RANDOM LINEAR NETWORK CODING BASED MULTIPATH TRAFFICS OVER HETEROGENEOUS CLOUD RADIO ACCESS NETWORK

2445

AN

ALGORITHM

FOR

GENERATING

PERMUTATIONS

IN

SYMMETRIC

GROUPS

USING

SOFT

SPACES

WITH

GENERAL

STUDY

AND

BASIC

PROPERTIES

OF

PERMUTATION

SPACES

1_{SHUKER MAHMOOD KHALIL,} 2_{FATIMA HAMEED KHADEER}

Department of Mathematics, College of Science, Basrah University, Basrah 61004, Iraq. E-mail: [email protected]

ABSTRACT

In this paper we introduced algorithm to find permutation in symmetric group using soft topological space to structure permutation topological space. Moreover, this class of permutation topology is called even (odd) permutation topology if its permutation is even (odd). Further, new notions in permutation topological spaces are investigated like splittable permutation spaces and ambivalent permutation spaces.

Keywords: Soft Set Theory, Symmetric Groups, Splitting, Ambivalent, Permutation Topological Spaces. MSC 2010: 54A10, 54C10, 54A05, 20G05, 20D06, 20B30

1. INTRODUCTION

Soft sets were originally shown by Molodtsov [1]. Then they applied in many fields where they have rich possibility for applications. In 2011, the connotation of soft topological spaces (STSs) is shown by Muhammad and Munazza [2]. Some notions of (STS) and of its applications fundamental connotations of fuzzy soft topology and Intuitionistic fuzzy soft topology are studied by many mathematicians see ([3-9]). In 2014, Mahmood [10] introduced the notion of permutation topological space (PTS) using permutation  in symmetric group S_n, where

each permutation



S_n can be represented as a product of disjoint (separate) cycles. In other

words, 1 )( ₁2, ₂2,

1 ,..., 1 2 , 1 1

(b b b_ b b

  2 )

2 ...,b_

, ) ( 1

...(bc  b₂c(),..., ( ) )

) (

  c_c

b and



i j

satisfy {₁, ₂,..., _ } { ₁, ₂,..., _j } j b j b j b i

i b i b i

b _

[11]. That means,  can be represented as

) ( ... 2 1  

 _c , where



_i separate cycles of

length



_i





_i and c() refers to the number of the product of separate cycles with the 1-cycles

of . In 2015, some methods are introduced to generate fuzzy soft set and intuitionistic fuzzy soft set using different sets [12]. Now, the interesting question is there any an algorithm shows the relation between permutation space and soft space. In this work, this algorithm is given. We generate permutation topological space using soft topological space (STS). This class of permutation topology is called even (odd) permutation topology if its permutation is even (odd). Further, new notions in permutation topological spaces are investigated like splittable permutation spaces and ambivalent permutation spaces.

2. PRELIMINARIES AND BASIC RESULTS

We will show some past results and basic definitions in this section.

Definition 2.1 ([13])

A "cycle type" of



is the partition 

2446 Definition 2.2 ([13])

of cycle n

S ations in We refer to all the permut

.  C by  type

Definition 2.3 ([13])

Let  _{be a permutation in Alternating group}

n

A and



C



. A() _{conjugacy class of}  in

n

A is defined by

  

 

{ | 1;for some } )

( A_n t t t A_n

A    

   

 





) if

( ,

) if

( ,

n H C

or C

n H C

 



 

where C are two classes of equal order in

Alternating group

A

_n such that

 

  



C C

C  and H_n {

C

 of S_n|

1



n

, with all parts



_k of



different and odd}.

Proposition 2.4 ([14])

The conjugacy classes

C

 of

A

n are

ambivalent if 4|(



_i1) for each part _i of .

Definition 2.5 ([10])

Let  S_n with {1,2,...,n} and the

"cycle type" of  is ()(₁,₂,...,_c()),

and {_i}c_i(₁) be a composite of "pairwise separate cycles" of  where _i (b₁i,b₂i,

), ..., i

i

b_ 1ic(). Each  (b₁,b₂,...,b_k), 

k cycle in S_n we put  set as

}

,...,

,

{

b

1

b

2

b

k







and is said to be  set of

cycle  . Also,  sets of {_i}_ic_(₁) are investigated by { {₁, ₂,..., i }|

i b i b i b

i 



 

)} ( 1ic  . Remark 2.6

Suppose that



_i _and



_j are  sets in,

where  _i  and _j . Then the known definitions will be written as following:

Definition 2.7 ([10])

Let



_i _and



_j be  sets in, they are called separate iff there exists 1d , for each

   r

1 such that bi_d b_rj and

   



 

1

1 k

j k b k

i k

b .

