An algorithm for generating permutation algebras using soft spaces

ABSTRACT

Soft set theory has recently gained significance for finding rational and logical solutions to various real-life problems, which involve uncertainty, impreciseness and vagueness. In this paper, we introduced an algorithm to find permutation algebras using soft topological space. Moreover, this class of permutation algebras is called even (odd) permutation algebras if its permutation is even (odd). Furthermore, new concepts in permutation algebras are investigated such as splittable permutation algebra and ambivalent permutation algebra. Furthermore, several examples are given to illustrate the concepts introduced in this paper.

KEYWORDS:

• Soft set theory
• symmetric groups
• splitting
• ambivalent
• permutation topological spaces

1. Introduction

The concept of soft sets is a novel notion was introduced by Molodtsov [1]. Next, Shabir and Naz [2] introduced the notion of soft topological spaces. Some notions of this concept with its applications fundamental concepts of fuzzy soft topology and Intuitionistic fuzzy soft topology are studied by many mathematicians see the following references [3–15]. BCK-algebra as a class of abstract algebras is introduced by Imai and Iseki [16,17]. Next, the concept of d-algebras, which is another useful generalization of BCK-algebras, is introduced (see refs [18–20]). Also, the notion of d*-algebras is investigated [21]. After then, the concepts of ρ-algebras and -ideals are introduced and studied [22]. In 2009, the concept of soft d-algebras is introduced (see Jun et al. [23]). Also, some extensions using power set are introduced such as soft ρ-algebra and soft edge ρ-algebra of the power set [24], soft BCL-algebras of the power set [25] and soft BCH-algebra of the power set [26]. Next, the notion of -fuzzy d-algebra is given [27]. In this paper, we introduced an algorithm to find permutation algebras using soft topological space. Moreover, this class of permutation algebras is called even (odd) permutation algebras if its permutation is even (odd). Furthermore, new concepts in permutation algebras are investigated such as splittable permutation algebra and ambivalent permutation algebra. Furthermore, several examples are given to illustrate the concepts and introduced in this paper.

2. Preliminaries

In this section, we recall basic definitions and results that are needed later.

Definition 2.1:

We call the partition the cycle type of [28].

Definition 2.2:

Let be a partition of . We define to be the set of all elements with cycle type [28].

Definition 2.3:

Let where is a permutation in alternating group . conjugacy class of in is defined by [29]where are two classes of equal order in alternating group such that and {of | , with all parts of different and odd}.

Proposition 2.4:

The conjugacy classes of are ambivalent if for each part of [30].

Definition 2.5:

Suppose is permutation in a symmetric group on the set and the cycle type of is , then composite of pairwise disjoint cycles where ,. For any k-cycle in , we define -set as and is called -set of cycle . So the -sets of are defined by [31].

Remark 2.6:

Suppose that and are -sets in , where and . Then, the known definitions will be written as follows.

Definition 2.7:

We call and are disjoint -sets in , if and only if and there exists , for each such that [31].

Definition 2.8:

We call and are equal -sets in , if and only if for each there exists such that [31].

Definition 2.9:

We call is contained in , if and only if [31].

Definition 2.10:

Let and be two subsets of [32]. Then, we call and are similar -sets in , if and only if and one of them contains at least two points say such that and . Let and be similar -sets in and , where . Then , , and . For any and two subsets of . Then, and

Definition 2.11:

For any collection of not disjoint -sets . We define the union (respectively, intersection) of by where and  where [31].

Definition 2.12:

Let be permutation in a symmetric group and composite of pairwise disjoint cycles where then permutation topological space (PTS), where and is a collection of -set of the family union and empty set [31].

Definition 2.13:

Let be a PTS. Then is called permutation single space (PSS) if and only if each proper open -set is a singleton [32].

Definition 2.14:

Let be a permutation topological space. Then is called permutation indiscrete space (PIS) if and only if each open -set is trivial -set [32].

Definition 2.15:

Let be an initial universe set and let be a set of parameters [1]. A pair is called a soft set (over where and is a multivalued function . In other words, the soft set is a parameterized family of subsets of the set Every set , , from this family may be considered as the set of e-elements of the soft set , or as the set of -approximate elements of the soft set. Clearly, a soft set is not a set. For two soft sets and over the common universe we say that is a soft subset of if and for all and are identical approximations. We write . is said to be a soft superset of , if is a soft subset of . Two soft sets and over a common universe are said to be soft equal if is a soft subset of and is a soft subset of . A soft set over is called a null soft set, denoted by (,), if for each . Similarly, it is called universal soft set, denoted by , if for each . The collection of soft sets over a universe and the parameter set is a family of soft sets denoted by .

