# (m, n)-Ideals in Semigroups Based on Int-Soft Sets

Authors: G. Muhiuddin; Abdulaziz M. Alanazi
Journal: Journal of Mathematics
 (2021)

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## Abstract

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, and topological spaces. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. In this paper, we introduce the notions of int-softm,n-ideals, int-soft m,0-ideals, and int-soft 0,n-ideals of semigroups by generalizing the concept of int-soft bi-ideals, int-soft right ideals, and int-soft left ideals in semigroups. In addition, some of the properties of int-soft m,n-ideal, int-soft m,0-ideal, and int-soft 0,n-ideal are studied. Also, characterizations of various types of semigroups such as m,n-regular semigroups, m,0-regular semigroups, and 0,n-regular semigroups in terms of their int-soft m,n-ideals, int-soft m,0-ideals, and int-soft 0,n-ideals are provided.


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## Body

## 1. Introduction

Soft set theory of Molodtsov [1] is an important mathematical tool to dealing with uncertainties and fuzzy or vague objects and has huge applications in real-life situations. In soft sets, the problems of uncertainties deal with enough numbers of parameters which make it more accurate than other mathematical tools. Thus, the soft sets are better than the other mathematical tools to describe the uncertainties. Aktaş and Çaǧman [2] show that the soft sets are more accurate tools to deal the uncertainties by comparing the soft sets to rough and fuzzy sets. The decision-making problem in soft sets had been considered by Maji et al. [3]. In [4], Maji et al. investigated several operations on soft sets. The notions of soft sets introduced in different algebraic structures had been applied and studied by several authors, for example, Aktaş and Çaǧman [2] for soft groups, Feng et al. [5] for soft semirings, and Naz and Shabir [6, 7] for soft semi-hypergroups.Song [8] introduced the notions of int-soft semigroups, int-soft left (resp. right) ideals, and int-soft quasi-ideals. Afterthat, Dudek and Jun [9] studied the properties of int-soft left (resp. right) ideals, and characterizations of these int-soft ideal are obtained. Moreover, they introduced the concept of int-soft (generalized) bi-ideals, and characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals are obtained. Dudek and Jun [9] introduced and characterized the notion of soft interior ideals of semigroups. The concept of union-soft semigroups, union-soft l-ideals, union-soft r-ideals, and union-soft semiprime soft sets have been considered by [10]. In addition, Muhiuddin et al. studied the soft set theory on various aspects (see, for example, [11–21]). For more related concepts, the readers are referred to [22–31].The results of this paper are arranged as follows. Section2 summarises some concepts and properties related to semigroups, soft sets, and int-soft ideals that are required to establish our key results, while Section 3 presents the principle of int-soft m,n-ideals. We prove that the int-soft bi-ideals are int-soft m,n-ideals for each positive integer m,n, but the converse is not necessarily valid. Then, we prove that the A subset of the S semigroup is m,n-ideal of S if and only if χA^,S over U is an int-soft m,n-ideal over U. Also, we prove that a soft set K^,S over U is an int-soft m,n-ideal over U if and only if K^m°χS^°K^n,S⊆K^,S. Moreover, we characterize m,n regular semigroups in terms of int-soft m,n-ideals over U. In this respect, we prove that a semigroup S is m,n-regular if and only if K^,S=K^m°χS^°K^n,S for each int-soft m,n-ideal K^,S over U. In Section 4, first, we present the idea of int-soft m,0-ideal and 0,n-ideal over U. After that, we obtain some analogues’ results to the previous section. Furthermore, we prove that a semigroup S is m,n-regular if and only if K^∩G^,S=K^m°G^∩K^°G^n,S for each int-soft m,0-ideal K^,S and for each int-soft 0,n-ideal G^,S over U. At the end of this section, we provide the existence theorem for int-soft m,n-ideal over U and for the minimality of int-soft m,n-ideal over U. We also provide a conclusion in Section 5 that contains the direction for certain potential work.


