of (∈ , ∈ ∨ qk)-fuzzy ideals, Shabir et al. [19] characterized regular semigroups. The concept of (∈ , ∈ ∨ (κ*, qκ))-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*, qκ))-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*, qκ))-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*, qκ))-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*, qκ))-fuzzy (m, n)-ideal. In Section 5, directions for some potential work are given. The paper finishes with a list of references.
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 AmSAn ⊆ 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:
Throughout this paper, 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 functionχΩ is defined as:
For any fuzzy subsets η and ξ of S, the η ∩ ξ, η ∪ ξ and η ∘ ξ are defined as:
and
where Ωx = {(r, s) ∈ S × S | x ≤ rs}. Define an order relation ⪯ on the set of all fuzzy subsets of S by
If such that η ⪯ ξ, then , 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 ⇔ ηA ⪯ ηB; ηA ∩ ηB = ηA∩B; ηA ∘ ηB = η(AB].
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) aω of S is defined by
For , aω ∈ η will stand for aω ⊆ η. Thus aω ∈ η ⇔ η (a) ≥ ω.
An OFP aω of S, for any κ* ∈ (0, 1], is said to be (κ*, q)-quasi-coincident with a fuzzy subset η of S, written as aω (κ*, q) η, if η (a) + ω > κ* .
Let 0 ≤ κ < κ* ≤ 1. For any OFP ℏω, we say that
ℏω (κ*, qκ) η if η (ℏ) + ω + κ > κ*;
ℏω ∈ ∨ (κ*, qκ) η if ℏω ∈ η or ℏω (κ*, qκ) η;
if ℏωαη does not hold for α ∈ {(κ*, qκ), ∈ ∨ (κ*, qκ)}.
is called an (∈ , ∈ ∨ (κ*, qκ))-fuzzy subsemigroup of S if ℏω, ℓ ɛ ∈ η ⇒ (ℏ ℓ) min{ω,ɛ} ∈ ∨ (κ*, qκ) η ∀ ω, ɛ ∈ (0, 1] , ℏ , ℓ ∈ S. It is called an (∈ , ∈ ∨ (κ*, qκ))-fuzzy left (resp. right) ideal of S if ℏ≤ ℓ, ℓω ∈ η ⇒ ℏ ω ∈ ∨ (κ*, qκ) η, and ℏ ∈ S, ℓω ∈ η ⇒ (ℏ ℓ) ω ∈ ∨ (κ*, qκ) η (resp. (ℓ ℏ) ω ∈ ∨ (κ*, qκ) η). If it is both an (∈ , ∈ ∨ (κ*, qκ))-fuzzy left and right ideal of S, then it is called an (∈ , ∈ ∨ (κ*, qκ))-fuzzy ideal. An (∈ , ∈ ∨ (κ*, qκ))-fuzzy subsemigroup of S is called an (∈ , ∈ ∨ (κ*, qκ))-fuzzy bi-ideal if ℏ≤ ℓ, ℓω ∈ η ⇒ ℏ ω ∈ ∨ (κ*, qκ) η and ℏω ∈ η, ıɛ ∈ η ⇒ (ℏ ℓ ı) min{ω,ɛ} ∈ ∨ (κ*, qκ) 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*, qκ))-fuzzy subsemigroup of S if and only iffor all x, y ∈ S.
Theorem 2.2. [7] Let η be a fuzzy subset of S. Then η is an (∈ , ∈ ∨ (κ*, qκ))-fuzzy bi-ideal of S if and only if
x ≤ y, and
for all x, y ∈ S, and
for all x, y, z ∈ S.
(∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals
Definition 3.1. A fuzzy subset η of S is called an (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideal of S if
for all ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn ∈ (0, 1] and x1, x2, …, xm, y, z1, z2, …, zn ∈ S
Example 3.2. Define a binary operation ′ · ′ and an order ′ ≤ ′ on the set S = {ℏ , ı , 𝚥 , ℓ} as follows:
·
ℏ
ı
𝚥
ℓ
ℏ
ℏ
ℏ
ℏ
ℏ
ı
ı
ı
ı
ı
𝚥
𝚥
𝚥
𝚥
𝚥
ℓ
ℏ
ℏ
ı
ℏ
Then S is an ordered semigroup. Now define a fuzzy subset η of S as follows:
Take κ* = 0.9 and κ = 0.1. It is easy to verify that η is an (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideal of S for each .
Throughout the paper, the set of all (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals and fuzzy bi-ideals of S will be denoted by and .