Definition 2.16:

The union of two soft sets and over is the soft set , where and for all if , if if [33]. We write . The intersection of and over , denoted is defined as and for all [21].

Definition 2.17:

Let be a soft set over [2]. The complement of with respect to the universal soft set denoted by is defined as where and for all ,

Definition 2.18:

Let be the collection of soft sets over [2]. Then is called a soft topology on if satisfies the following axioms:

1. and belong to .

2. The union of any number of soft sets in belongs to .

3. The intersection of any two soft sets in belongs to .

The triplet is called a soft topological space over . The members of are called soft open sets in and complements of them are called soft closed sets in . Furthermore, is said to be a soft indiscrete space over , if . Also, is said to be a soft discrete space over , if is the collection of all soft sets which can be defined over .

Some results on permutations 2.19: [29]

1. is odd.

2. is even,

3. .

Remark 2.20:

In this work, for any set of distinct objects and for any cycle , we will use the same symbol () to refer to the cardinality of set and to refer to the length of the cycle . Hence and .

Definition 2.21:

A -algebra is a non-empty set with a constant 0 and a binary operation* satisfying the following axioms [18]:

3. and imply that for all x, y in X.

Definition 2.22:

A -algebra is said to be algebra if satisfies the following additional axioms [21]:

1. ,

2. , for all .

Definition 2.23:

A -algebra is said to be -algebra if satisfies for all imply that [22].

Definition 2.24:

Let be a -algebra [21]. Then is called a -algebra if it satisfies the identity , for all .

Definition 2.25:

Let be a soft topological space, where , and [34]. Now, let . For each , let . For each , let be a map from X into natural numbers N defined by , for all and where . Then is called permutation in a symmetric group where for all , is permutation which is the product of cyclic factors of the length , where and . Furthermore, if . Then is called permutation topological space induced by soft topology , where , and is a collection of -set of the family together with and the empty set. Also, if is a soft indiscrete topological space. Then is called PIS induced by soft topology , where . Finally, for any non-indiscrete soft topological space where and we can generate permutation topological space as follows:

1. Find for all ,

2. Find , for all ,

3. Find , for all and ,

4. Find for all where

5. Find

6. Find the disjoint cycle factors including the 1-cycle of say

7. Find the -sets of

8. Find , where and

9. Find , where .

10. Then is PTS.

Remark 2.26:

For all , let . Here we used normal union (), normal intersection () and empty set () then we have is a topological space for each () [34].

1. We consider that is isomorphic, where and . In other words, for each , we have , where .

2. For any and , we have (since )

3. If , for some and . Then for all .

Definition 2.27:

Let be a permutation space induced by soft topology , and for each proper then [34]

Let non-empty open -set}. For each , let and . Let where . Here we used normal intersection () between pairwise sets to find the set . For each we have is cycle in . Then the disjoint cycles decomposition of new permutation in a symmetric group induced by saying where and whenever . Hence is called permutation subspace induced by soft topology where is a family of all -sets of disjoint cycles decomposition of together with and the empty set.

3. An algorithm to generate permutation algebras from soft spaces

In this section, we will introduce an algorithm to generate permutation algebra by analysis soft topological space and this class of permutation algebra is called even (odd) permutation algebra if its permutation is even (odd). Moreover, some basic properties of permutation spaces are studied.

Steps of the work 3.1:

Let be a soft topological space, where , and . Now, let . For each , let . For each , let be a map from X into natural numbers N defined by , for all and where . Then is called permutation in a symmetric group where for all , is permutation which is the product of cyclic factors of the length , where and . Furthermore, if . Hence, by Definition 2.5, we consider that:

is permutation topological space induced by soft topology , where , and is a collection of -set of the family together with and the empty set. Now, we need to follow these steps:

1. Defend binary operation on permutation topology by

2. Find which is holed as follows:

   • (i) (ii) (iii) and imply that for all .

3. Then is a permutation -algebra induced by soft topology.

Definitions 3.2:

1. Let be a permutation d-algebra induced by soft topology . Then is called permutation -algebra induced by soft topology , if satisfying the following axioms: (i) (ii) , for all .

2. Let be a permutation d-algebra induced by soft topology . Then is called permutation -algebra induced by soft topology , if it satisfies the identity , for all .

3. Let be a permutation d-algebra induced by soft topology . Then is called permutation -algebra induced by soft topology , if it satisfies the identity , for all .