## 2. Preliminaries

LetS be a semigroup. For ∅≠Ω,℧⊆S, Ω℧ is defined as Ω℧=υℏ|υ∈Ω,ℏ∈℧. A subset ∅≠Ω of S is called a sub-semigroup of S if υℏ∈Ω∀υ,ℏ∈Ω. A subset ∅≠Ω of S is called a left (resp. right) ideal of S if SΩ⊆Ωresp.ΩS⊆Ω and is called an ideal of S if Ω is both left and right ideal of S. A sub-semigroup ℧ of S is called a bi-ideal of S if ℧S℧⊆℧.LetU be a universal set and let E be a set of parameters. Let PU denote the power set of U and let Ω⊆E. A pair K^,Ω is called a soft set (over U) [32] if F:Ω⟶PU is a mapping. We denote the set of all soft sets over U with parameter set S by SSU.LetK^,Ω and G^,℧ be soft sets over U. Then, G,℧ is called a soft subset of K^,Ω if ℧⊆Ω and G^υ⊆K^υ, ∀υ∈℧.LetK^,Ω and G^,Ω be two soft sets. Then, for each υ∈Ω, the union and intersection are defined as(1)K^⋓G^υ=K^υ∪G^υ,K^⋒G^υ=K^υ∩G^υ.For any two soft setsK^,Ω and G^,Ω of S, the int-soft product K^°G^ is defined as(2)K^°G^υ=∪υ=ℏκK^ℏ∩G^κ,if there existℏ,κ∈Ssuch thatυ=ℏκ,∅,otherwise.A soft setK^,S over U is called an int-soft right (resp. Left) ideal over U if K^υκ⊇K^υresp.K^υκ⊇K^κ for all υ,κ∈S. It is called an int-soft ideal over U if it is both int-soft left and int-soft right ideal over U. An int-soft sub-semigroup K^,S over U is called an int-soft bi-ideal over U if K^υκℏ⊇K^υ∩K^ℏ for all υ,κ,ℏ∈S. The set of all int-soft left (resp. Right) ideals and int-soft bi-ideals over U will be denoted by ℐLU (resp. ℐRU) and ℐBU.More concepts related to our study in different aspects have been studied in [33–39].For∅≠Ω⊆S, the characteristic soft set over U is denoted by χΩ^,S and defined as(3)χΩ^υ=U,ifυ∈Ω,∅,ifυ∉Ω.Let∅≠Ω,℧⊆S. Then, we have (1) χΩ°χ℧=χΩ℧ and (2) χΩ∩χ℧=χΩ∩℧.The concept ofm,n-ideals of semigroups was introduced by Lajos [40] as follows. Let S be a semigroup and m,n be nonnegative integers. Then, a sub-semigroup Ω of S is said to be an m,n-ideal of S if ΩmSΩn⊆Ω. After that, the concept of m,n-ideals in various algebraic structures such as ordered semigroups, LA-semigroups, and fuzzy semigroups had been studied by, for instance, Akram et al. [41], Bussaban and Changphas [42], Changphas [43], Mahboob et al. [44], and many others.We denote byυm,n the principal m,n-ideal, υm,0 the principal m,0-ideal, and υ0,n the principal 0,n-ideal generated by an element υ of S, respectively. They were given by Krgovic [45] as follows:(4)υm,n=∪i=1m+nυi∪υmSυn,υm,0=∪i=1mυi∪υmS,υ0,n=∪i=1nυi∪Sυn.In whatever follows,ℳm,n, ℳm,0, and ℳ0,n denote the set of all m,n-ideals, m,0-ideals, and 0,n-ideals of S.