Lemma 3.3. Let . Then .
Proof. Let . Then, first two conditions of (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideal hold. Suppose that , for any x1, x2, …, xm, y, z1, z2, …, zn ∈ S and ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn ∈ (0, 1]. By Theorem 2.2, we have η (x1x2 ⋯ xmyz1z2 ⋯ zn)
Since and , we have η (x1) ≥ ω1, η (x2) ≥ ω2, …, η (xm) ≥ ωm and η (z1) ≥ ɛ1, η (z2) ≥ ɛ2, …, η (zn) ≥ ɛn. So η (x1x2 ⋯ xmyz1z2 ⋯ zn) ≥ min {η (x1) , η (x2) , … , η (xm) , η (z1) , η (z2) , … If , then
So, (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} ∈ η. If , then So η (x1x2 ⋯ xmyz1z2 ⋯ zn) + min {ω1, ω2, …, ωm, ɛ1, ɛ2, …, which implies that (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} (κ*, qκ) η. So (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} ∈ ∨ (κ*, qκ) η. Hence . □
Remark 3.4. In general, .
Example 3.5. Let S be an ordered semigroup of Example 3.2. Define a fuzzy subset η of S as follows:
Take κ* = 0.7 and κ = 0.1. Then ∀ m, n ≥ 2, but because
Theorem 3.6.Let . Then ⇔
,
∀ x1, z1 ∈ S, and
η (x1x2 ⋯ xmyz1z2 ⋯ zn) ≥ min {η (x1) , η (x2) , …, η (xm) , η (z1) , η (z2) , …, η (zn) , ∀ x1, x2, …, xm, y, z1, z2, …, zn ∈ S.
Proof. (⇒) Let x1, z1 ∈ S such that x1 ≤ z1. Now on contrary suppose that for some x1, z1 ∈ S. So ∃ ω ∈ (0, 1] such that . Then but , which is a contradiction. Hence . Thus (1) holds.
Next, suppose to the contrary that for some x1, z1 ∈ S. So ∃ ω ∈ (0, 1] such that . Then and , but , which is a contradiction. Thus which proves (2).
Finally, to show that (3) holds, suppose to the contrary that for some x1, x2, …, xm, y, z1, z2, …, zn ∈ S. So ∃ ω ∈ (0, 1] such that η (x1x2 ⋯ xmyz1z2 ⋯ zn) < ω and . Then and , but , which is a contradiction. Hence η (x1x2⋯ xmyz1z2 ⋯ zn) ≥ min {η (x1) , η (x2) , …, η (xm) , η (z1) , η (z2) , … Thus (3) holds.
(⇐) Assume that the conditions (1), (2) and (3) hold. By (1), we have for any x1, z1 ∈ S with x1 ≤ z1. Let (ω ∈ (0, 1]). Then η (z1) ≥ ω. Thus . Now, if , then η (x1) ≥ ω. So . Again, if , then . So , which implies that . Hence .
Next, assume that and (ω1, ɛ1 ∈ (0, 1]). Then η (x1) ≥ ω1 and η (z1) ≥ ɛ1. Now, as , we have . If , then η (x1z1) ≥ min {ω, ɛ} implies that (x1z1) min{ω1,ɛ1} ∈ η . In the other case when , we have . So implies (x1z1) min{ω,ɛ} (κ*, qκ) η. Thus (x1z1) min{ω,ɛ} ∈ ∨ (κ*, qκ) η.
Finally, take any x1, x2, …, xm, y, z1, z2, …, zn ∈ S and ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn in (0, 1] such that , . By (3), η (x1x2 ⋯ xmyz1z2. As and for all ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn ∈ (0, 1], we have η (x1) ≥ω1, η (x2) ≥ ω2, …, η (xm) ≥ ωm and η (z1) ≥ ɛ1, η (z2) ≥ ɛ2, …, η (zn) ≥ ɛn. So η (x1x2 ⋯ xmyz1z2 ⋯ zn) If , then η (x1x2 ⋯ xmyz1z2, ⋯ zn) ≥ min {ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn} implies that
Again, if , then So , which implies that (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} (κ*, qκ) η . Therefore (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} ∈ ∨ (κ*, qκ) η. Hence .□
Definition 3.7.S is said to be (m, n)-regular if ∀ a ∈ S ∃ x ∈ S such that a ≤ amxan.
Lemma 3.8 If S is (m, n)-regular, then .
