American Journal of Mathematics and Statistics 2017, 7(5): 183-198 DOI: 10.5923/j.ajms.20170705.01
Extension Permutation Spaces with Separation Axioms in Topological Groups Shuker Mahmood*, Marwa Abud Alradha Department of Mathematics, College of Science, University of Basra, Iraq
Abstract Some notations in permutation topological spaces is given in this paper and some new permutation spaces like (PSS), (PIS), (PHS), ( T0 ), ( T1 ), (EPTS), (IEPTS), (DEPTS), ( E( ) T0 ), ( E ( ) T1 ), ( E ( ) T2 ), permutation homogeneous space, E ( ) -connected space, E ( ) -disconnected space and others are introduced and discussed. The aim of this work is to introduce and study new classes of the topological groups they are called permutation topological groups, extension permutation topological groups, permutation homogeneous topological group, Lindelof permutation topological group, E ( ) -connected group, E ( ) -disconnected topological group, (EPTG), (IEPTG), (DEPTG), ( E( ) T0 group), ( E ( ) T1 group), ( E ( ) T2 group) and others. Moreover, several examples are given to illustrate the concepts introduced in this paper.
Keywords Permutation topological space, Symmetric group, Cycle type, Permutation homogeneous, 𝛽 −Connectedness, Permutation topological groups
1. Introduction be a permutation in symmetric group S n with letter n . The support of , is the set {i | (i ) i} where {1,2,..., n} and is not identity in S n . So we say and are disjoint cycles iff supp( ) supp( ) [10]. There are many applications on Let
permutations, in recent years they are used to solve equations (see [8-11]). Permutation topological space (, tn ) 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
A {b1 , b2 ,, br } of in
(, t n ) 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 A {b1 , b2 , , br } of in (, tn ) as
set. That means we can put A . However * Corresponding author:
[email protected] (Shuker Mahmood) Published online at http://journal.sapub.org/ajms Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved
{i }ic(1 ) where i ( 1 i c( ) ) disjoint cycles of also we denote to its cycle by (b1 b2 br ) 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), ( T0 ), ( T1 ), (EPTS), (IEPTS), (DEPTS), ( E( ) T0 ), ( E ( ) T1 ), ( E ( ) T2 ),
permutation homogeneous space, E ( ) -connected space,
E ( ) -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, E ( ) -connected group, E ( ) -disconnected topological group, (EPTG), (IEPTG), (DEPTG), ( E( ) T0
group),
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Extension Permutation Spaces with Separation Axioms in Topological Groups
( E ( ) T1 group), ( E ( ) T2 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.
{b1 , b2 ,..., bk } and is called set of cycle . So
sets of {i }ic(1 )
the
are
defined
by
{i {b , b ,...,b } | 1 i c( )} . i 1
i 2
i
i
Remark 2.6 [7]
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
(1 , 2 ,...) with 1 2 ... and length l ( ) and the size
i .
The
i 1
of are defined as
k , Further, in , where
i 1
├ n { partition ; n} for n N . An element of ├ n is called a partition of n. Remark 2.2 [15] We only write the non zero components of a partition. Choose any S n and write it as 1 2 .... c ( ) . With disjoint cycles of length
i
and c ( ) is the number
of disjoint cycle factors including the 1-cycle of . Since disjoint cycles commute, we can assume that 1 2 ... c ( ) . Therefore ( 1 , 2 ,..., c ( ) ) is a partition of n and each i is called part of . We call the partition ( )
(1( ),2 ( ),...,c( ) ( )) the cycle type of . Definition 2.4 [14]
S n {e} .