## 3. Int-Softm,n-Ideals

Definition 1.
An int-soft sub-semigroupK^,S over U is called an int-soft m,n-ideal over U if(5)K^ℏ1ℏ2,…,ℏmκυ1υ2,…,υm⊇K^ℏ1∩K^ℏ2∩⋯K^ℏm∩K^υ1∩K^υ2∩⋯∩K^υn,for all ℏ1,ℏ2,…,ℏn,κ,υ1,υ2,…,υm∈S.
The set of all int-softm,n-ideals over U will be denoted by ℐm,nU.Example 1.
LetS=0,υ,ℏ. Define the binary operation ′⋅′ on S as follows.
Then,S,⋅ is a semigroup. Define K^,S∈SSU as(6)K^κ=U1,ifκ∈0,υ,U2,ifκ=ℏ,where U1,U2⊆U such that U2⊆U1. It is straightforward to verify that K^,S∈ℐm,nU.Lemma 1.
InS, K^,S∈ℐBU⇒K^,S∈ℐm,nU.Proof (straightforward).Remark 1.
In general, in a semigroupS, K^,S∈ℐm,nU⇏K^,S∈ℐBU.Example 2.
LetS=0,υ,ℏ,κ. Define the binary operation ′⋅′ on S as follows.
Then,S is a semigroup. Define K^,S∈SSU as(7)K^ω=U,ifω∈0,κ,∅,ifω∈υ,ℏ.
Then,K^,S∈ℐm,nU, ∀m,n≥2, but K^∉ℐBU because ∅=K^υ=K^κℏ0⊉K^κ∩K^0=U.Theorem 1.
LetK^,S,ℱ^,S∈ℐm,nU. Then, K^∩˜ℱ^,S∈ℐm,nU.Proof.
Letυ,ℏ∈S. We have(8)K^∩˜ℱ^υℏ=K^υℏ∩ℱ^υℏ,=K^υ∩K^ℏ∩ℱ^υ∩ℱ^ℏ=K^∩˜ℱ^υ∩K^∩˜ℱ^ℏ.
Letυ1,υ2,…,υm,κ,ℏ1,ℏ2,…,ℏn∈S. Now, we have(9)K^∩˜ℱ^υ1υ2,…,υmκℏ1ℏ2,…,ℏn⊇K^υ1υ2,…,υmκℏ1ℏ2,…,ℏn∩ℱ^υ1υ2,…,υmκℏ1ℏ2,…,ℏn,⊇K^υ1∩K^υ2∩⋯∩K^υm∩K^ℏ1∩K^ℏ2∩⋯∩K^ℏn∩ℱ^υ1∩ℱ^υ2∩⋯∩ℱ^υm∩ℱ^ℏ1∩ℱ^ℏ2∩⋯∩ℱ^ℏn⊇K^∩˜ℱ^υ1∩K^∩˜ℱ^υ2∩⋯∩K^∩˜ℱ^υm∩K^∩˜ℱ^ℏ1∩K^∩˜ℱ^ℏ2∩⋯∩K^∩˜ℱ^ℏn.
Therefore,K^∩˜ℱ^,S∈ℐm,nU.Theorem 2.
Let∅≠Ω⊆S. Then, ∅≠Ω∈ℳm,n⇔χΩ^,S∈ℐm,nU.Proof.
⇒ Let υ1,υ2,…,υm,κ,ℏ1,ℏ2,…,ℏn∈S. Below are the cases we have:Case 1. Ifxk∉Ω for some k∈1,2,…,m, then(10)χΩ^υ1υ2,…,υmκℏ1ℏ2,…,ℏn⊇χΩ^υ1∩χΩ^υ2∩⋯∩χΩ^υm∩χΩ^ℏ1∩χΩ^ℏ2∩⋯∩χΩ^ℏn.Case 2. Ifyl∉Ω for some l∈1,2,…,n, then(11)χΩ^υ1υ2,…,υmκℏ1ℏ2,…,ℏn⊇χΩ^υ1∩χΩ^υ2∩⋯∩χΩ^υm∩χΩ^ℏ1∩χΩ^ℏ2∩⋯∩χΩ^ℏn.Whenxk∉Ω and yl∉Ω for k∈1,2,…,m and l∈1,2,…,n are used in previous cases.Case 3. Ifxk,yl∈Ω, ∀k∈1,2,…,m and l∈1,2,…,n, then υ1υ2,…,υmzℏ1ℏ2,…,ℏn∈ΩmSΩn⊆Ω. Therefore,(12)χΩ^υ1υ2,…,υmcℏ1ℏ2,…,ℏn=1,⊇χΩ^υ1∩χΩ^υ2∩⋯∩χΩ^υm∩χΩ^ℏ1∩χΩ^ℏ2∩⋯∩χΩ^ℏn.
Hence,χΩ^,S∈ℐm,nU.