Definition 2.8 ([10])

Let



_{i and}



_j be  sets in. We say

they are equal iff there exists 1d , for each

   r

1 such that bi_d b_rj.

Definition 2.9: ([10])

We say



_{i is contained in}



_j and denoted





_



_i ˆ _j,    



j

k j k b i

k i k b

 

1 1

iff

.

Definition 2.10: ([15])

For any {b₁,b₂,...,b_r} and  {a₁,a₂,

} ,a_

 two subsets of , we call



 and  are similar sets in, iff

   





1 1 k ak r

2447 and there are two points say

b

_i

,

b

_j





such

that b_i  and b_j. Assume

 _} , , 2 , 1

{b b _ b_r 



 _,  {a₁,a₂,,a_}

are similar  sets in  and

       }, { {

{Max Max

Max 



 }}, 

_where

  }. , , 2 , 1 { } , , 2 , 1 { _

 b b  b_r  a a  a

 

Then  ˆ, if







. Also,

 _,

if ˆ        







      _            if if , , and  







      _            if if , ,

For any  {b₁,b₂,_,b_r} and  

} , , 2 , 1

{a a  a_ two subsets of . Then,

















                  disjoint are & f , if , ) and similar are & ( Or ) 1 1 ( if , ) and similar are & ( Or ) 1 1 ( if ,                                    i k ak r

k bk k ak r

k bk

and

























                disjoint are if f and similar are Or 1 1 if and similar are Or 1 1 if β &η β λ Ω, β μ β η β λ i , β μ ) β η Δ β &η β (λ ) υ

k ak r

k bk ( , β η ) β λ Δ β &η β (λ ) υ

k ak r

k bk ( , β λ β η β λ

Definition 2.11: ([10]) of not family a be I i i i b i b i

b₁, ₂,..., }}_

{ _   _i { Let ) union ( intersection We define the

. sets   rate sepa , where  

_



i j

I i 

by respectively, I

i i

}



{



 of

,





_



i j

I i 

and

} ; 1 inf{

1 i I

i k i k b j k j k

b  

      . } ; 1 sup{

1 i I

i k i k b j k j k

b  

      where

Definition 2.12: ([10])

Assume  is permutation in S_n, and let

) (

1

}

{



c

i

i  be a family of product of separate

cycles with the 1-cycles of , where _i 



_i,

) (

1ic  _{, then permutation topology}

t

_n is a

family of  sets of the collection {_i}c_i(₁)

together with  {1,2,...,n} and empty set. We

say (,t_n) is permutation space.

Definition: 2.13: ([15])

Assume (,_n)is a permutation topological space. We say (,_n) is a Permutation Single Space (PSS) iff all their proper open sets are singleton.

Definition: 2.14: ([15])

Assume (,_n)is a permutation topological space. We say (,_n) is a Permutation Indiscrete Space (PIS) iff each open  set is trivial set.

Definition 2.15: ([1])

2448 and K is a subset of E. We say (F,K)is a soft set overX if F is a multi-valued function of

Kinto P(X).

Definition 2.16 ([16])

Assume (F,A)and (M,L) are soft sets over X , their union (H,C) is a soft set such that for all eC A L, H(e)F(e) if

L A

e  , H(e) M(e) if eLA,

if ) ( ) (e M e

F  eAB_{. we write} (F,A)

) ,

(M L



(H,C). Also, their intersection is a soft set, denoted by (H,C) (F,A)  (M,L)

where C  AL, and it is defined as

) ( ) ( )

(e F e M e

H   for all eC.

Definition 2.17 ([2])

A soft set (G,B)c is the complement of

) ,

(G B in (X,E) _{and it is defined by} (Gc, K),

where K  E\{eB |G(e)  X}and eK  





  

. ,

), ( \ ) (

B e if X

B e if e G X e c G

Definition 2.18: ([2])

Assume is a family of soft sets over X . We say  is a soft topology on X if  satisfies the following axioms:

(i)The intersection of any two soft sets in  belongs to the family.

(ii) The union of any number of soft sets in 

belongs to the family. (iii) (X,E)and belong to the family .