Then supp( ) , the
, is the set {i | (i ) i} where {1,2,..., n} . So we say and are disjoint cycles iff supp( ) supp( ) . support of
Definition 2.5 [7] Suppose
is permutation in symmetric group
the set {1,2,..., n} and the cycle type of
( ) (1 , 2 ,..., c ( ) ) , then pairwise
disjoint
cycles
S n on
is
composite of
{i }ic(1 )
where
i (b , b ,...,b ) , 1 i c( ) . For any k cycle (b1 , b2 ,...,bk ) in S n we define set as i 1
i 2
i
i
j .
and
We call
We will give some
only if
i and j are disjoint sets in , if and
k 1
k 1
bki bkj
and there exists 1 d , for
each 1 r such that bd br . i
j
Definition 2.8 [7]
i and j are equal sets in , if and only if for each 1 d there exists 1 r such i j that bd br . We call
Definition 2.9 [7] We call
ˆ i is contained in j and denoted by i
j , if and only if
i
k 1 Definition 2.10 [7]
bki
j
bkj . k 1
We define the operations and as followers:
Definition 2.3 [8]
Suppose first that
i
definitions needed in this work. Definition 2.7 [7]
l ( ) Max{i N ; i 0} and i We set
i
(b1 , b2 ,..., bk ) in S n we put suppose that i and j are sets
For any k cycle
2. Preliminaries
on
sets in
j i i , if bk bk k 1 k 1 j , if bki b j k i j k 1 k 1 , if i j , if & are disjo int i j j i i , if bk bk k 1 k 1 j , if bki b j k and i j k 1 k 1 , if i j , if & are disjo int i j
American Journal of Mathematics and Statistics 2017, 7(5): 183-198
Remarks 2.11 [7]
Definition 2.17 [7]
1. The intersection of 2. The union of
, and be three permutations in symmetric group S n , and let : (, t n ) (, t n ) be a function,
i and j is i j .
Let
i and j is i j .
i is i . 4. The intersection and union of and i are and i , respectively. 5. The intersection and union of and i are i and , respectively. 3. The complement of
Definition 2.12 [7] Let
S n , and
be permutation in symmetric group
set in the space , and is smallest closed set containing or equal , and any is called closed
ˆ is called closed set iff . set
The set ( ) o
is called the interior of
set in the permutation space .
set {b1 , b2 ,..., bk } iff x b j , for some j {1,2,..., k} .
1. We call x belong to
2. The condition
inverse image of under defined
by
the
rule
another set, the
is called set and
1 ( ) { 1 (a1 ), 1 (a2 ),
(, t m ) , a function
: (, t n ) (, t m ) is permutation continuous if the inverse image under of any open set in t m is an open set in t n (i.e
1 ( ) t n
whenever
t m ).
Lemma 2.19 [7] The identity permutation e (1) in symmetric group S n
Lemma 2.20 [7]
Remark 2.21 [7] A base for a permutation topological space (, t n ) is a
can be written as i , where each
of t n
i belongs to
iI
D . Further, the subbase M of t n such that each proper
set of t n can be written as a union of finite intersections of elements of M . In another word, the family of open sets consisting of all finite intersections of elements of M , together with the set , forms D . Let open
containing x and such that
r ( ) . If i and j are disjoint sets
(, tn ) .
sub-collection D of t n such that each member
x means that
x . Therefore, x is an interior point of set if and only if there is an open set
neither
..., (bk )} . In ( ) { (b1 ), (b2 ), direction, let {a1 , a2 ,..., ar } be
A composition of permutation continuous functions is permutation continuous.
Remarks 2.15 [7]
3.
is called set and defined by the rule
under
is a permutation continuous on a permutation space
Definition 2.14 [7]
Given permutation topological spaces (, t n ) and
t n is set in the space , then
r
sets of the family {i }ic(1 ) union
and empty set. Definition 2.13 [7]
the
of
set {b1 , b2 ,...,bk } , the image
is a
topological space where {1,2,..., n} and t n
If
where for each
..., 1 (ar )}. The usual properties relating images and inverse images of subsets of complements, unions, and intersections also hold for permutation sets. Definition 2.18 [7]
c( )
composite of pairwise disjoint cycles {i }i 1 , where i i , 1 i c( ) , then (, t n ) is a permutation
collection of
185
in
, then
ˆ j nor j ˆ i . i
Remark 2.16 [7] Any map between two permutation topological spaces is called permutation map.
