1. Introduction

Let be a permutation in symmetric group with letter . The support of , is the set where and is not identity in . So we say and are disjoint cycles iff [10]. There are many applications on permutations, in recent years they are used to solve equations (see [8-11]). Permutation topological space is one of the more interesting applications was first introduced by Shuker [7] in 2014, where each set in the permutation space is either open or closed. That means it's not necessary any subset of in is set. Therefore in this paper we will solve this problem in section three by give more definitions and notations of permutation space and hence we can deal with any subset of in as set. That means we can put . However where disjoint cycles of also we denote to its cycle by and hence in this paper after we give some new definition we will consider that all the notations and definitions are hold except it is not necessary every set in the permutation space is either open set or closed set. In another direction, new construction is called similar set with some notations are recalled that is required to be set for any subset of .

A topological group is a set that has both an algebraic structure and a topological structure. Further, many notations of topological group are discussed by many researchers (see [1-6]).

In section four and five, some new permutation spaces like (PSS), (PIS), (PHS), , (EPTS), (IEPTS), (DEPTS), , permutation homogeneous space, -connected space, -disconnected space and others are introduced and discussed. Further, in this paper many interesting properties and examples of permutation topological groups and extension permutation topological groups will be explored. Also, the notations of permutation homogeneous topological group, Lindelof permutation topological group, -connected group, -disconnected topological group, (EPTG), (IEPTG), (DEPTG), ( group), ( group), ( group) and others are defined and illustrated. In other words, separation axioms, connectedness and related properties of permutation topological groups and of extension permutation topological groups are discussed.

2. Preliminaries

In this section we recall the basic definition and information which are needed in our work.

Definition 2.1 [11]

A partition is a sequence of nonnegative integers with and . The length and the size of are defined as and We set for . An element of is called a partition of n.

Remark 2.2 [15]

We only write the non zero components of a partition. Choose any and write it as . With disjoint cycles of length and is the number of disjoint cycle factors including the 1-cycle of . Since disjoint cycles commute, we can assume that . Therefore is a partition of n and each is called part of .

Definition 2.3 [8]

We call the partition the cycle type of .

Definition 2.4 [14]

Suppose first that . Then , the support of , is the set where . So we say and are disjoint cycles iff .

Definition 2.5 [7]

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

Remark 2.6 [7]

For any cycle in we put , Further, suppose that and are sets in , where and . We will give some definitions needed in this work.

Definition 2.7 [7]

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

Definition 2.8 [7]

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

Definition 2.9 [7]

We call is contained in and denoted by , if and only if .

Definition 2.10 [7]

We define the operations and on sets in as followers:

and

Remarks 2.11 [7]

1. The intersection of and is .

2. The union of and is .

3. The complement of is .

4. The intersection and union of and are and , respectively.

5. The intersection and union of and are and , respectively.

Definition 2.12 [7]

Let be permutation in symmetric group , and composite of pairwise disjoint cycles , where , then is a permutation topological space where and is a collection of sets of the family union and empty set.

Definition 2.13 [7]

If is set in the space , then is called closed set in the space , and is smallest closed set containing or equal , and any set is called closed set iff .

Definition 2.14 [7]

The set is called the interior of the set in the permutation space .

Remarks 2.15 [7]

1. We call belong to set iff , for some .

2. The condition means that . Therefore, is an interior point of set if and only if there is an open set containing and such that .

3. If and are disjoint sets in , then neither nor .

Remark 2.16 [7]

Any map between two permutation topological spaces is called permutation map.

Definition 2.17 [7]

Let and be three permutations in symmetric group , and let be a function, where for each set , the image of under is called set and defined by the rule . In another direction, let be set, the inverse image of under is called set and defined by the rule . The usual properties relating images and inverse images of subsets of complements, unions, and intersections also hold for permutation sets.

Definition 2.18 [7]

Given permutation topological spaces and , a function is permutation continuous if the inverse image under of any open set in is an open set in (i.e whenever ).

Lemma 2.19 [7]

The identity permutation in symmetric group is a permutation continuous on a permutation space .