We say (X,E,) is a soft topological space (STS) overX . Also, each member in the familyis said to be a soft open set and its complement is said to be a soft closed set. Further, (X,E,) is said to be a soft topological indiscrete space (SITS) _over X , if

)} , ( , { X E 

 . Also, (X,E,) is called a soft discrete topological space (SDTS) overX , if  contains of all soft sets overX .

Some Results on Permutations 2.19:([17, 18])

(1)

 (b₁,b₂,...,b_r) iseven



r

is odd.

(2)

 S_n iseven  nc() is even,

(3)

  (1) ( Identity )  (b)for some b .

Remark 2.20:

In this work, for any set

} , , 2 , 1

{d d d_k

D _ of

k

distinct objects and for

any cycle B(b₁b_2b_m)we will use B m and D k to refer to the length of the cycle

B

and the cardinality of set D.

3. AN ALGORITHM TO GENERATE PERMUTATION SPACES FROM SOFT SPACES

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

Steps of the work 3.1:

Assume (X,E,) is a (STS), where

, 2 , 1 {s s

X  _,s_k}, E {e₁,e₂,...,e_n}and

, 1 )} 1 ( { 1 let Now, }. 1 )} , {( ), , ( ,

{ X E F_i E _im_ T  F_i e _im_

 

. 1 )} ( { , , 1 )} 2 ( { 2

m i n e i F n T m i e i F

T  _   _ For

any 1 i  n, let T_i{BT_i /B }. (1in), _let _i : X  N _{be a map from} X into natural numbers N defined by

, ) 1 ( )

(s_j j i k

i   

 for all s_jX and (1in)

where k X . Then _i

n

i 



1

 

 is called

permutation in symmetric group S_nk where for

all 1in, )

2 ( ) 1 ( (

1 i sg i sg

i T

L

i  





  

)) (

L g s

i _



 is permutation which is product of



i

2449

 

 T_i

L g s g s g s

L { ₁, ₂, , _ }

  and 1L

 i

T . Further,  _i (_i(s_i)) if T_i . Then



, )

( t_h is called permutation topological space

induced by soft topology , where

} , , 2 , 1

{ _ h



 , hnk and t_h is a family of



 set of the family {_i}n_i1 together with and empty set. Also, if (X,E,) is a soft indiscrete space. Then (,t_h )_ is called permutation

indiscrete space (PIS) induced by soft topology , where  (1 2 3_h)_{. Finally, for any}

) , ,

(X E  non-indiscrete (STS), where

}, 1 )} , {( ), , ( ,

{ X E F_i E _im_ 

 E{e₁,e₂, ,e_n}

and X{s₁,s₂,,s_k}, we can generate permutation topological space (,t_h )_as

follows:

(1)- Find T_i {F_j(e_i)}m_j1,for all 1in,

(2)- Find T_i{T_i /}, for all 1in, (3)- Find _i(s_j) j(i1)k,  (1 jk) _and

 (1in),

(4)- Find _i for all 1 i  n, where

)), ( ( _i s_i i





   if T_i  and 

)), (

) 2 ( ) 1 ( (

1 i sg i sg i sg _L

i T

L

i    _

 _



 

  

where if T_i 

 

 T_i

L g s g s g s

L { ₁, ₂, , _ }

 

(5)-Find ,

1 i n

i  

  

(6)-Find the "separate cycle factors including the 1-cycle" of  _say



1



2

...



c()

,

(7)-Find the  sets of {_i}c_i(₁)

(8)- Find , where {1,2,h}andh nk,

(9)- Find

t

_h , where t_h {,,₁,₂, }

) ( ,_c

 .

(10) Then (,t_h )_ is (PTS).

Remarks 3.2:

(1) For all 1  i  n_{, let} T_i { ,X}T_i

*

 .

Here we used normal union (), normal intersection () and empty set (



) then we have

) * ,

(X T_i is a topological space for each (1in).

(2) We consider that )

/ * , ( ) * , (

: X T_i X_i T_i

i 

 is an

isomorphic, where X_i_i(X) and

/ *

i

T }.

* ); ( |

{Y Y _i Z ZT_i

  In other words,

1tmk, we have _i1(t)s_c, where

) 1 (  

t k i

c .

(3) For any (1i qn)and(1 j k), we have_i(s_j)X_q[Since_i(s_j)  j(i1)k  j

, ) 1 (q k

 j1,2,..,k]

(4) If 

} { / * ) (

i X i T j s i

  





 , for some

) 1

(  i  n

and (1 jk). Then 

} { / * ) (

q X q T j s i

  





 for all

) 1

( qn .