Proof. Suppose that and x, a, y ∈ S . Since S is (m, n)-regular, there exist p, q ∈ S such that xay ≤ xmpxnaymqyn. Now η (xay) = η (xmpxnaymqyn) = η (xm (pxnaymq) yn)
Hence by Theorem 2.2, .□
Lemma 3.9 Let . Then .
Proof. Take any x, y ∈ S such that x ≤ y and yω ∈ ⋂ i∈Iηi. Then yω ∈ ηi for each i ∈ I. As each , xω ∈ ηi for each i ∈ I. Thus xω ∈ ⋂ i∈Iηi.
Next take any x, y ∈ S and ω, ɛ ∈ (0, 1] such that xω, yɛ ∈ ⋂ i∈Iηi. Then xω, yɛ ∈ ηi for each i ∈ I. So ηi (x) ≥ ω, ηi (y) ≥ ɛ. Thus, we have ⋂i∈Iηi (xy) = ⋀ i∈Iηi (xy) ≥ ⋀ i∈I min {ηi (x) , ηi (y)} ≥ min {ω, ɛ} . So (xy) min{ω,ɛ} ∈ ⋂ i∈Iηi (xy).
Finally, take any and for each x1, x2, ⋯ , xm, z, y1, y2, ⋯ , yn ∈ S and ω1, ω2, ⋯ , ωm, ɛ1, ɛ2, ⋯ , ɛn ∈ (0, 1]. Therefore and for i ∈ I. As each , so (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} ∈ ∨ (κ*, qκ) ηi ∀ i ∈ I. Thus (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} ∈ ηi or ηi (x1x2 ⋯ xmyz1z2 ⋯ zn) + min {ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn} + κ ≥ κ*. If (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2, …, ɛn} ∈ ηi, then
Remark 3.10. Let . Then in general. The following example validates the above claim:
Example 3.11. Let S = {ℏ , ı , 𝚥 , ℓ}. Define a binary operation ′ · ′ and an order ≤ on S as follows:
·
ℏ
ı
𝚥
ℓ
ℏ
ℏ
ℏ
ℏ
ℏ
ı
ℏ
ℏ
ℓ
ℏ
𝚥
ℏ
ℏ
ℏ
ℏ
ℓ
ℏ
ℏ
ℏ
ℏ
Then S is an ordered semigroup. Define by
Then , but because .
Definition 3.12. [9] Let . Then the set
where ω ∈ (0, 1] , is called an (∈ ∨ (κ*, qκ))-level subset of η.
Theorem 3.13.Let η be any strongly convex fuzzy subset of S. Then ⇔ the (∈ ∨ (κ*, qκ))-level subset , ∀ ω ∈ (0, 1].
Proof. (⇒) Take any x ∈ S and y ∈ [η] ω such that x ≤ y. As y ∈ [η] ω, we have yω ∈ ∨ (κ*, qκ) 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 xω ∈ ∨ (κ*, qκ) η. So x ∈ [η] ω.
Next, take any x1, x2, …, xm, z1, z2, …, zn ∈ [η] ω for ω ∈ (0, 1]. Then and ; that is η (xi) ≥ ω or η (xi) + ω + κ > κ* and η (zj) ≥ ω or η (zj) + ω + κ > κ* for each i ∈ {1, 2, …, m} and j ∈ {1, 2, …, n}. Since , by Theorem 3.6, .
Case (1). Let η (xı) ≥ ω, η (z𝚥) ≥ ω for each ı ∈ {1, 2, . . , m} , 𝚥 ∈ {1, 2, …, n}. If , then η (x1x2 ⋯ xmyz1z2 ⋯ zn) and, thus, (x1x2 ⋯ xmyz1z2 ⋯ zn) ω (κ*, qκ) η. If , then η (xy) ≥ min {η (x1) , η (x2) , …, η (xm) , η (z1) , η (z2) , …, = ω and, so, (x1x2 ⋯ xmyz1z2 ⋯ zn) ω ∈ η. Hence (x1x2 ⋯ xmyz1z2 ⋯ zn) ω ∈ ∨ (κ*, qκ) η.