4. Let be a permutation -algebra (-algebra, -algebra, -algebra) induced by soft topology and be a collection of some random subsets of . Then is called permutation -subalgebra (resp. -subalgebra, -subalgebra, -subalgebra) induced by soft topology , if , for any .

Permutation subalgebras induced by soft topology 3.3:

Let be a permutation space induced by soft topology and . By Definition 2.7, we consider that is called permutation subspace induced by soft topology where is a family of all -sets of disjoint cycles decomposition of together with and the empty set. Also, is called permutation -subalgebra of induced by soft topology if is permutation -algebra and with

Remarks 3.4:

If there are two permutation subspaces and of and is permutation -subalgebra of induced by soft topology , then need not be permutation -subalgebra of too.

Example 3.5:

Let the set of houses under consideration be . Let E={big ; with pool swimming ; with garden ; old } be the set of parameters framed to buy the best house. Suppose that the soft set describing the Mr opinion to choose the best house was defined by In addition, we assume that the “best house” in the opinion of his friend, say Mr Y, is described by the soft set , where Consider that:

is a soft topology. Find permutation space induced by soft topology . Also, find and where and

Solution: We consider that . Hence we have is a topological space for each (). Now, we consider that is a topological space for each (). where Moreover, for all , is permutation which is the product of cyclic factors of the length , where and . Hence we have Then is a permutation in a symmetric group . For any permutation can be decomposed essentially uniquely into the product of disjoint cycles. So we can write as . With disjoint cycles of length and is the number of disjoint cycle factors including the 1-cycle of . Hence

Therefore, is a permutation space induced by soft topology where and . Now to find the permutation subspace for , we consider that . Then is a permutation in a symmetric group induced by and is a permutation subspace induced by soft topology where . Now to find the permutation subspace for , we consider that: and Therefore, we have is a permutation in a symmetric group induced by and is a permutation subspace induced by soft topology where Now, define by . Then is permutation -algebra, since satisfies conditions for -algebra. Also, is permutation -subalgebra of . But is not permutation -subalgebra of since .

4. Some notions of permutation algebras

Definition 4.1:

Let be a permutation -algebra induced by soft topology . Then is called even (odd) permutation -algebra if its permutation is even (odd) in .

Example 4.2:

Let be a permutation d-algebra induced by soft topology , where in Example 3.1.7. Then is an even permutation d-algebra since is an even.

Definition 4.3:

Let be a permutation -algebra induced by soft topology . Then is called splittable permutation -algebra if its permutation satisfies .

Example 4.4:

Let be a permutation topological space induced by soft topology , where and , let be a binary operation defined in the following table:

Table

Then is splittable permutation -algebra since .

Remark 4.5:

It is clearly that every even (odd) permutation -algebra is even (odd) permutation -algebra, but the converse need not be true, see Example 3.5 is an even permutation -algebra but is not even permutation -algebra.

Definition 4.6:

Let be a permutation -algebra induced by soft topology . Then is called ambivalent permutation space if is splittable permutation -algebra and for each part of satisfies .

Example 4.7:

Let be a permutation topological space induced by soft topology , where and . Let be a binary operation defined in the following table:

Table

Then is an ambivalent permutation -algebra induced by soft topology .

The maps induced by soft topologies 4.8:

Suppose that and are three soft topological spaces over the common universe X the parameter set E with their permutations and in a symmetric group and let be a map, where for each -set , the image of under is said to be -set and defined by the rule . In another side, let be -set, the inverse image of under is called -set and defined by the rule . Then is called a permutation map induced by soft topology .

Definition 4.9:

Suppose that and are three soft topological spaces over the common universe X the parameter set E with their permutations and in a symmetric group Then is called a permutation continuous induced by soft topology , if under of any open -set in is an open -set in whenever .

Theorem 4.10:

Let be a soft topological space. If for any pair such that and  for any and . Then , where is a permutation space induced by soft topology .

Proof:

Let , , and . Since any pair such that . Then and are disjoint cycles in a symmetric group (since , for any ). Also, for any and , , we consider that and are disjoint cycles in a symmetric group (since , for any and ). But is a permutation for the permutation space . Therefore, we consider that is the number of disjoint cycle factors of . Furthermore, is the number of disjoint cycle factors including the 1-cycle of . Hence in general. That means either or . If , then there is at least 1-cycle say for some (, with and this contradiction with our hypothesis. Hence .

Theorem 4.11:

Let be a permutation -algebra induced by soft topological space and for any pair such that and for any and Then is an even permutation -algebra, if .