Permutation Subspaces Induced by Soft Topology 3.3

Let (,t_n)_ be a permutation space induced

by soft topology _,   and 

   

i i

T   , for each proper _i t_n,











separate are & if ,

separate not

are & if }, ,..., 2 , 1 {

    

    

i i i

2450 Let  {T_i |T_i nonempty open set}. T_i



 , let {₁, ₂,..., i }

k i b i b i b Max i k

b  and mMax{b_ki;

}

 



i

T . Let _

       i T i T

B ( ) where

} ,..., 2 , 1 { m 

 and (

_

) is a normal intersection.

T_i  we consider ( ₁, ₂,..., i ) k i b i b i b i

T  is

 k

i cycle in S_m. In other words, the product of separate cycles of other permutation in symmetric group S_m induced by  _say



 _where



 



     t r br i

T i T

1( )



and







    i T i T

wheneverB. Moreover, we say (, )_    m t

is a permutation subspace induced by soft

topology  where

  

m

t

is a family of all

  

 sets of product of separate cycles of 



 together with  and empty set.

Lemma 3.4:

If each pair of different members in



are separate, then B{b₁,b₂,...,b_t}has exactly t

points where tms and

T

s

i T i





   . Proof:

Suppose that T_i _T_j , for any

    j T i

T , and B{b₁,b₂,...,b_k}withk t

where t ms and s

i T i T    

 . Since

B         i T i T )

( where {1,2,...,m}. Thus

|

|B |  ( )|

      i T i

T . sinceT_i _T_j , for

any T_i,T_j . This implies

that|B|   

     |  ( )| | |   i T i T t s m i T i

T   

  

 . But |B| k t and this

contradiction. Therefore B{b₁,b₂,...,b_t}has exactly t points.

Example 3.5

Let the set of cars under consideration be



X {a₁, a₂, a₃}.Let E{cheap (e₁); dark

color (e₂); modern (

e

₃); beautiful (e₄ )} be set parameter set. Now, to buy a good car. Let

) ,

(F A be soft set describing the Mr.

Z

opinion

and it is defined by

} 4 , 3 , 1 {e e e A , } 3 , 1 { ) 1

(e a a

F  F(e₃){a₂},F(e₄) X Further, we suppose that the good car in the opinion of his friend, say Mr.W, is described by

) ,

(G B and it is defined by

} 4 , 1 {e e B , } 3 { ) 1 (e a

G  G(e₄){a₂,a₃}

We have: )} , ( ), , ( ), , ( ,

{ U E F A G B 

 is a soft topology.

Find permutation space (,t_n )_ induced by

soft topology. Also, find (₁, )_

   m t and 

₂, )

(

   m

t where  {1,4} and

}. 12 {   

Solution: We consider that

     )} 4 ( ), 4 ( { 4 )}, 3 ( ), 3 ( { 3 )}, 2 ( ), 2 ( { 2 )}, 1 ( ), 1 ( { 1 e G e F T e G e F T e G e F T e G e F T       }} 3 , 1 { , { 4 }}, 3 , 2 , 1 { }, 3 , 1 {{ 3 }, , { 2 }}, 3 { }, 3 , 1 {{ 1 a a T a a a a a T T a a a

T  

       

 {{₁, ₃},{₃}}, ₂ , ₃ { ₂}, ₄ { ,{ ₂, ₃}} 1 a a a T T a T X a a

T  }}, 2 { , , { * 3 }, , { * 2 }}, 3 , 1 { }, 3 { , , { *

1 X a a a T X T X a

T      

, , { *

4 X

2451 a topological space for each (1i4).Now, we

consider that ) / * ,

(X_i T_i is a topological space for each (1in).

where ) ({1,2,3},{ ,{1,2,3},{3},{1,3}}), /

* 1 , 1

(X T  

}}), 6 , 5 , 4 { , { }, 6 , 5 , 4 ({ ) / * 2 , 2

(X T  

}}), 8 { }, 9 , 8 , 7 { , { }, 9 , 8 , 7 ({ ) / * 3 , 3

(X T  

}}). 12 , 11 }{ 12 , 11 , 10 { , { }, 12 , 11 , 10 ({ ) / * 4 , 4

(X T  

Moreover, for all 1in,

)) ( ) 2 ( ) 1 ( ( 1 L g a i g a i g a i i T L

i    _

 



 

 is permutation

which is product of T_i cyclic factors of the

length _L , where _LT_i and 1L T_i .