Case (2). Let η (xı) ≥ ω for each ı ∈ {1, 2, . . , m}, η (z𝚥) ≥ ω for each 𝚥 ∈ {1, 2, . . , n - 1}, and η (zn) + ω + κ > κ*. If , then
η (x1x2 ⋯ xmyz1z2 ⋯ zn) ≥ min {η (x1) , η (x2) , …, η (xm) , η (z1) , η (z2) , = κ* - ω - κ ; i.e., η (x1x2 ⋯ xmyz1z2 ⋯ zn) + ω + κ > κ*, and, so, (x1x2 ⋯ xmyz1z2 ⋯ zn) ω (κ*, qκ) η. Again, if , then
η (x1x2 ⋯ xmyz1z2 ⋯ zn) ≥ min {η (x1) , η (x2) , …, η (xm) , η (z1) , η (z2) , …, and, thus, (x1x2 ⋯ xmyz1z2 ⋯ zn) ω ∈ η. Hence (x1x2 ⋯ xmyz1z2 ⋯ zn) ω ∈ ∨ (κ*, qκ) η.
Case (3). Let η (xı) ≥ ω for each ı ∈ {1, 2, …, m}, η (z𝚥) ≥ ω for each 𝚥 ∈ {1, 2, …, n - 2}, and η (zκ) + 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 ⋯ xmyz1z2 ⋯ zn) ω ∈ ∨ (κ*, qκ) η.
Thus in each case, we have (x1x2 ⋯ xmyz1z2 ⋯ zn) ω ∈ ∨ (κ*, qκ) η, and so x1x2 ⋯ xmyz1z2 ⋯ zn ∈ [η] ω. Hence . (⇐) Let x1, z1 ∈ S such that x1 ≤ z1. Suppose to the contrary that . Then , for some , implies that z1 ∈ [η] ω, but x1 ∉ [η] ω, a contradiction. Thus .
Next, we assume that if, for some x1, y1 ∈ S. Then such that . 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 for all x1, z1 ∈ S. Using the similar argument, we get for all x1, x2, …, xm, y, z1, z2, …, zn ∈ S. Hence by Theorem 3.6, .□
Theorem 3.14.Let and let η be defined byThen
η is an ((k*, q) , ∈ ∨ (κ*, qκ))-fuzzy (m,n)-ideal of S; and
η is an (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m,n)-ideal of S.
Proof. (1). Let x1, z1 ∈ S with x1 ≤ z1 and let ω ∈ (0, 1] be such that . Then z1 ∈ B, η (z1) + ω > k*. Since and x1 ≤ z1 ∈ B, we have x1 ∈ B. Thus . If , then η (x1) ≥ ω. So . If , then . Therefore .
Next, we assume that x1, z1 ∈ S and ω1, ɛ1 ∈ (0, 1] be such that and . Then x1, z1 ∈ B, η (x1) + ω1 > k* and η (z1) + ɛ1 > k*. As , we have x1z1 ∈ B. Thus . If , then η (x1z1) ≥ min {ω1, ɛ1} and, so, (x1z1) min{ω1,ɛ1} ∈ η. Again, if , then
So (x1z1) min{ω1,ɛ1} (κ*, qκ) η. Therefore (x1z1) min{ω1,ɛ1} ∈ ∨ (κ*, qκ) η .
Finally, assume that x1, x2, …, xm, y, z1, z2, …, zn ∈ S and ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn ∈ (0, 1] be such that and . Then x1, x2, …, xm, y, z1, z2, …, zn ∈ B, η (x1) + ω1 > k*, η (x2) + ω2 > k*, …, η (xm) + ωm > k* and η (z1) + ɛ1 > k*, η (z2) + ɛ2 > k*, …, η (zn) + ɛn > k*. As B is an (m, n)-ideal, x1x2 ⋯ xmyz1z2 ⋯ zn ∈ B. Thus . If , then η (x1x2 ⋯ xmyz1z2 ⋯ zn) ≥ min {ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn} , and, so, (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} ∈ η. Again, if min {ω1, ω2, …, ωm, , then Thus (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} ∈ ∨ (κ*, qκ) η. Hence η is a ((k*, q) , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideal of S.
(2). Let x1, z1 ∈ S with x1 ≤ z1 and let u ∈ (0, 1] be such that . Then z1 ∈ B and η (z1) ≥ ω. As and x1 ≤ z1 ∈ B, we have x1 ∈ B. Thus . If , then η (x1) ≥ ω. So . If , then . So . Therefore .
Next, assume that x1, z1 ∈ S and ω1, ɛ1 ∈ (0, 1] be such that and . Then x1, z1 ∈ B, η (x1) > ω1 and η (z1) > ɛ1. Since , x1z1 ∈ B. Thus . If , then η (x1z1) ≥ min {ω1, ɛ1} and, so, (x1z1) min{ω1,ɛ1} ∈ η. Again, if , then So (x1z1) min{ω1,ɛ1} (κ*, qκ) η. Hence (x1z1) min{ω1,ɛ1} ∈ ∨ (κ*, qκ) η.