Lemma 2.20 [7]

A composition of permutation continuous functions is permutation continuous.

Remark 2.21 [7]

A base for a permutation topological space is a sub-collection of such that each member of can be written as , where each belongs to . Further, the subbase of such that each proper open set of can be written as a union of finite intersections of elements of . In another word, the family of open sets consisting of all finite intersections of elements of , together with the set , forms . Let be the collection of permutation topological spaces. Then subbase for the product permutation topology is given by , so that a base can be taken to be .

Definition 2.22 [7]

Let be permutation topological space for each index . The product permutation topology on the set is the coarsest permutation topology on making all the projection mappings permutation continuous.

Lemma 2.23 [7]

If the spaces are permutation topological spaces, then have a countable base.

Remark 2.24

If is an algebraic (a topological) property, we say that the topological group has property , if the group (the topological space ) has property .

Definition 2.25 [13]

Let be a group, and be subsets of , we let and denote and . The subset is called symmetric if .

Definition 2.26 [7]

(Permutation subspaces):

Suppose permutation space, and , for each proper , then

Let nonempty open set}. For each , let and . Suppose , and , then we have this set has exactly points where where . Here we used normal intersection between pairwise sets to find the set . For each we have is cycle in . Then {, } are disjoint cycles decomposition of new permutation in symmetric group induced by say .

Definition 2.27 [7]

Let be a permutation space and , then we denote to permutation subspace of by where , and .

Definition 2.28 [7]

(Connectedness): Let be permutation topological space. The collection of sets is said to be a decomposition of the set if and if the members of are all nonempty and pairwise disjoint cycles in . Then is called decomposition of we also say that has been decomposed into the sets of . Assume the permutation topological space has been decomposed into two open sets and . In this form the permutation space is called disconnected.

Definition 2.29 [7]

A permutation space and its topology are both said to be connected if cannot be decomposed into two open sets. A subset of is said to be connected whenever the permutation subspace is connected, and is said to be disconnected if is decomposed into two open sets.

3. New Notations in Permutation Topological Space

Let be permutation topological space. Each set in the permutation space is either open or closed. Therefore in this paper we will deal with any subset of in as set. That means we can put . However where disjoint cycles of also we denote to its cycle by and hence in this paper after we give some new definition we consider that all the notations and definitions are hold except it is not necessary everyset in the permutation space is either open set or closed set.

Definition 3.1

Let and be two subset of . 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 .

Definition 3.2

Let and be similar sets in and , , where . Then if and . Also, , and .

Definition 3.3

For any and two subset of .

Then,

and

Remark 3.4

In permutation topological space any subset such that and is called an open set iff . Also, it is called closed set iff .

4. Permutation Topological Group

Definition 4.1

Let be a permutation topological space. Then is called Permutation Single Space (PSS) if and only if each proper open set is a singleton.

Definition 4.2

Let be a permutation topological space. Then is called Permutation Indiscrete Space (PIS) if and only if each open set is trivial set.

Definition 4.3

Given permutation topological spaces and , a function is permutation open map if the image under of any open set in is an open set in .

Lemma 4.4

Let be a permutation topological space. Then is permutation single space (PSS) if and only if .

Proof:

Suppose that is a (PSS). Then each proper open set is a singleton. That means, , for some . Let , then (since ), and hence . However, , where , , . Then for some . This implies that contains more one element, but this contradiction since is an open set and each open set is singleton. Therefore we consider that .

Conversely, if . Then we consider that . However, , where , , . Then for all . This implies that contains only one element for each , but . Thus each proper open set is a singleton and hence is (PSS).

Lemma 4.5

Let be a permutation topological space. Then is permutation indiscrete space (PIS) if and only if .

Proof:

Suppose that is a (PIS). Then each open set is trivial set and hence . This implies that . Hence , where . Then .

Conversely, if . Then we consider that and hence , but . Then and this implies that is (PIS).