Hence we have

, ) 3 )( 3 1 ( )) 3 ( 1 ))( 3 ( 1 ), 2 ( 1 (

1  a  a  a 

 ), 5 ( )) 2 ( 2 (

2



a 



), 8 ( ) ) 2 ( 3 (

3  a 

 ). 12 11 10 )( 12 11 ( )) 3 ( 4 ) 2 ( 4 (

4   a  a 

 Then     _i i   4 1  ) 12 11 10 )( 12 11 )( 8 )( 5 )( 3 )( 3 1 (













10 11 12 9 8 7 6 5 4 1 2 3 12 11 10 9 8 7 6 5 4 3 2 1

is a permutation in symmetric group

S

₁₂. Now, we can write





S

_n as



₁



₂

...



c₍₎. Hence

 

_











10 11 12 9 8 7 6 5 4 1 2 3 12 11 10 9 8 7 6 5 4 3 2 1  ) 12 10 )( 11 ( ) 9 ( ) 8 ( ) 7 ( ) 6 ( ) 5 ( ) 4 ( ) 2 ( ) 3 1 ( .

Therefore (,t_n ) is a permutation space induced by soft topology



,

where



n

t

=t₁₂ {,,{1,3},{2},{4},{5},{6},{7},{8},

}} 12 , 10 { }, 11 { }, 9 { and }. 12 , 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 { 

 Now to find the

permutation subspace for, we consider that

}, 3 , 1 { 1 

T T₂ {2},T₃ {4},T₄ ,

}, 4 , 1 { 5 

T T₆ {1,4},T₇ {1,4},   

{1,4}, ₉ {1,4}, ₁₀ {1,4} 8 T T

T

 

 {{1,3},{2},{4},{1,4}}

 }} 4 , 1 { }, 4 { }, 2 { }, 3 , 1 {

{Max Max Max Max

Max

 

 m

Max {3,2,4} 4

}, 4 , 3 , 2 , 1 { 1    _       _   i

T Ti

B ( ₁ )

Then ) 2 )( 3 4 1 ( 3 1 2 4 4 3 2 1 ) 4 1 ( ) 4 ( ) 2 ( ) 3 1 (   

_











  

is a permutation in symmetric group

S

₄ induced

by







{

1

,

4

}

and ( ₁, ₄ )

  

t

 is a permutation subspace induced by soft topology



,

where

}} 4 , 3 , 1 { }, 2 { , , 1 { 4      

 _{t  }

m

t . Moreover,

lemma 3.4 is not hold for  {1,4}.

Since s

i

TTi 

 

 

 (13) (2) (4) (14) 21126a nd hence ms462. But

t



0

_(since





B ), thus t ms. That means, lemma 3.4

is not hold for  {1,4} (since there are two different members {1,3},{1,4}are not separate). Now to find the permutation subspace for







{

12

}

,

2452

}, 3 , 1 { 1 

T T₂ {2}, T₃ {4}, T₄ {5}, },

6 { 5 

T T₆ {7}, T₇ {8}, T₈ {9},

}, 10 { 9 

T T₁₀ {12}

}} 12 { }, 10 { }, 9 { }, 8 { }, 7 { }, 6 { }, 5 { }, 4 { }, 2 { }, 3 , 1 {{  



m 12   ₂  ,

T

s

i T

i





 



11

 and  ( ₂ ) {11}.



   

 _ 

i

T Ti

B This

implies that t B 1, also

. 1 11

12 

 s

m That means tms.

Further,

) 12 )( 11 )( 10 )( 9 )( 8 )( 7 )( 6 )( 5 )( 4 )( 2 )( 3 1 (



  

is a permutation in symmetric group

S

₁₂ induced

by   {12} and ( ₂, ₁₂ )_

  

t is a

permutation subspace induced by soft

topology



,

where

}, 7 { }, 6 { }, 5 { }, 4 { }, 2 { }, 3 , 1 { , , { ₂

12 

    

 _{  }

t m t

}, 10 { }, 9 { }, 8

{ {11},{12}}. Moreover, (lemma 3.4) is hold for  {12}(since each pair of different members in



are separate).

4. Some Notions of Permutation Spaces

Definition 4.1:

Assume (,t_n) is a (PTS), we say (,t_n) is even (odd) permutation space if its permutation

 _{is even (odd) in}S_n.

Example 4.2:

Assume (,t₉) is a (PTS), where

) 2 6 1 4 (



 (7 3 8 9)(5). Then (,t₉) is an even permutation space since S₉ is an even.