⇐ Let υ1,υ2,…,υm,ℏ1,ℏ2,…,ℏn∈Ω and κ∈S. Then, χΩ^υ1υ2,…,υmcℏ1ℏ2,…,ℏn⊇χΩ^υ1∩χΩ^υ2∩⋯∩χΩ^υm∩χΩ^ℏ1∩χΩ^ℏ2∩⋯∩χΩ^ℏn=1 implies χΩ^υ1υ2,…,υmcℏ1ℏ2,…,ℏn=1. Therefore, υ1υ2,…,υmκℏ1ℏ2,…,ℏn∈Ω. Thus, ΩmSΩn⊆Ω, as required.Theorem 3.
LetK^,S∈SSU. Then, K^,S∈ℐm,nU⇔K^m°χS^°K^n,S⊆K^,S.Proof.
⇒ Let a∈S. If K^m°χS^°K^na=∅, then K^m°χS^°K^n,S⊆K^,S. In the other case, when fm°S°fna≠∅, then there exist elements r,s∈S such that a=rs, K^m°χS^r≠∅ and K^ns≠∅. As K^m°χS^s≠∅, there exist u1,v1∈S such that x=u1v1, K^mu1≠∅ and χS^v1=U. It is easy to show that there exist u2,v2,…,um,vm∈S such that, for any l∈2,…,m, we have ul−1=ulvl, K^ul≠0 and K^m−l+1vl≠∅. As K^ny≠∅, there exist u1′,v1′∈S such that y=u1′v1′, K^u1′≠∅ and K^n−1v1′≠∅. Similarly, there exist u2′,v2′,…,un−1′, vn−1′∈S such that, for l∈2,…,n−1, we have ul−1=ul′vl′,K^ul′≠∅ and K^n−lvl′≠∅. Now, we have(13)K^m°χS^°K^na=∪a=rsK^m°χS^r∩K^ns,=∪a=rs∪r=u1v1K^mu1∩χS^v1∩∪s=u1′v1′K^u1′∩K^n−1v1′=∪a=rs∪r=u1v1∪s=u1′v1′K^mu1∩Sv1∩K^u1′∩K^n−1v1′=∪a=rs∪r=u1v1∪s=u1′v1′K^mu1∩K^u1′∩K^n−1v1′=∪a=rs∪x=u1v1∪y=u1′v1′∪u1=u2v2K^u2∩K^m−1v2∩∪u1′=u2′v2′K^u2′∩K^n−2v2′∩K^v1′=∪a=xy∪x=u1v1∪y=u1′v1′∪u1=u2v2∪u1′=u2′v2′K^u2∩K^m−1v2∩K^u2′∩K^n−2v2′∩K^v1′=∪a=xy∪x=u1v1∪y=u1′v1′∪u1=u2v2∪u1′=u2′v2′⋯∪um−1=um,vm∪un−2′=un−1′vn−1′K^u2∩K^u3∩⋯∩K^um∩K^vm∩K^un−1′∩K^vn−1′∩⋯∩K^v2′∩K^v1′⊆∪a=xyK^u2u3,…,umvmv1u1′u2′,…,un−1′vn−1′=∪a=xyK^xy,sincex=u2u3,…,umvmv1andy=u1′u2′,…,un−1′vn−1′=K^a.
⇐ For any υ1,υ2,…,υm,κ,ℏ1,ℏ2,…,ℏn∈S, let a=υ1υ2,…,υmκℏ1ℏ2,…,ℏn. Since K^m°χS^°K^n,S⊆K^,S, we have(14)K^υ1υ2,…,υmκℏ1ℏ2,…,ℏn=K^a,⊇K^m°χS^°K^na=∪a=pqK^m°χS^p∩K^nq⊇K^m°χS^υ1υ2,…,υmκ∩K^nℏ1ℏ2,…,ℏn⊇∪υ1υ2,…,υmκ=uvK^mu∩χS^v∩∪ℏ1ℏ2,…,ℏn=u′v′K^u′∩K^n−1v′⊇K^mυ1υ2,…,υm∩χS^κ∩K^ℏ1∩K^n−1ℏ2,…,ℏn−1ℏn⊇K^mυ1υ2,…,υm∩K^n−1ℏ1ℏ2,…,ℏn−1∩K^ℏn⋮⊇K^υ1∩K^υ2∩⋯∩K^υm∩K^ℏ1∩K^ℏ2∩⋯∩K^ℏn.
Hence,K^,S∈ℐm,nU.Definition 2.
A semigroupS is called the m,n-regular if, ∀a∈S∃x∈S such that a=amxan.Lemma 2.
IfS is m,n-regular, K^,S∈ℐm,nU⇏K^,S∈ℐBU.Proof.
