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

. 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-soft ( m, 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.


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. us, 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. e 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. e 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 intsoft 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 intsoft bi-ideals are obtained. Dudek and Jun [9] introduced and characterized the notion of soft interior ideals of semigroups. e concept of union-soft semigroups, unionsoft 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][12][13][14][15][16][17][18][19][20][21]). For more related concepts, the readers are referred to [22][23][24][25][26][27][28][29][30][31]. e results of this paper are arranged as follows. Section 2 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. en, 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 (K, S) over U. In Section 4, first, we present the idea of intsoft (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 intsoft (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.

Preliminaries
Let S be a semigroup.
Let U be a universal set and let E be a set of parameters. Let P(U) 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 ⟶ P(U) is a mapping. We denote the set of all soft sets over U with parameter set S by S S (U).
Let (K, Ω) and (G, [) be soft sets over U. en, Let (K, Ω) and (G, Ω) be two soft sets. en, for each υ ∈ Ω, the union and intersection are defined as (1) For any two soft sets (K, Ω) and (G, Ω) of S, the int-soft product K°G is defined as 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 subsemigroup (K, S) over U is called an int-soft bi-ideal over U if K(υκZ) ⊇ K(υ) ∩ K(Z) for all υ, κ, Z ∈ S. e set of all int-soft left (resp. Right) ideals and int-soft bi-ideals over U will be denoted by I L (U) (resp. I R (U)) and I B (U).
For (∅ ≠ )Ω ⊆ S, the characteristic soft set over U is denoted by (χ Ω , S) and defined as e concept of (m, n)-ideals of semigroups was introduced by Lajos [40] as follows. Let S be a semigroup and m, n be nonnegative integers. en, a sub-semigroup Ω of S is said to be an (m, n)-ideal of S if Ω m SΩ n ⊆ 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.
Define the binary operation ′ · ′ on S as follows.
where U 1 , U 2 ⊆ U such that U 2 ⊆ U 1 . It is straightforward to verify that (K, S) ∈ I (m,n) (U).
Define the binary operation ′ · ′ on S as follows.

Proof. As (F, S) is an int-soft sub-semigroup over U, by eorem 3, it is sufficient to show that
Hence, (F, S) ∈ I (m,n) (U).
as required.  Journal of Mathematics erefore, (K°F, S) ∈ I (m,n) (U). By Lemmas 11 and 12, we have the following. □ Theorem 11. Let S be a (m, n)-regular and (K, S) ∈ S S (U).
Dually, a minimum int-soft (m, 0)-ideal and minimal int-soft (0, n)-ideal over U can be described.  ere is at least one minimal int-soft (m, n)-ideal over U in (m, n)-regular semigroup S ⇔ S has at least one minimal int-soft (m, 0)-ideal and one minimal intsoft (0, n)-ideal over U.

Conclusion
e main purpose of this article is to present in semigroups the ideas of int-soft (m, 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. e ideas proposed in this paper can also be seen to be more general than int-soft biideals, 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.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.