{(i , tn )}iI be the collection of permutation topological i
i
spaces. Then subbase for the product permutation topology i
i
i
i
( i , tn ) is given by M { i1 (k ) | k tn , iI
iI
i
i
i I , k 1,2,..., c( )} , so that a base can be taken to be i
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ij
d
D { i1 (i ) | j 1
j
ij
ij
Extension Permutation Spaces with Separation Axioms in Topological Groups
{ {Ti }T , {(br )}tr 1 } are disjoint cycles decomposition
i
tn , i I , d N }.
i
i
i
Let ( , t ) be permutation topological space for each i i
index i I . The product permutation topology t ti iI
on making all the projection mappings permutation continuous. Lemma 2.23 [7]
i : i
we denote to permutation subspace of
(, tm
spaces, then 1 2 ... have a countable base. If P is an algebraic (a topological) property, we say that the topological group G has property P , if the group (G ,) (the topological space (G , ) ) has property
P. Definition 2.25 [13] Let (G ,) be a group, F and K be subsets of
G , we let FK and F denote FK { f k | f F , k K } 1 1 and F { f | f F} . The subset F is called 1 symmetric if F F . 1
Ti
, {br }tr 1}
Connectedness): Let (, tn ) be permutation topological space. The collection of sets (
{1,2,..., n} if i and if the members i of iI
are all nonempty and {i }iI pairwise disjoint cycles in S n . Then is called decomposition of we also say that has been decomposed into the sets of . Assume the permutation topological space (, tn ) has been decomposed into two open sets k and
j . In this form the permutation space is
disconnected.
called
Definition 2.29 [7]
(Permutation subspaces):
Suppose (, t n ) permutation space,
ˆ and
Ti i , for each proper i t n , then
be
A permutation space and its topology are both said to connected if cannot be decomposed into
two open
{b1i , b2i ,..., bii }, if & i are not disjo int k Ti , if & i are disjo int
let
bki
nonempty open
set}. For each
i i i Max {b1 , b2 ,..., bik }
m Max{bki ; Ti } . Suppose
Ti
Ti s
and
B
points where
Ti
( Ti )
where
{1, 2,..., m} . Here we used normal intersection ( ) between pairwise sets to find the set we have Ti (b1 , b2 ,..., bik ) is i
sets.
A
subset of
is said to be
connected whenever the permutation subspace
(, tm
) is connected, and
disconnected
is said to be
if
is decomposed into two
open
sets.
, and
t m s , then we have this set B {b1 , b2 ,...,bt } has
i
{, , {Ti }
and {1, 2,..., m} .
Definition 2.26 [7]
i
{i }iI is said to be a decomposition of the set
Remark 2.24
exactly t
) where tm
(, tn ) by
Definition 2.28 [7]
If the spaces 1 , 2 ,... are permutation topological
Ti ,
say .
ˆ , then Let (, tn ) be a permutation space and
iI
on the set i is the coarsest permutation topology
Definition 2.27 [7]
i
Let {Ti | Ti
S m induced by
of new permutation in symmetric group
Definition 2.22 [7]
B . For each Ti
ik cycle in S m . Then
3. New Notations in Permutation Topological Space Let (, t n ) 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 A {b1 , b2 ,, br } of in (, t n ) as set. That means we can put
A . However {i }ic(1 ) where
American Journal of Mathematics and Statistics 2017, 7(5): 183-198
i ( 1 i c( ) ) disjoint cycles of also we denote to its cycle by (b1 b2 br ) 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 every set in the permutation space is either open
set or closed set. Definition 3.1
, br } and {a1 , a 2 , , a }
Let {b1 , b2 , be two subset of
. Then, we call and are
sets in , if and only if
similar
r
k 1
k 1
bk ak and
one of them contains at least two points say bi , b j such that bi
and
b j .