Definition 4.6 [12]

(Multiplication Permutation Map)

Let and be two permutations in symmetric group . Then and are two permutation maps from onto . Further, is a product map of permutation maps where . In another side, the map is a permutation in as this form

Now, let be a binary operation on and be a map defined by . Then the permutation map from permutation space into for any permutation in symmetric group is called multiplication permutation map. Further, it is called multiplication permutation continuous iff the inverse image under of any open set in is an open set in (i.e whenever ).

Example: 4.7 Suppose that and are permutations in symmetric group with , and let be a binary operation on where ,. We consider that the multiplication permutation map , where is a multiplication permutation continuous map.

Remark 4.8

By above example we consider the following:

(1)-For any , if . Then there is a multiplication permutation continuous map from permutation space into satisfies .

(2)-For any , the mathematical system is a commutative group.

(4)-For any and in , the multiplication permutation map such that:

(5)-For any , there is an inversion permutation map such that .

Where with

Lemma 4.9

For any even positive integer , the commutative group has proper symmetric subgroup.

Proof:

Let , then for any even positive integer . This implies that the set is a proper subset of . New, to prove that is a symmetric subgroup of it is enough to show that . That means is a group with the following table:

Since . Then and hence . Therefore is a proper symmetric subgroup of .

Lemma 4.10

For any even positive integer , the commutative group has proper normal subgroup.

Proof:

By lemma (4.9) we consider that is a proper subgroup of , where

and . Now, we need to show that is a normal. In other words, we want to prove that , for any , . It is clear if , then . Also, if , we have and . Then . Since Then we consider that put . Therefore, we getThus, or x+g , for any even positive integer . This implies that Then . Hence is a proper normal subgroup of .

Definition 4.11

Let be a permutation topological space and be a group. Then we say that is a permutation topological group (PTG) if and the multiplication permutation map is multiplication permutation continuous map and the inversion permutation map is permutation continuous map.

Example 4.12

Let be a permutation in symmetric group . Then is permutation topological space, where and . Also, let be a group with the following table:

Table (1)

It is clear that is an indiscrete permutation space. Thus is multiplication permutation continuous map and is inversion permutation continuous map, . Then is a permutation topological group.

Lemma 4.13

Let be a permutation function. Then,

(a) is a permutation continuous and permutation open map, if is (PIS).

(b) is a permutation continuous and permutation open map, if is (PSS).

Proof:

(a) Let be a (PIS). Then each open set is trivial set and hence . It is clear and . Also, and (since each permutation map is bijection). Then is a permutation continuous and permutation open map

(b) Let be (PSS). Then each proper open set is a singleton. This implies that, we have , for some . In another side, if is not singleton for some proper open set . That means this map send one point to more than one point and hence is not permutation, but this contradiction. Therefore, for any open set in we consider that , for some . Also, by similarity we consider that , for some . Thus and are open sets. Then is a permutation continuous and permutation open map.

Definition 4.14

Let be a permutation function, then is called a permutation homeomorphism if it has the following properties:

(1)-is a bijection,

(2)- is permutation continuous,

(3)- is permutation continuous ( is permutation open map).

Definition 4.15

A permutation topological space is called a permutation homogeneous space (PHS), if for any there exists a permutation homeomorphism such that .

Example 4.16

Let be an identity permutation in symmetric group . Then is permutation topological space, where and . It is clear that is (PSS). Define as follows: for any , let . Therefore we get is a permutation in symmetric group and such that . Moreover, is a bijection map (since each permutation is bijection). Also, is a permutation continuous and permutation open since is (PSS). Then is a permutation homogeneous.

Definition 4.17

A permutation topological group is called a permutation homogeneous topological group, if its permutation space is a permutation homogeneous.

Remark 4.18

Let be a permutation topological group, and , Define , , . Then the map is a permutation homeomorphism. Also, define , . Then the map is a permutation homeomorphism.

Lemma 4.19

Every permutation topological group is a permutation homogeneous topological group.