Finally, assume that x1, x2, …, xm, y, z1, z2, …, zn ∈ S and ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn ∈ (0, 1] be such that and . Then x1, x2, …, xm, z1, z2, …, zn ∈ B, η (x1) > ω1, η (x2) > ω2, …, η (xm) > ωm and η (z1) > ɛ1, η (z2) > ɛ2, …, η (zn) > ɛn. As , x1x2 ⋯ xmyz1z2 ⋯ zn ∈ B. Thus . If , then η (x1x2 ⋯ xmyz1z2 ⋯ zn) ≥ min {ω1, ω2, …, ωm, ɛ1, ɛ2, …, ɛn} . Therefore (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1, ɛ2, …, ɛn} ∈ η . Again, if , then Therefore (x1x2 ⋯ xmyz1z2 ⋯ zn) min{ω1,ω2,…,ωm,ɛ1,ɛ2,…,ɛn} ∈ ∨ (κ*, qκ) η, as required. □
Theorem 3.15.Let (∅ ≠) B ⊆ S. Then ⇔ , where
.
Proof. (⇒) Let . Then, by Theorem 3.14, .
(⇐) Let x, y ∈ S with x ≤ y. If y ∈ B, then . Since and x ≤ y, we have . Thus , and, so, x ∈ B.
Next, we show that B is subsemigroup of S. To show this, take any x, y ∈ B. Then and . Now Therefore and, so, xy ∈ B. Thus B is a subsemigroup of S.
Finally, take any x1, x2, …, xm, z1, z2, …, zn ∈ B and y ∈ S. Then . Now
Theorem 3.16.Let . Then ⇔ the level subset U (η ; ω) of η which is defined asfor is an (m,n)-ideal of S.
Proof. (⇒) Suppose that x, y ∈ S such that x ≤ y ∈ U (η ; ω), where . Then η (y) ≥ ω. By Theorem 3.6, . Therefore x ∈ U (η ; ω).
Next, take any x, y ∈ U (η ; ω). Then η (x) ≥ ω and η (y) ≥ ω. So, by Theorem 3.6, Therefore xy ∈ U (η ; ω). Thus U (η ; ω) is a subsemigroup of S.
Finally, take any x1, x2, …, xm, z1, z2, …, zn ∈ U (η ; ω) and y ∈ S. Then min {η (x1) , η (x2) , …, η (xm) , η (z1) , η (z2) , …, η (zn)} ≥ ω. Now, by Theorem 3.6, we have
η (x1x2 ⋯ xmyz1z2 ⋯ zn)
Thus x1x2 ⋯ xmyz1z2 ⋯ zn ∈ U (η ; ω). Hence .
(⇐) Let x1, z1 ∈ S be such that x1 ≤ z1. Suppose to the contrary that . Then for some . This implies that z1 is in U (η ; ω), but x1 ∉ U (η ; ω), a contradiction. Thus .
Suppose next to the contrary that for some x1, z1 ∈ S. Then there exist such that . This implies that x1 ∈ U (η ; ω) and z1 ∈ U (η ; ω), but x1z1 ∉ U (η ; ω), a contradiction. Therefore for all x1, z1 ∈ S.
Finally, assume to the contrary that for some x1, x2, … xm, y, z1, z2, …, zn ∈ S. Then such that . Thus x1, x2, …, xm ∈ U (η ; ω) and z1, z2, …, zn ∈ U (η ; ω), but x1x2 ⋯ xmyz1z2 ⋯ zn ∉ U (η ; ω), which is a contradiction. Therefore for all x1, x2, …, xm, y, z1, z2, …, zn ∈ S. Hence, by Theorem 3.6, . □
Example 3.17. Define a binary operation ′ · ′ and an order ′ ≤ ′ on the set S = {0, ℏ , ℓ , 𝚥} as follows:
·
0
ℏ
ℓ
𝚥
0
0
0
0
0
ℏ
0
ℏ
ℏ
ℏ
ℓ
0
ℏ
ℏ
ℏ
𝚥
0
ℏ
ℏ
ℏ
≤ : = {(0, 0) , (ℏ , ℏ) , (ℓ , ℓ) , (0, ℏ) , (ℏ , ℓ) , (ℓ , 𝚥)} Then (S, · , ≤) is an ordered semigroup. Now define a fuzzy set η of S as: η (0) =0.5, η (ℏ) =0.4, η (ℓ) =0.1 and η (𝚥) =0.3. Therefore
Since for all , where κ* = 1 and κ = 0, U (η ; ω) is bi-ideal of S. Therefore by Theorem 3.16, η is (∈ , ∈ ∨ (κ*, qκ))-fuzzy bi-ideal of S.