Definition 4.3:

Assume (,t_n) is a (PTS), we say (,t_n)

is splittable permutation space if its permutation satisfies H_n.

Example 4.4:

Assume (,t₁₀ ) is a (PTS), _where

) 10 8 1 7 3 4 6 9 5 (



 (2). Therefore,

we have (,t₁₀ ) is splittable permutation spacesinceH_n.

Definition 4.5:

Assume (,t_n) is a (PTS), we say (,t_n)

is ambivalent permutation space if (,t_n) is splittable permutation space and for each part _i

of () satisfies 4|(_i 1).

Example 4.6:

Assume (,t₆) be a (PTS), where (6 1 5 3 2) (4). Then (,t₆) is ambivalent permutation space since (,t₆) is

splittable permutation space and for each part _i

of () satisfies 4|(_i 1).

The Maps induced by soft topologies 4.7

Suppose that (X,E,₁), (X,E,₂) and

) 3 , ,

(X E  are three (STSs) over the common universe X and the parameter set E with their permutations ,  and  in symmetric group

n

S where n (X )(E). Hence, we consider that

2 ) , ( 1 ) , (

:  _  _

  t_n   t_n is a map, and 

} ,..., 2 , 1

{b b b_k

 

 set, () is said to be 

 set and it is defined as

)} ( ),..., 2 ( ), 1 ( { )

(  b  b  b_k

  . We say  is a

permutation map induced by soft topology ₃.

Definition 4.8

Suppose that (X,E,₁), (X,E,₂) and

) 3 , ,

2453 permutations ,  and  in symmetric group

n

S where n(X )(E). We say

2 ) , ( 1 ) , (

:  _  _

  t_n   t_{n is a permutation}

continuous induced by soft topology ₃, if 

 ) ( 1 _

 

n

t

whenever







t

_n. Theorem: 4.9

Let (X {x_j}k_j1,E{e_r}n_r1,) be a (STS). If for any pair _i  _i T_i

2

1 

 (1i n) such that

 

 

2 1 i

i  and 

} { / *

, )

(

i X i T j x i

  





 for

any (1in) and (1 jk). Then

 

  n

i Ti c

1 )

( , where (,t_nk )_ is a permutation space induced by soft topology .

Proof:

{ 1

Let

_i  , , } 2

,

1 xg xg _ii g

x

  ,

} 2 , , 2 , 1 {

2 xh xh xh i

i _

   T_i,(1 i  n) and

,

j i j i 

  j  1,2 . Since any pair

 

 _i T_i

i1 2

 (1in) such that

 

 

2 1 i

i  . Then( ( ) ( ) ( ))

1 2

1 g g_i

g i x i x

x

i   

 

)) ( ) ( ) ( (

1 2

1 g g_i

g i x i x

x

i   

  and( ( ) ( )

2 1 i h

h

i x  x



)) (

2

i h

i x_



 are separate cycles in symmetric

group

S

_nk [since ( ) ( )

r h x i t g x

i 

  , for any

]. X r h x t g x  

Also, for any  T_i

i g x g x g x

i { ₁, ₂, , _ }

 _ and

, } , , 2 , 1

{  

 T_l

l h x h x h x

l _

  (1 i  l  n).

Hence, we consider that

)) ( ) 2 ( ) 1 ( (

i g x i g

x i g x

i   _

  and

)) ( ) 2 ( ) 1 ( (

l h x l h

x l h x

l   _

 _ are separate

cycles in symmetric group S_nk [since

) ( ) (x_{j l} x_f

i 

  _{, for any} 1 i  l  n and

X f x j

x ,  ]. In other side,

nk S i

T

j i xg i xg i xg ij

n

i  



 



 )

1( ( 1) ( 2) ( ))

1(    

 

is a permutation for the permutation space



, )

( t_nk . Thus, we consider that  



n

i 1Ti is the

number of the product of separate cycles of

) 1( ( 1) ( 2) ( )) 1(



 



i T

j i xg i xg i xg i_j

n

i     . Further, c(



)

is the number of the product of separate cycles

with the 1-cycles of



. Thus c()  



n

i 1Ti in

general. Therefore either c()  



n

i 1Ti or

 ) (

c 

 

n

i 1Ti . If c()  

n

i 1Ti , then there is at

least 1-cycle say (_i(x_j)) for some (x_jX, )

1in with 

} { / * ) (

i X i T j x i

  





 and this

contradiction with our hypothesis. Hence 

) (

c 

 

n

i 1Ti .