Proof:

Let be a permutation topological group, we need to show that it's permutation space is a permutation homogeneous. That means we have to show that for any there exists a permutation homeomorphism such that . Since is permutation topological group. Then there is a permutation homeomorphism such that , and hence for any there exists a permutation homeomorphism such that . Moreover, we consider that this permutation homeomorphism such that and . Hence is a permutation homogeneous topological group.

Lemma 4.20

Let be a permutation topological group, and is a subset of . Then is an open set if and only if is an open set.

Proof:

Since the map is a permutation homeomorphism. Then the proof is obvious.

Lemma 4.21

Let be a permutation topological group, and . Then is an open set if and only if is an open set.

Proof:

Since the map is a permutation homeomorphism. Then the proof is obvious.

Theorem 4.22

A permutation topological group is an Lindelof permutation topological group.

Proof:

Let be permutation topological group where , and , then for each we have the proper open set is a countable set, and for each base for permutation space we have where , but is a countable set (each finite set is a countable), (see Runde, 2005), so is a countable base, since only the union of a countable collection of a countable sets is countable. Therefore permutation space with countable base, then we have permutation space is an Lindelof space (see Bourbaki; 1989. Page 144). Hence is an Lindelof permutation topological group.

Remark 4.23

If is a collection of permutation topological groups, then is an Lindelof permutation topological group.

Definition 4.24

Let be a permutation topological space, and . The connected component of in is the largest connected subset of containing .

Example 4.25

Let be a permutation in symmetric group . Find connected component of in permutation topological space .

Solution:

where . Hence is a permutation topological space, let be the family of all subsets of which are contain point . Then we consider that each one of the permutation subspaces{} of is decomposed, for all into two open sets and and hence are disconnected, where , for all , and . Further only is connected where . Hence is connected component of 3 in permutation topological space . In another side, is not connected component of all its points and then is not connected.

Definition 4.26

A permutation topological group is connected topological group iff is connected component of all its points.

Example: 4.27 Let be a permutation topological group in example (4.12). Then is connected component of all its points and hence the permutation topological group is connected topological group.

Lemma 4.28

Let be a permutation in symmetric group . Then the connected component of in permutation topological space is , if .

Proof:

Since , then every proper open set in is a singleton (i.e, satisfies ). Moreover, for any subset of , we have . Now, we looking for the largest subset of contains 1 with permutation subspace is connected. Thus we first discuses with . Here

If . Then is a collection of all non-empty open sets and such that , for any and . Also, If . Then is a collection of all non-empty open sets and such that , for any . Then, cannot be decomposed into two open sets. Because there exist no two open sets are disjoint sets. Also, for each subset of with we have . Therefore, is the largest connected subset of containing 1. Then the connected component of in permutation topological space is .

Definition 4.29

A permutation topological space is said to be if for any two distinct points , there is an open set in such that , but .

Definition 4.30

A permutation topological space is said to be if for any two distinct points , there are two open sets , in such that and .

Example 4.31

Let be a permutation topological space in example (4.16), where and . It is clear that is (PSS) and hence each singleton set is an open set. Then, for any two distinct points , there are two open sets , in such that , …(1) and …(2). Hence from (1) we get is . Also, from (1) and (2) we have is .

Lemma 4.32

Let be a permutation in symmetric group . Then if and only if is space.

Proof:

Assume , then by lemma(4.4) we have is a (PSS) and hence any singleton set is open set. Hence for any two distinct points , there are two open sets in such that and .

Conversely, suppose that is space and . Hence , for some (since ), and hence . However, , where, , . Then for some . This implies that contains more one element. That means there are two distinct elements . However, is space, then there are two open sets , in such that , and ,. Thus . But this contradiction since the cycles for any pair of open sets are disjoint and hence we consider that . Then .

5. Extension Permutation Topological Space (EPTS)

Suppose that is a permutation topological space. Now, we define new set by Here we used the normal union between open sets to generate the new topology on with two operations and (see definition 3.3). In another side,

Definition 5.1

Let be a permutation topological space. Then is called an Extension Permutation Topological Space (EPTS), and each is called an Extension Permutation set and denoted by set.