(κ*, κ)-lower parts of (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals
Definition 4.1. The (κ*, κ)-lower part of fuzzy subset η is defined [7] as follows:
∀ x ∈ S and 0 ≤ κ < κ* ≤ 1. For any (∅ ≠) A ⊆ S and , the (κ*, κ)-lower part of the characteristic function ηA, will be denoted by .
Definition 4.2. [7] Let . Define , and as follows:
∀ x ∈ S and 0 ≤ k < k* ≤ 1.
Lemma 4.3. Let l ∈ Z+ and x ∈ S . Then
for any .
Proof. Now
⋮ □
Lemma 4.4. Let (∅ ≠) B ⊆ S. Then ⇔ .
Proof. Follows from Theorem 3.15. □
Lemma 4.5. Let . Then
η is an (∈ , ∈ ∨ (κ*, qκ))-fuzzy subsemigroup of S ⇔ .
η is an (∈ , ∈ ∨ (κ*, qκ))-fuzzy subsemigroup of S ⇔ .
Proof. (1). (⇒) Let a ∈ S. If Aa =∅, then . Suppose that (u, v) ∈ Aa. Then, we have
(⇐) Assume that . Now, for any x, y ∈ S, we have
Hence η is an (∈ , ∈ ∨ (κ*, qκ))-fuzzy subsemigroup of S.
(2). (⇒) By (1). we have . Therefore Thus
(⇐) For any x, y ∈ S, we have η (xy)
Hence η is an (∈ , ∈ ∨ (κ*, qκ))-fuzzy subsemigroup of S. □
Theorem 4.6.Let . Then ⇔
If x ≤ y, then ;
; and
.
Proof. (⇒) Let . Then condition (1) follows by Lemma 3.6. If Aa =∅, then . Suppose that (u, v) ∈ Aa. Then, we have
≤η (a) . Thus (2) holds.
Let a ∈ S. If , then (3) is obvious. So assume that ηn) (a) ≠0. Then ∃ (x, y) ∈ Aa such that (ηm ∘ 1) (x) ≠0 and ηn (y) ≠0. Now the following cases arise:
Case 1. When (ηm ∘ 1) (x) ≠0. In this case ∃ (u1, v1) ∈ Ax such that ηm (u1) ≠0, ⇒ ∃ (u2, v2) ∈ Au1 such that ηm-1 (u2) ≠0 and η (v2) ≠0. ⇒ ∃ (u3, v3) ∈ Au1 such that ηm-2 (u3) ≠0 and η (v3) ≠0. ⋮ ⇒ ∃ (um-1, vm-1) ∈ Aum-2 such that η2 (um-1) = ηm-m-2 (um-1) ≠0 and η (vm-1) ≠0. ⇒ ∃ (um, vm) ∈ Aum-1 such that η (um) ≠0 and η (vm) ≠0.
Case 2. When ηn (y) ≠0. Then such that and . ⇒ such that and . ⇒ such that and . ⋮ ⇒ such that and . ⇒ such that and . So ⋮
(⇐) Assume that (1) , (2) and (3) hold in S. By condition (2), it follows from Lemma 4.5 that η is an (∈ , ∈ ∨ (κ*, qκ))-fuzzy subsemigroup of S. Therefore, by Theorem 2.1, .
Next, take any x1, x2, …, xm, z, y1, y2, …, yn ∈ S and let a = x1x2 ⋯ xmzy1y2 ⋯ yn. Now
η (x1x2 ⋯ xmzy1y2 ⋯ yn) = η (a) ⋮ Hence . □
Corollary 4.7.Let . Then ⇔
If x ≤ y, then ;
; and
.
Theorem 4.8.The following assertions hold in S:
S is (m, n)-regular ⇔ .
S is (m, n)-regular ⇔ .
Proof. (⇒) Let a ∈ S . Then a ≤ amxan, it follows that (amx, an) ∈ Aa. Now, we have
Therefore .