Theorem: 4.10

Let (X {x_j}k_j1,E{e_r}n_r1,) be a (STS)

and for any pair _i  _i T_i

2

1 

 (1i n) such

that   

2 1 i

i  and 

} { / *

, )

(

i X i T j x i

  





 for

any (1in) and (1 jk).Then



, )

( t_nk is an even permutation space, if

) 1

( |

2 

 

 n

i Ti

nk

2454 By [Theorem (4.9)], we consider that



, )

( t_nk is a permutation space induced by soft

topology



and its permutation  in symmetric

group S_nk satisfies c() . 1

 



n

i Ti However,

) 1

( |

2 

 

 n

i Ti

nk and hence nkc() is even.

Then (,t_n)_ is an even permutation space.

Lemma: 4.11

Assume (X,E,) is a (STS). Then(,t_n)_ is odd permutation space, if (X,E,)is a (SITS) and 2|n.

Proof:

Assume (X,E,) is a (SITS) and let



, )

( t_n be a permutation space induced by soft topology . Hence  (12 3_n)[since

) , ,

(X E  is (SITS)], thus c()1_{. Also, since} n

|

2 , Then there is a positive integer number

q

such that n2q and hence nc()=(even) -

(odd)=(odd). Hence



is odd permutation in

S

_n.

Then

(



,

t

_n

)

_ is odd permutation space.

Lemma: 4.12

Assume (X {x_j}k_j1,E{e_r}n_r1,) is a soft discrete topological space. Then





 



j T n

n

i 1Ti for any (1 jn).

Proof:

Assume (X,E,) is a soft discrete topological space. Then, we consider that

,

  

j T i

T (1i,j n). So T_i T_j ,

) , 1

( i j n

 and hence

         



n T T

T n

i 1Ti 1 2 

       

j T n j T j

T j

T  , (1 jn).

Lemma: 4.13

Let (X {x_j}k_j1,E{e_r}n_r1,) be a (STS).

Then (,t_n)_ is (PSS), if _i 1,_iT_i, where (1in).

Proof:

Let(X,E,)be a (STS). Then(,t_n)_ is

(PSS), if _i 1,_iT_i, and (,t_n)_ be a permutation space induced by soft topology .

Since



_i



1

,





_i



T

_i



, this implies that

}

{

g

i



x



for some

x

g



X

. Therefore

) )) ( (

1 1(



 



 i

T

j i gj

n

i  x

 for some

x

_gj



X

. Then

 , )

( t_n is (PSS) [since each proper open





set is a singleton].

Lemma: 4.14

Let (X {x_j}k_j1,E{e_r}n_r1,) be a (STS).

Then(,t_n )_ is an even permutation

space, if _i 1,_iT_i, where (1i n).

Proof:

is 

, )

( t_n By [Lemma, (4.13)] we get

of the be the family

) (

1

} {_i c_i

Let . (PSS)

 cycles of

-s with the 1 product of separate cycle

 

where ( ( ))

j g i

i  x

  , for some x X

j

g  and

) (

1ic  _{[since each proper open}   set is

2455

) 1( ( )) 1( 



 

 

i T

j i xg j

n

i 





e



S

_nk

.

But

e

is an identity element in

S

_nk. Thus e



 =(1)(2)(3)_(nk) and hencec() nk. This implies nk c()0 (even). Hence



, )

( t_n is an even permutation space.

Theorem: 4.15

Let (X {x_j}k_j1,E{e_r}n_r1,) be a (STS)

and for any pair _i



_i



T

_i



2

1





(1in)

such that



_i1

_



_i2





and



} { / *

, )

(

i X i T j

i x

 







 for any (1in) and

) 1

(  jk . Then (,t_nk )_ is a splittable permutation space, if the following are hold.

(1) 2|(_i 1),_i T_i(1in),

(2) If _i _j , then_i  _j , where 

T_i i

 and _j T_j(1 j,in).

Proof:

By [Theorem (4.9)], we consider that (,t_nk )_

is a permutation space induced by soft topology  and its permutation  in symmetric group

nk

S

satisfies c() . 1

 



n

i Ti Moreover, i

n

i  

1

  

where for all 1in, 



  

L i i T

L i  

1

)) ( ) 2 ( ) 1 ( ( 1

L i g x i g x i g x i i T

L    _





 is permutation

which is product of T_i cyclic factors of the

length

L i

 , where T_i L i

 and 1L T_i .