Suppose thatK^,S∈ℐm,nU and υ,κ,ℏ∈S. Since S is m,n-regular, υκℏ=υmpυnκℏmqℏn for some p,q∈S. Therefore,(15)K^υκℏ=K^υmpυnκℏmqℏn,=K^υmpυnκℏmqℏn⊇K^υ∩K^ℏ,as required.Lemma 3.
LetK^,S∈SSU. Then, K^υ⊆K^lυl, ∀l∈ℤ+ and υ∈S.Proof.
Letυ∈S. As υl=υυl−1, we have(16)K^lυl=∪υl=ℏκK^ℏ∩K^l−1κ,⊇K^υ∩K^l−1υl−1=K^υ∩∪υl−1=ℏ′κ′K^ℏ′∩K^l−2κ′⊇K^υ∩K^υ∩K^l−2υl−2⋮⊇K^υ∩⋯∩K^υ∩K^υ=K^υ.Theorem 4.
S is m,n-regular ⇔K^,S⊆K^m°χS^°K^n,S, ∀K^,S∈SSU.Proof.
⇒ Let υ∈S. Then, υ=υmxυn for some x∈S. We have(17)K^m°χS^°K^nυ=∪υ=rsK^m°χS^r∩K^ns,⊇K^m°χS^υmx∩K^nυn=∪υmx=pqK^mp∩χS^q∩K^nυn⊇K^mυm∩χS^x∩K^nυn=K^mυm∩K^nυn⊇K^υ∩K^υ,by Lemma3=K^υ.
Therefore,K^,S⊆K^m°χS^°K^n,S.
⇐ Let υ∈S. Since χυ^,S∈SSU, so by Theorem 2, χυ^,S⊆χυ^m°χS^°χυ^n,S. Therefore, χυ^x⊆χυ^m°χS^°χυ^nx=χυmSυnx. It follows that υ∈υmSυn, and so, S is m,n-regular.Theorem 5.
S is m,n-regular ⇔K^,S=K^m°χS^°K^n,S∀K^,S∈ℐm,nU.Proof.
⇒ Suppose that S is m,n-regular and K^,S∈ℐm,nU. Then, by Theorems 3 and 4, K^m°χS^°K^n,S⊆K^,S and K^,S⊆K^m°χS^°K^n,S. Hence, K^,S=K^m°χS°K^n,S.
⇐ Suppose that ω∈S. As ωm,n∈ℳm,n, by Theorem 2, χωm,n,S∈ℐm,nU. Thus, by hypothesis, we have(18)χωm,n=χωm,nm°χS^°χωm,nn=χωm,nmSωm,nn.
Therefore,ωm,n=ωm,nmSωm,nn. By Lemma 1 in [4], ωm,n=ωmSωn. Thus, ω∈ωmSωn, as required.Lemma 4.
IfK^,S∈ℐm,nU and ℱ^,S is an int-soft sub-semigroup over U, such that(19)K^m°χS^°K^n,S⊆ℱ^,S⊆K^,S,then ℱ^,S∈ℐm,nU.Proof.
Asℱ^,S is an int-soft sub-semigroup over U, by Theorem 3, it is sufficient to show that ℱ^m°χS^°ℱ^n,S⊆ℱ^,S. Now,(20)ℱ^m°χS^°ℱ^nυ⊆ℱ^m°χS^°ℱ^nυ⊆ℱ^υ.
Hence,ℱ^,S∈ℐm,nU.Lemma 5.
LetK^,S∈ℐm,nU and ℱ^,S∈SSU. If K^°ℱ^,S⊆K^,S or ℱ^°K^,S⊆K^,S, then(1)
K^°ℱ^,S∈ℐm,nU(2)
ℱ^°K^,S∈ℐm,nUProof.
WhenK^°ℱ^,S⊆K^,S, then we have(21)K^°ℱ^°K^°ℱ^υ⊆K^°K^°ℱ^υ=K^°K^°ℱ^υ⊆K^°ℱ^υ.
It follows thatK^°ℱ^,S is an int-soft sub-semigroup over U. Also, we have(22)K^°ℱ^m°χS^°K^°ℱ^nυ=K^°ℱ^m°χS^°K^°ℱ^n−1°K^°ℱ^υ,⊆K^m°χS^°K^n−1°K^°ℱ^υ⊆K^m°χS^°K^n°ℱ^υ⊆K^°ℱ^υ.
Thus,K^°ℱ^,S∈ℐm,nU. Similarly, when ℱ^°K^,S⊆K^,S, then K^°ℱ^,S∈ℐm,nU. Similar to (1), it can be verified.