(⇐) Take any a ∈ S. Since is fuzzy subset of S, by hypothesis, implies Therefore ∃ (x, y) ∈ Aa such that and . We have following cases:
Case 1. When . Then ∃ (u1, v1) ∈ Ax such that . ⇒ ∃ (u2, v2) ∈ Au1 such that and implies v2 = a . ⇒ ∃ (u3, v3) ∈ Au2 such that and implies v3 = a . ⋮ ⇒ ∃ (um-1, vm-1) ∈ Aum-2 such that and implies vm-1 = a . ⇒ ∃ (um, vm) ∈ Aum-1 such that and implies um = a and vm = a .
Case 2. When . Then such that and implies . ⇒ such that and implies . ⇒ such that and implies . ⋮ ⇒ such that and implies . ⇒ such that and implies and . Therefore by above two cases we have , as required. □
Theorem 4.9The following assertions hold in S:
S is (m, n)-regular ⇔ .
S is (m, n)-regular ⇔ .
Proof. (1) . (⇒) Let . Then by Theorems 4.6 and 4.8, and . Hence .
(⇐) Let a ∈ S. Then . By hypothesis, we have implies . Therefore ∃ (x, y) ∈ Aa such that and (χ[a](m,n)) n (y) ≠0. Again, ∃ (u, v) ∈ Ax such that (χ[a](m,n)) m (u) ≠0. Now we have (χ[a](m,n)) n (y) ≠0 and (χ[a](m,n)) m (u) ≠0, it follows that y ∈ (([a] (m,n)) n] and u ∈ (([a] (m,n)) m]. Since (x, y) ∈ Aa and (u, v) ∈ Ax, therefore a ≤ xy ≤ uvy ∈ (([a] (m,n)) m] S (([a] (m,n)) n] ⊆ (([a] (m,n)]) mS (([a] (m,n)]) n ⊆ (amSan]. Thus [a] (m,n) ⊆ (amSan]. Since (amSan] ⊆ [a] (m,n). Therefore [a] (m,n) = (amSan], and hence by Theorem [1, Theorem 2.4] S is (m, n)-regular. □
Lemma 4.10. Let with m ≥ 2 or n ≥ 2. Then S is (m, n)-regular ⇔ B = (B2] .
Proof. (⇒) Let . Then B = (BmSBn]. Now, we have B = (BmSBn] = (BmS ((BmSBn]) n] ⊆ (BmSBnBmSBn] = (BB] .
Also, (BB] ⊆ B. Hence B = (B2].
(⇐) Take any x ∈ S . Then, as , we have [x] m,n = ([x] m,n [x] m,n] = ([x] m,n [x] m,n [x] m,n] = … = (([x] m,n) m+n+1] ⊆ (xmSxn] .
Since x ∈ [x] m,n, we have x ∈ (xmSxn]. Hence S is (m, n)-regular. □
The conditions m ≥ 2 or n ≥ 2 in Lemma 4.10 is necessary, we show it by the following example:
Example 4.11. Let S = {ℏ , ı , 𝚥 , ℓ , ϑ}. Define operation ′ · ′ and ordered ≤ on S as follows:
·
ℏ
ı
𝚥
ℓ
ϑ
ℏ
ℏ
ℏ
ℏ
ℏ
ℏ
ı
ℏ
ı
ℏ
ℓ
ℏ
𝚥
ℏ
ϑ
𝚥
𝚥
ϑ
ℓ
ℏ
ı
ℓ
ℓ
ı
ϑ
ℏ
ϑ
ℏ
𝚥
ℏ
≤ : = {(ℏ , ℏ) , (ı , ı) , (𝚥 , 𝚥) , (ℓ , ℓ) , (ϑ, ϑ) , (ℏ , ı) , (ℏ , 𝚥) , (ℏ , ℓ) , (ℏ , ϑ)} . Then S is regular ordered semigroup. It is a routine to verify that B = {ℏ , ϑ} is a bi-ideal of S, but B ≠ (B2].
Theorem 4.12.Let with m ≥ 2 or n ≥ 2. Then
S is (m, n)-regular ⇔ .
S is (m, n)-regular ⇔ .
Proof. (1). (⇒) Let . Then by Theorem 4.9(2), , which implies
Thus . Also, because . Hence .
(⇐) Let and b ∈ B. Then by hypothesis Since , implies b ∈ (B2]. Therefore B ⊆ (B2]. Also, (BB] ⊆ B. Therefore B = (B2], and hence by Lemma 4.10, S is (m, n)-regular. □
Theorem 4.13.Let with m ≥ 2 or n ≥ 2. Then S is (m, n)-regular ⇔ .