For each S_n can be written as uniquely product of separate cycles. Thus 

) ( ... 2 1  

 _c , where _i  _i , {_i}c_i(₁ ) are separate cycles and c() is the number of the product of separate cycles with the 1-cycles of. Hence  () (₁,₂,...,c()

)

is the

"cycle type" of . Since

L i i T

L n

i 



     

1 1

and

 ) (

c 

 

n

i 1Ti . Then for any i there exists iL

satisfies

L i

i 

  , where (1in). The length

of any cycle

(

(

)

(

)

(

))

2 1

L L

i g i g i g i

i

x

x

x













_

is

L

i



and this implies that

 



) ) ( ,..., 2 , 1

(  _{c } ,

1 2 , 1 1 , , 2 1 , 1 1

(   



T 

) ,

, 2 , 1 , , 2 2 , , 2

2 _{ }

n T n n

n

T   



    No

w, if

L i

 is even number for some

) 1

(  in and (1LT_i), thus (  1)

L i



is odd and this contradiction with (1) of our hypothesis which state that 2 | ( _i  1),

) 1

( i n

i T

i    

 . Then

L

i



are odd

numbers for all (1i n) and(1LT_i).

Also, for any (i j) or (gh) we have

h

g j

i







where (1i,jn), (1gT_i) and

) 1

( hT_j . Then { |1in;1L T_i}

L i 

are different sets and by (2) of our hypothesis we

consider that { |1in;1L T_i}

L i

 are

different too. Therefore H_n and hence



, )

2456 Theorem: 4.16

Let (X {x_j}k_j1,E{e_r}n_r1,) be a (STS)

and for any pair _i  _i T_i

2

1 

 (1i n) such

that   

2 1 i

i  and 

} { / *

,

) (

i X i T j

i x

 







 for

any (1in) and (1 jk). Then



, )

( t_nk is an ambivalent permutation space, if the following are hold.

(1) 2|(_i 1),_i T_i(1in),

(2)If _i _j, then



_i 



_j , where



 _i

i T

 and

jTj (1 j,in)



 ,

(3)If 4|(_i 1),_iT_i(1in).

Proof:

From (1) and (2) we consider that permutation space (,t_nk )_ is a splittable [ By Lemma, (4.13)]. Also, from (1) and (2) that easy to show that

 (1,2,...,c()) 

(



11

,

 

,

,

,

,

,

,...,

,

,

,

,

_{1 2}

2 2

1 1

2 1 2 2 2

1





n n



T

T















_{ }

)



n T n



. That means for any part _{i of} (),

there exists _i



T_i



L

 for some (1in)and

) 1

(  L T_i satisfies

L

i

i





 and hence

from (3) we have 4|(_i1),(1ic()).

Then (,t_nk )_ is an ambivalent permutation space.

5. PROPOSED IMPROVEMENTS

Due to the difficulties of finding link between two different topological spaces in different strictures,

the present algorithm is the first link between soft space (X,E,)and permutation space

(

,t_n), where they are two different topological spaces in different strictures. This algorithm will help us to study all of the notions in (STSs) that are given in past work in permutation topological spaces when they are induced by some soft topologies. This algorithm is highly recommended.

6. CONCLUSIONS AND OPEN PROBLEMS

In this work, an algorithm has been introduced to find the permutation in symmetric group using soft topological space to structure permutation topological space. Further, suppose that

), 1 , ,

(X E (X,E,₂) and (X,E,₃)are three (STSs) over the common universe

X

the parameter set

E

with their permutations



,



and



in symmetric group S_n where

) ( ) (X E

n  . Hence the questions can be summarized as follows:

1)

Is necessarily true, if

1

) ,

(t_n _ is a even (res. odd, splittable, ambivalent) permutation space and

2

1 ( )

, (

: ) ,  

  t_n   t_n is continuous permutation map. Then the image under



is even (res. odd, splittable, ambivalent) permutation space, too.

2)

Is necessarily true, if (X,E,



₁) _and )

, ,

(X E



₂ are two (STSs). Then (,t_n)_T is a permutation space induced by soft topology

T

, where T



₁



₂.

3)

Is necessarily true, if (X,E,T) is a soft connected. Then (,t_n)_T is a permutation connected induced by soft topology

T

.

4)

Is necessarily true, if (X,E,T) is a soft

compact space. Then (,t_n)_T is a permutation compact space induced by soft topology

T

.

ACKNOWLEDGEMENT

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