Example 5.2

Let be a permutation in symmetric group . Hence is a permutation topological space, where and . Thus is (EPTS), where . Moreover, and are all closed subset of for example and are similar sets and (since). Further, and are disjoint sets, thus neither nor . In another side, and

Remarks 5.3

(1) For any permutation in symmetric group there is extension permutation topological space (EPTS).

(2) If is open (closed) set in then is open (closed) set in However, the converse is not true in general.

(3) Any pair of subsets are similar (disjoint) sets in if and only if they are similar (disjoint) sets in

(4) Any pair of subsets are disjoint sets if and only if their complements are disjoin sets or disjoin sets.

(5) If are similar sets in or similar sets in then their complements it is not necessary to be similar sets or similar sets.

Example 5.4

Let be a permutation topological space in example (5.2), where and Thus is (EPTS), where Let Then and their complements are similar sets in and similar sets in However, are similar sets in and similar sets in but their complements are neither similar sets nor similar sets.

Lemma 5.5

Let be a permutation topological space. Then is (EPTS) if

Proof:

Let be a permutation topological space and Then is (PIS) by lemma (3.5). This implies that However, thus Then is (EPTS).

Lemma 5.6

Let be a permutation map. Then,

(a) is a permutation continuous, if is permutation continuous map.

(b) is a permutation open, if is permutation open map.

(c) is a permutation homeomorphism, if is permutation homeomorphism.

Proof:

(a) Suppose that is a permutation continuous map. Let , then , for some but is a permutation continuous map, thus and hence Since Then this implies that Hence is a permutation continuous map.

(b) Assume is a permutation continuous map and let Then for some but is a permutation open map, thus and hence Further, since Then this implies that Hence is a permutation open map.

(c) By (a) and (b), it is clear the proof is obvious.

Definition 5.7

A permutation topological space is said to be if for any two distinct points there is an open set in such that but

Definition 5.8

A permutation topological space is said to be if for any two distinct points there are two open sets in such that and

Definition 5.9

A permutation topological space is said to be if for any two distinct points there are two open disjoint sets in such that and

Definition 5.10

Let be (EPS) and be a group. Then we say that is an Extension Permutation Topological Group (EPTG) if and the multiplication permutation map is multiplication permutation continuous map and the inversion permutation map is permutation continuous map.

Lemma 5.11

If is (PTG), then is (EPTG).

Proof:

Suppose that is (PTG). Then there are two permutation continuous maps and such that and . By lemma (5.6) we have and are permutation continuous maps and hence is (EPTG).

Definition 5.12

Let be (EPTS). Then is called an Indiscrete Extension Permutation Topological Space (IEPTS) if and only if each open set is trivial set.

Definition 5.13

Let be (EPTS). Then is called a discrete Extension Permutation Topological Space (DEPTS) if and only if each subset in is open set.

Remark 5.14

Let be (EPTG), then is said to be group iff is space. Also, is said to (DEPTG) [(IEPTG)] iff is (DEPTS) [(IEPTS)].

Lemma 5.15

Let be a permutation topological space. Then is (DEPTS) if and only if

Proof:

Suppose that is a (DEPTS). Then for each we have and hence is an open set [since is a (DEPTS)]. That means there are two open sets and such that but is singleton and this implies that either or or That means each open singleton set is open set. Then is a (PSS). Thus by lemma (4.4) we have

Conversely, if Then by lemma (4.4) we have is a (PSS). Thus each singleton is an open set and hence an open set. For any subset of and we have Therefore, is an interior point of thus and hence but in general Thus and hence That means any subset of is open set. Hence is (DEPTS).

Lemma 5.16

is (DEPTS) if and only if is (PSS).

Proof:

Let be (DEPTS). Then by lemma (5.15) we have and hence by lemma (4.4) we have is (PSS).

Conversely, if is (PSS), then by lemma (5.15) and Lemma (4.4) we have is (DEPTS).