{b1 , b2 ,, br } and {a1 , a 2 , , a } be similar sets in and Max{Max{ }, Max{ }} , where {b1 , b2 ,, br } ˆ if and {a1 , a2 ,, a } . Then Let
, if ˆ if . Also, , , if , if and . , if Definition 3.3
Then,
, br } and {a1, a2 ,
, if [( bk ak ) Or k 1 k 1 ( & are similar and )] r , if [( b ak ) Or k k 1 k 1 ( & are similar and )] , if [ ] , if [ & are disjo int ] r
and
r , if [( bk ak ) Or k 1 k 1 ( & are similar and )] r , if [( b ak ) Or k k 1 k 1 ( & are similar and )] , if [ ] , if [ & are disjo int]
Remark 3.4 In permutation topological space (, t n ) any subset
ˆ and is called an open set iff A such that A
Ao A . Also, it is called closed set iff A A .
Definition 3.2
For any {b1 , b2 , two subset of .
187
, a }
4. Permutation Topological Group Definition 4.1
Let (, n ) be a permutation topological space. Then
(, n ) is called Permutation Single Space (PSS) if and only if each proper open set is a singleton. Definition 4.2
Let (, n ) be a permutation topological space. Then
(, n ) is called Permutation Indiscrete Space (PIS) if and only if each open set is trivial set. Definition 4.3
Given permutation topological spaces (, t n ) and
(, tm )
,
a
function
: (, tn ) (, tm )
permutation open map if the image under
is
of any open
set in t is an open set in t . n
m
Lemma 4.4
Let (, n ) be a permutation topological space. Then
(, n ) is permutation single space (PSS) if and only if c( ) n . Proof:
Suppose that (, n ) is a (PSS). Then each proper open
set
is
a
singleton.
That
means,
A & A n A {bi } , for some bi . Let c ( ) k n , then
k n (since 1 c( ) n ),
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12 k .
and hence
( ) (1 , 2 ,, k )
i i 1
Then that
Extension Permutation Spaces with Separation Axioms in Topological Groups
, (1 i k ) .
i
that
is an open
12 n
n
. However,
i i 1
, where
i 1
( ) (1 , 2 ,, n ) Then
i n
,
i i
, (1 i n) .
for all (1 i n) . This implies that
i contains only one element for each (1 i n) , but n {i | 1 i n} { , } . Thus each proper open set is a singleton and hence (, n ) is (PSS). Lemma 4.5
Let (, n ) be a permutation topological space. Then
(, n ) is permutation indiscrete space (PIS) if and only if c( ) 1 .
Suppose that (, n ) is a (PIS). Then each open set is trivial
set and hence n { , } . This implies
A & A n
A . Hence (b1 b2 b3 bn ) , where ( ) (n) . Then c( ) 1 . Conversely, if c( ) 1 . Then we consider that
1 (b1 b2 b3 bn ) and hence n {1 } { , } , but {1 } . Then n { , } and this implies that (, n ) is (PIS). Definition 4.6 [12] (Multiplication Permutation Map) Let
1
(2,1)
(i, j )
. ( 1 (n), 2 (n))
(n, n)
Now, let : be a binary operation on and (1 2 ) : be a map defined by
(1 2 ) (( x, y)) 1 ( x) 2 ( y), ( x, y ) . Then the permutation map (1 2 ) from permutation space ( , t n t n ) into (, n ) for any permutation in symmetric group S n is called multiplication permutation map. Further, it is called multiplication permutation continuous iff the inverse image under (1 2 ) of any open set in t n is an open
set in t n t n
(i.e
1
( 1 2 ) ( )
t n t n whenever t n ). Example: 4.7 Suppose that (5 1 2 4 3) and
1
2 (1) are permutations in symmetric group S n with
n 5 , and let : be a binary operation on
x y 1, if x y 1 n, ( x, y ) , ( x y 1) n, if x y 1 n. ( x, y ) . We consider that the multiplication