## 4. Int-Softm,0-Ideals and Int-Soft 0,n-Ideals

Definition 3.
An int-soft sub-semigroupK^,S over S is called an int-soft m,0-ideal over U if(23)K^υ1υ2,…,υmκ⊇K^υ1∩K^υ2∩⋯∩K^υm,for all υ1,υ2,…,υm,κ∈S.
An int-soft0,n-ideal can be described dually.
Whatever follows, we denote the set of all int-softm,0-ideals and 0,n-ideals over U by ℐm,0U and ℐ0,nU.Example 3.
LetS=0,υ,ℏ,κ. Define the binary operation ′⋅′ on S as follows.
Then,S is a semigroup. Define K^,S,ℱ^,S∈SSU as(24)K^ω=U,ifω∈0,ℏ,∅,ifω∈υ,κ,ℱ^ω=V,ifω∈0,υ,∅,ifω∈ℏ,κ.
It is straightforward to verify thatK^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU.Lemma 6.
InS, K^,S∈ℐRUresp.K^,S∈ℐLU⇒K^,S∈ℐm,0Uresp.K^,S∈ℐ0,nU.Proof (straightforward).Remark 2.
In general,K^,S∈ℐm,0Uresp.K^,S∈ℐ0,nU⇏K^,S∈ℐRUresp.K^,S∈ℐLU.Example 4.
In Example3, K^,S∈SSU⇒K^,S∈ℐm,0U,ℐ0,nU∀m,n≥2, but K^,S∉ℐRU,ℐLU.Definition 4.
A semigroupS is called the m,0-regular (resp. 0,n-regular) if ∀υ∈S∃ℏ∈S such that υ=υmℏresp.υ=ℏυn.Lemma 7.
The following assertions hold:(1)
Inm,0-regular S, K^,S∈ℐm,0U⇒K^,S∈ℐRU(2)
In0,n-regular S, K^,S∈ℐ0,nU⇒K^,S∈ℐLUProof.
Letυ,ℏ∈S. Since S is m,0-regular, so ∃κ∈S such that υℏ=υmκℏ. Therefore, we have(25)K^υℏ=K^υmκℏ=K^υmκℏ⊇K^υ.
Hence,K^,S∈ℐRU. (2). Similarly, this can be proved.Lemma 8.
Let∅≠Ω⊆S. Then, Ω∈ℳm,0 (resp. Ω∈∈ℳ0,n) ⇔ the χΩ^,S∈ℐm,0U (resp. χΩ^,S∈ℐ0,nU).Proof.
⇒ Let υ1,υ2,…,υm,κ∈S. If xk∉Ω, for some k∈1,2,…,m, then χΩ^υ1υ2,…,υmκ⊇χΩ^υ1∩χΩ^υ2∩⋯∩χΩ^υm. If xk∈Ω for each k∈1,2,…,m, then υ1υ2,…,υmκ∈ΩmS⊆Ω. Therefore,(26)χΩ^υ1υ2,…,υmc=1⊇χΩ^υ1∩χΩ^υ2∩⋯∩χΩ^υm.
Hence,χΩ^,S∈ℐm,0U.
⇐ Let υ1,υ2,…,υm∈Ω and κ∈S. Then, χΩ^υ1υ2,…,υmc⊇χΩ^υ1∩χΩ^υ2∩⋯∩χΩ^υm=1 implies χΩ^υ1υ2,…,υmc=1. Therefore, υ1υ2,…,υmc∈Ω. Thus, ΩmS⊆Ω, as required.Theorem 6.
LetK^,S be any int-soft sub-semigroup over U. Then, K^,S∈ℐm,0U (resp. K^,S∈ℐ0,nU) ⇔K^m°χS^,S⊆K^,S (resp. χS^°K^n,S⊆K^,S).Proof.
It is similar to the proof of Theorem3.Lemma 9.
LetS be m,n-regular, K^,S∈ℐm,0U, and ℱ^,S∈ℐ0,nU. Then, K^,S=K^°K^,S and ℱ^,S=ℱ^°ℱ^,S.Proof.
LetK^,S∈ℐm,0U. Then, K^°K^,S⊆K^,S. We have(27)K^x⊆K^m°χS^°K^nx=K^m°χS^°K^n−1°K^x,⊆K^m°χS^°K^n−1°K^m°χS^°K^nx⊆K^m°χS^°K^m°χS^x⊆K^°K^x,so we obtain K^,S⊆K^°K^,S. Hence, K^,S=K^°K^,S.Theorem 7.
InS, the following assertions are true:(1)
S is m,0-regular ⇔K^,S⊆K^m°χS^,S, ∀K^,S∈SSU(2)
S is 0,n-regular ⇔K^,S⊆χS^°K^n,S, ∀K^,S∈SSUProof.
⇒ Let υ∈S. Then, ∃ℏ∈S such that υ=υmℏ. Now, we have(28)K^m°χS^υ=∪υ=κsK^mκ∩χS^s,⊇K^mυm∩χS^ℏ=K^mυm⊇K^υ.
Therefore,K^,S⊆K^m°χS^,S.
⇐ Take any υ∈S. Since χυ^,S∈SSU, χυ^,S⊆χυ^m°S,S. Therefore, χυ^ℏ⊆χυ^m°Sℏ=χυmS^ℏ. It follows that υ∈υmS and so, S is m,0-regular. Similar to (1), (2) can be verified.Theorem 8.
The following assertions are true inS:(1)
S is m,0-regular ⇔K^,S=K^m°χS^,S, ∀K^,S∈ℐm,0U(2)
S is 0,n-regular ⇔K^,S=χS^°K^n,S, ∀K^,S∈ℐ0,nUProof.
(1)⇒ Suppose that S is m,0-regular and K^,S∈ℐm,0U. Then, by Theorems 7 and 6, we have K^,S⊆K^m°χS^,S and K^m°χS^,S⊆K^,S. Hence, K^,S=K^m°χS^,S.