Proof. (⇒) Let . Then . Since , by Theorem 4.12, . Similarly . Therefore, Now
and similarly . Also for any x, y ∈ S such that x ≤ y implies . Hence .
Similarly . Therefore . Again, by Theorem 4,
Similarly Therefore Hence .
Converse is obvious. □
Lemma 4.14. Let with m ≥ 2 or n ≥ 2. Then S is (m, n)-regular ⇔ A ∩ B = (AB] ∩ (BA] .
Proof. (⇒) Let . Then . So, by Lemma 4.10, A ∩ B = ((A ∩ B) 2] = ((A ∩ B) (A ∩ B)] ⊆ (AB]. Similarly A ∩ B ⊆ (BA]. Therefore A ∩ B ⊆ (AB] ∩ (BA] . Now ((AB]) mS ((AB]) n = ((AB) (ABAB ⋯ ABSABAB ⋯ A) B] ⊆ (ABSB] = (ABmSBn] ⊆ (AB] . Thus ((AB]) mS ((AB]) n ⊆ (AB], and similarly (AB] (AB] ⊆ (AB]. Therefore . Similarly . So . Again, by Lemma 4.10, (AB] ∩ (BA] = (((AB] ∩ (BA]) 2] ⊆ ((AB] (BA] ⊆ ((ASA] ⊆ A. Similarly ((AB] (BA] ⊆ B . Therefore (AB] ∩ (BA] ⊆ A ∩ B . Hence A ∩ B = (AB] ∩ (BA] .
Converse is obvious. □
Lemma 4.15. Let such that m ≥ 2 or n ≥ 2. Then for any such that (AB] ∩ (BA] ⊆ C implies A ⊆ C or B ⊆ C ⇔ S is (m, n)-regular and the set of all (m, n)-ideals of S forms a chain.
Proof. (⇒) Take x ∈ S. Since of S and (([x] m,n) 2] ⊆ (([x] m,n) 2], by hypothesis, [x] m,n ⊆ (([x] m,n) 2]. Now [x] m,n ⊆ (([x] m,n) 2] ⊆ (([x] m,n) 3] ⊆ … ⊆ ((([x] m,n) m] H (([x] m,n) n]] ⊆ (xmSxn].
Since x ∈ [x] m,n, x ∈ (xmSxn] . Hence S is (m, n)-regular. Next, we show that a chain. Let . By Lemma 4.14, A ∩ B = (AB] ∩ (BA]. Since , by hypothesis, A ⊆ A ∩ B or B ⊆ A ∩ B. If A ⊆ A ∩ B, then A ⊆ B. In the other case if B ⊆ A ∩ B, then B ⊆ A, as required.
(⇐) Assume that such that (AB] ∩ (BA] ⊆ C. As S is (m, n)-regular, by Lemma 4.14, A ∩ B = (AB] ∩ (BA] ⊆ C. Now, by hypothesis, either A ⊆ B or B ⊆ A. Therefore either A ∩ B = A or A ∩ B = B. Hence, either A ⊆ C or B ⊆ C. □
Theorem 4.16.Let either m ≥ 2 or n ≥ 2. Then for any such that implies or ⇔ S is (m, n)-regular and the set forms a chain.
Proof. (⇒) Let such that (AB] ∩ (BA] ⊆ C. By Lemma 4.4, . Let x ∈ (AB] ∩ (BA]. Then x ∈ (AB] , x ∈ (BA]. Now Since x ∈ (AB] ∩ (BA] and (AB] ∩ (BA] ⊆ C, .
(⇐) Assume that for any such that . As S is (m, n)-regular, by Theorem 4.13,
Now, by hypothesis, either or . Therefore either or . Hence either or . □
Conclusion
This paper’s primary objective is to introduce the notion of (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideal in the ordered semigroups, and to improve the concept of (m, n)-regular ordered semigroups by considering the (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals. In this article, the special cases are as follows:
(1). By taking m = 1 = n, all results of [7] will be special cases of this article results.
(2). If we take m = 1 = n, κ* = 1, then most of the results from [8] are obtained as direct corollaries of the present paper results.
(3). If we put m = 1 = n, κ* = 1 and κ = 0, then most of the results of the paper [2] are deduced as direct corollaries of the results of this paper.
Conflict of interest
The authors have no conflict of interest.
Footnotes
Acknowledgments
The authors are grateful to the anonymous referees for a careful checking of the details and for helpful comments that improved this paper. This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.
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