Lemma 5.17

Every permutation topological space is if and only if its extension is

Proof:

Suppose that is Then for any two distinct points there are two open sets in such that and However, every open set is open set. Then is

Conversely, if is Then for any two distinct points there are two open sets in such that and Moreover, there are open set and such that and Thus & & & Hence, there are four cases cover all probabilities which are holed as following:

However, & Then is space.

Lemma 5.18

Let be an identity element in extension permutation topological group then is a topological group if and only if is open set.

Proof:

Let be a group, then by lemma (5.17) we have is a and hence [by lemma (4.32)]. Then is a (PSS) [by lemma (4.4)]. Hence any singleton set is open set. Then is open set and hence is open set [ since each open set is open set].

Conversely, suppose that is open set. That means there are two open sets and such that , but is singleton and this implies that either or or That means each open singleton set is open set. Then by Lemma (4.21) we have is open set for any because is open set. Hence is (PSS) and hence [by lemma (4.4)]. Therefore is space [by lemma (4.32)]. Then is [by lemma(5.17)].

Lemma 5.19

Let be extension topological group. Then is , if is

Proof:

Let be a topological group, then for any two distinct points there is open set in such that and Define Then the map is a permutation homeomorphism. Put and then is open set in and Then is open set in such that Thus (since ). Let and where Now, if This implies that and for some Thus But this contradiction since the cycles for any pair of open sets are disjoint and hence we consider that Then is a topological group.

Lemma 3.20

If is (DEPTG), then is group.

Proof:

Assume is (DEPTG). Then is (DEPTS). Let be any two distinct points in Then, either or Thus, if Let and Hence are two open sets [since is (DEPTS)]. Also, Then there are two open disjoint sets in such that and Also, if we have and are two open disjoint sets in such that and Hence is group.

Definition 5.21:

(Connectedness)

Let be extension permutation topological space. The collection of sets is said to be a decomposition of the set if and if the members of are all nonempty and disjoint sets. Then is called decomposition of we also say that has been decomposed into the sets of Assume the extension permutation topological space has been decomposed into two open sets and In this form the permutation space is called disconnected. Moreover, and its topology are both said to be connected if cannot be decomposed into two open sets.

Lemma 5.22

Let be permutation topological space. Then is connected, if its extension space is connected.

Proof:

Suppose that is connected. Then cannot be decomposed into two open sets. That means for any pair of non empty open sets we have and hence for any we have . Thus cannot be decomposed into two open sets. Then is connected.

Definition 5.23

An extension permutation topological group is called connected topological group, if is connected.

Lemma 5.24

If is (DEPTS), then is disconnected space.

Proof: Assume is (DEPTS). Then there are two open disjoint sets and , where [since and is (DEPTS)], and . Thus is decomposed into two open sets and hence is disconnected space.

Lemma 5.25

If is open set, where is an identity element in extension permutation topological group , then is disconnected topological group.

Proof:

Assume is open set. Then by lemma (5.18) we get is a topological group and hence by (5.17) we have is . This implies that [by lemma (4.32)]. Then is (DEPTS) [by lemma (5.15)]. Hence is disconnected space [by lemma (5.24)].

Example 5.26

Let be an identity permutation in symmetric group . Thenis (DEPTS), where {1,2,3,4,5,6,7,8,9} and . Also, let be a group with the following table:

Table (2)

Thus , and , are permutation continuous maps. Then is group, group, group and disconnected group.

Remark 5.27

Finally, our new notations are given and hence these notations of permutation topological group can be considered a special case of topological group using permutation in symmetric group.

6. Conclusions

In this paper, the concepts of permutation topological groups, extension permutation topological groups, permutation homogeneous topological group, Lindelof permutation topological group, -connected group, -disconnected topological group, (EPTG), (IEPTG), (DEPTG), ( group), ( group), ( group) and others are introduced. Assume is permutation space and , where is a d-algebra (resp. BCK-algebra, BCL-algebra). The question we are concerned with is: what is the possible conditions we need to be is permutation topological d-algebra (resp. permutation topological BCK-algebra, permutation topological BCL-algebra).?