⇐ Take R∈∈ℳm,0. By Lemma 8, χR,S∈ℐm,0U. By hypothesis χR,S=χRm°χS^,S. So, χRℏ=χRm°χS^ℏ=χRmSℏ, and it follows that RmS=R. Therefore, by Theorem 1 in [45], S is m,n-regular. Similar to (1), (2) can be verified.Theorem 9.
S is m,n-regular ⇔K^∩ℱ^,S=K^m°ℱ^n,S, ∀K^,S∈ℐm,0U, and ℱ^,S∈ℐ0,nU.Proof.
⇒ Suppose that K^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU. As S is m,n-regular, we have(29)K^∩ℱ^,S⊆K^∩ℱ^m°χS^°K^∩ℱ^n,S⊆K^m°χS^°ℱ^n,S.
By Theorem8 and Lemma 9, we have χS^°ℱ^n,S=ℱ^,S and ℱ^,S=ℱ^n,S. Therefore, K^∩ℱ^,S⊆K^m°ℱ^n,S. Also, K^m°ℱ^n,S⊆K^∩ℱ^,S. Therefore, K^∩ℱ^,S=K^m°ℱ^n,S.
⇐ Take R∈ℳm,0 and L∈ℳ0,n. By Lemma 2, χR^,S∈ℐm,0U and χL,S∈ℐ0,nU. By hypothesis, we have(30)χ^R∩L=χR^∧χL^=χR^m°χL^n=χ^RmLn,it follows that R∩L=RmLn. Thus, by Theorem 12 in [44], S is m,n-regular.Corollary 1.
IfS is m,n-regular, then K^∩ℱ^,S=K^°ℱ^,S, ∀K^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU.Theorem 10.
S is m,n-regular ⇔K^∩ℱ^,S=K^m°ℱ^∩K^°ℱ^n,S∀K^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU.Proof.
⇒ Suppose that K^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU. As S is m,n-regular, we have(31)K^∩ℱ^,S⊆K^∩ℱ^m°χS^°K^∩ℱ^n,S⊆K^m°χS^°ℱ^n,S⊆K^m°ℱ^,S,and so, K^∩ℱ^,S⊆K^m°ℱ^,S. Similarly, K^∩ℱ^,S⊆K^°ℱ^n,S. Thus, K^∩ℱ^,S⊆K^m°ℱ^∩K^°ℱ^n,S. Since K^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU, the reverse inclusion holds. Hence, K^∩ℱ^,S=K^m°ℱ^∩K^°ℱ^n,S.
⇐ Take R∈ℳm,0 and L∈ℳ0,n. By Lemma 8, χR^,S∈ℐm,0U and χL^,S∈ℐ0,nU. Observe that, by hypothesis, we have(32)χ^R∩L=χ^R∩χ^L=χR^m°χL^∩χR^°χL^n=χ^RmL∩RLn,and it follows that R∩L=RmL∩RLn. Therefore, by Theorem 3 in [45], S is m,n-regular.Lemma 10.
ForK^,S∈SSU, K^∪K^m°χS^,S∈ℐm,0U (resp. K^∪χS^°K^n,S∈ℐ0,nU).Proof (straightforward).Lemma 11.
Inm,n-regular semigroup S, for each K^,S∈ℐm,nU, there exist G^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU such that K^,S=G^°ℱ^,S.Proof.
Suppose thatK^,S∈ℐm,nU. Then, K^m°χS^°K^n,S⊆K^,S. As S is m,n-regular, K^,S⊆K^m°χS^°K^n,S. Therefore, K^,S=K^m°χS^°K^n,S. Let G^,S=K^∪K^m°χS^,S and ℱ^,S=K^∪χS^°K^n,S. By Lemma 9, G^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU. Since S is m,n-regular, G^,S=K^∪K^m°χS^,S=K^m°χS^,S and ℱ^,S=K^∪χS^°K^n,S=χS^°K^n,S, so(33)G^°ℱ^,S=K^m°S°S°K^n,S=K^m°S°K^n,S=K^,S,as required.Lemma 12.
Inm,n-regular semigroup S, ∀K^,S∈ℐm,0U and ℱ^,S∈SSU, K^°ℱ^,S∈ℐm,nU.Proof.
LetK^,S∈ℐm,0U and ℱ^,S∈SSU. Now,(34)K^°ℱ^m°χS^°K^°ℱ^nυ=K^°ℱ^°K^°ℱ^°⋯°K^°ℱ^︸m−times°χS^°K^°ℱ^°K^°ℱ^°⋯°K^°ℱ^︸n−timesυ,=K^°ℱ^°K^°ℱ^°K^°ℱ^°⋯°K^°ℱ^︸m−1−times°χS^°K^°ℱ^°K^°ℱ^°⋯°K^°ℱ^︸n−1−times°K^°ℱ^υ⊆K^°ℱ^°χS^°χS^°χS^°K^°ℱ^υ⊆K^°χS^°ℱ^υ⊆K^m°χS^°ℱ^υby Lemma9⊆K^°ℱ^υ.
Therefore,K^°ℱ^,S∈ℐm,nU.
By Lemmas11 and 12, we have the following.Theorem 11.
LetS be a m,n-regular and K^,S∈SSU. Then, K^,S∈ℐm,nU⇔ there exist ℱ^,S∈ℐm,0U and ℱ^,S∈ℐ0,nU such that K^,S=ℱ^°ℱ^,S.Definition 5.