Proof:

Let be a permutation -algebra induced by soft topological space and for any pair such that and for any and Then by Theorem 4.10), we consider that is a permutation space induced by soft topology and its permutation in a symmetric group satisfies . However, and hence is even. This implies that is even permutation in a symmetric group . Then is an even permutation -algebra.

Lemma 4.12:

Let be a permutation -algebra induced by soft topological space . Then is odd permutation -algebra, if is a soft indiscrete topological space and .

Proof:

Let be a permutation -algebra induced by soft indiscrete topological space and . Hence (since is soft indiscrete space), thus . Also, since , Then there is a positive integer number such that and hence = (even) (odd) = (odd). Hence is odd permutation in . Then is odd permutation space.

Lemma 4.13:

Let be a permutation -algebra and . Then is permutation -subalgebra induced by soft topology , if is permutation -subalgebra induced by soft topology .

Proof:

Suppose that is permutation -algebra and is permutation -subalgebra induced by soft topology . Then we consider that is permutation -algebra induced by soft topology . Also, satisfies whenever and . Hence is permutation -subalgebra of -algebra induced by soft topology .

Lemma 4.14:

Let be a permutation -algebra induced by soft topological space . Then is an even permutation -algebra, if , where .

Proof:

Let be a permutation -algebra induced by soft topological space and , where . This implies that for some . Therefore for some . Then is (PSS) (since each proper open -set is a singleton). Thus, composite of pairwise disjoint cycles where for some and (since each proper open -set is a singleton). However, . But is an identity element in Thus and hence This implies (even). Hence is an even permutation -algebra.

Theorem 4.15:

Let be a permutation -algebra induced by soft topological space and for any pair such that and for any and Then is a splittable permutation -algebra, if the following are hold.

1. ,

2. If , then , where and .

Proof:

Let be a permutation -algebra induced by soft topological space and for any pair such that and for any with the following are hold.

1. ,

2. If , then , where and .

Then by Theorem 4.10, we consider that is a permutation space induced by soft topology and its permutation in a symmetric group satisfies Moreover, where for all is permutation which is the product of cyclic factors of the length , where and . For any permutation can be decomposed essentially uniquely into the product of disjoint cycles. Thus, we can write as . With disjoint cycles of length and is the number of disjoint cycle factors including the 1-cycle of . Then is the cycle type of . Since and . Then for any there exists satisfies , where . The length of any cycle is and this implies that Now, if is even number for some and , thus is odd and this contradiction with (1) of our hypothesis which states that . Then are odd numbers for all and . Also, for any or we have where and Then are different sets and by (2) of our hypothesis we consider that are different too. Therefore and hence is a splittable permutation -algebra.

Theorem 4.16:

Let be a permutation -algebra induced by soft topological space and for any pair such that and for any and . Then is an ambivalent permutation -algebra, if the following are hold.

1. ,

2. If , then , where and ,

3. If .

Proof:

Let be a permutation -algebra induced by soft topological space and for any pair such that and for any with the following are hold.

1. ,

2. If , then , where and ,

3. If . Then from (1) and (2), we consider that is a splittable permutation -algebra (by Theorem 4.15). Also, from (1) and (2) that easy to show that That means for any part of , there exists for some and satisfies and hence from (3) we have Then is an ambivalent permutation -algebra.

5. Methodology flowchart

The flow of this work is explained by using the chart in Figure 1 to generate permutation algebras from soft topological spaces.

Figure 1. Generating permutation algebra from soft topological space.

6. Conclusions and open problems

In this work, an algorithm has been introduced to find the permutation algebra using soft topological space. Furthermore, the aim of this work was to delve into the aspects of a symmetric group that has a link with soft topological space in terms of permutation in symmetric. This would aid authors to appreciate the role it plays in the theories and applications of soft topological space and deal with these in permutation algebras. Assume that and are three soft topological spaces over the common universe the parameter set with their permutations and in a symmetric group . Hence the questions can be summarized as follows:

Is necessarily true, if is an even (res. odd, splittable, ambivalent) permutation algebra induced by soft topology and is continuous permutation map. Then is even (res. odd, splittable, ambivalent) permutation algebra induced by soft topology .

Is necessarily true, if and are two permutation algebras induced by soft topologies and respectively. Then is a permutation algebra induced by soft topology where and is defined by .

Acknowledgements

The authors would like to thank from the anonymous reviewers for carefully reading of the manuscript and giving useful comments, which will help to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Shuker Mahmood Khalil http://orcid.org/0000-0002-7635-3553

Fatima Hameed Khadhaer http://orcid.org/0000-0003-4143-676X