An int-softm,n-ideal K^,S over U is called minimal if, for all int-soft m,n-ideal K^′,S over U, K^′,S⊆K^,S implies K^′,S=K^,S.
Dually, a minimum int-softm,0-ideal and minimal int-soft 0,n-ideal over U can be described.Theorem 12.
Inm,n-regular semigroup S, a soft set K^,S over U is a minimal int-soft m,n-ideal over U⇔ there exist a minimal int-soft m,0-ideal G^,S and a minimal int-soft 0,n-ideal ℱ^,S over U such that K^,S=G^°ℱ^,S.Proof.
⇒ Let K^,S∈ℐm,nU be minimal. By Lemma 11, K^,S=K^∪K^m°χS^,S°K^∪χS^°K^n,S. We show that K^∪K^m°χS^,S∈ℐm,0U is minimal. To show this, let K^′,S∈ℐm,0U such that K^′,S⊆K^∪K^m°χS^,S. Since S is m,n-regular, so, by Corollary 1, K^∪K^m°χS^,S∩K^∪χS^°K^n,S=K^∪K^m°χS^,S°K^∪χS^°K^n,S. Again, by Corollary 1, K^′,S°K^∪χS^°K^n,S=K^′,S∩K^∪χS^°K^n,S⊆K^∪K^m°χS^,S∩K^∪χS^°K^n,S=K^,S. By Lemma 12, K^′,S°K^∪χS^°K^n,S∈ℐm,nU. Since K^′,S°K^∪χS^°K^n,S⊆K^,S, by minimality of the int-soft m,n-ideal K^,S over U, we have K^′,S°K^∪χS^°K^n,S=K^,S. Therefore, K^∪K^m°χS^,S∩K^∪χS^°K^n,S=K^′,S∩K^∪χS^°K^n,S. As K^,S⊆K^∪K^m°χS^,S∩K^∪χS^°K^n,S, we have K^,S⊆K^′,S. So, K^∪K^m°χS^,S⊆K^′,S. Hence, K^′,S=K^,S. Thus, K^∪K^m°χS^,S∈ℐm,0U is minimal. Similarly, K^∪χS^°K^n,S∈ℐ0,nU is minimal.
⇐ Assume that K^,S=G^°ℱ^,S for some minimal int-soft m,0-ideal G^,S and minimal int-soft 0,n-ideal ℱ^,S over U. By Lemma 11, K^,S∈ℐm,nU. To show that K^,S∈ℐm,nU is minimal, let W^,S∈ℐm,nU such that W^,S⊆K^,S. Then, W^m°χS^,S⊆K^m°χS^,S⊆G^°ℱ^m°χS^,S=G^°ℱ^°G^°ℱ^°⋯°G^°ℱ^°χS^⊆G^°ℱ^°G^°ℱ^°⋯°G^°ℱ^°χS^⊆G^°χS^⊆G^m°χS^°G^n°χS^⊆G^m°χS^⊆G^.
AsW^m°χS^,S∈ℐm,0U and G^,S∈ℐm,0U is minimal, W^m°χS^,S=G^,S. Similarly, χS^°W^n,S=ℱ^,S. Now, K^,S=G^°ℱ^,S=W^m°χS^°χS^°W^n,S⊆W^m°χS^°W^n,S⊆W^,S. Hence, K^,S∈ℐm,nU is minimal.Corollary 2.
There is at least one minimal int-softm,n-ideal over U in m,n-regular semigroup S⇔S has at least one minimal int-soft m,0-ideal and one minimal int-soft 0,n-ideal over U.


## 5. Conclusion

The main purpose of this article is to present in semigroups the ideas of int-softm,n-ideals, int-soft m,0-ideals, and int-soft 0,n-ideals. If we take m=1=n in the int-soft m,n-ideals, int-soft m,0-ideals, and int-soft 0,n-ideals in particular, then we get the int-soft bi-ideals, int-soft right ideals, and int-soft left ideals. The ideas proposed in this paper can also be seen to be more general than int-soft bi-ideals, int-soft right ideals, and int-soft left ideals. Also, if we place m=1=n in the results of this paper, then the results of [8] are deduced as corollaries, which is the main application of the results of this paper.In the future work, one can further study these concepts to various algebraic structures such as semi-hypergroups, semi-hyperrings, rings,LA-semigroups, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, EQ-algebras, and lattice implication algebras.



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