where
Proof:
that
(1,2)
( 1 (1), 2 (n)) ( 1 (2), 2 (1)) ( 1 (i ), 2 ( j ))
for some (1 i k ) . This implies
set and each open set is singleton. Therefore we consider that c( ) n . Conversely, if c( ) n . Then we consider
and
2
be two permutations in symmetric group
S n . Then 1 and 2 are two permutation maps from onto . Further, 1 2 : is a product map
(1, n)
i contains more one element, but this contradiction
since
(1,1)
1 2 ( 1 (1), 2 (1)) ( 1 (1), 2 (2))
i 1
i i
,
k
i n , where
However,
of
permutation
maps
( 1 2 )(( x, y)) ( 1 ( x), 2 ( y)), ( x, y)
another side, the map as this form
1 2
where . In
is a permutation in
Sn Sn
permutation map (1 2 ) : ( , t5 t5 ) (, 5 ) , where (1 2 ) (( x, y)) x y, ( x, y) is a multiplication permutation continuous map. Remark 4.8 By above example we consider the following: (1)-For any S n , if c( ) 1 . Then there is a multiplication permutation continuous map
(1 2 ) : from permutation space ( , tn tn ) into (, n ) satisfies
(1 2 ) (( x, y)) x y, ( x, y) . (2)-For any n 1 , the mathematical system (,) is a commutative group. (4)-For any n 1 and ( x, y ) in , the multiplication permutation map such that:
American Journal of Mathematics and Statistics 2017, 7(5): 183-198
189
( 1 2 ) (( x, y ))
is clear if 1 f F or 1 x , then x f x 1 1
1 ( 1 2 ) (( x , y )), if 2 x 2 or n 2 . ( 1 2 ) (( x, y 1 )), if 2 y 2 or n 2 ( ) (( x 1 , y 1 )), if (2 x 2 or n 2 ) 1 2 & ( 2 y 2 or n 2)
or f F . Also, if 1 f F & 1 x , we have
n 1 , there is an inversion permutation map : such that ( x) x 1 , x .
(5)-For any
2 n with Where 1 (1) (2) (n)
if x 1, x, ( x) n 2 x, if x 1. Lemma 4.9 For any even positive integer n 3 , the commutative group (,) has proper symmetric subgroup. Proof:
n n 1 n 2 x 1 n ( 2 x). 2 2 consider that f x 1
Since f x 1 1 n Then
we
n n n ( 2 x), if n ( 2 x) n, 2 2 put ( n 2 x), if n ( n 2 x) n. 2 2
g f x 1 .
n n n ( 1), if n ( 2 x) n, 2 2 Therefore, we get x g 1 ( n 1), if n ( n 2 x) n. 2 2
n n Thus, x g 1 n ( 1) n or x+ g 1 ( 1) n , 2 2 for any even positive integer n 3. This implies that
n ( 1), if x g 1 n, x f x xg 2 Then ( n 1), if x g 1 n. 2 n n x f x 1 ( 1) n 1 f F . Hence ( F , ) 2 2 is a proper normal subgroup of (, ) . 1
n 1 , then 1 f n for any even 2 positive integer n 3 . This implies that the set F {1, f } is a proper subset of . New, to prove that ( F ,) is a symmetric subgroup of (,) it is enough to show that f f 1 F . That means ( F ,) is a group Let
n
1 x 1 n 2 x and f n 2 1 . Then x f x F .
f n
Definition 4.11
Let (, n ) be a permutation topological space and
with the following table: *
1
f
1
1
f
f
f
1
n n Since f f 1 (n 1) (n 1) 1 2 2
n 1) 1 = 2n n 2 1 n 1 n . Then 2 f f ( f f 1) n 1 and hence F 1 {1, f } F . Therefore ( F , ) is a proper symmetric subgroup of (, ) . 2(n
Lemma 4.10 For any even positive integer n 3 , the commutative group (,) has proper normal subgroup. Proof: By lemma (4.9) we consider that ( F ,) is a proper subgroup of (,) , where
n F {1, f } and f n 1 . Now, we need to 2 show that ( F ,) is a normal. In other words, we want to prove that x f x 1 F , for any f F , x . It
(,) be a group. Then we say that (,, n ) is a permutation topological group (PTG) if q ( x, y ) x y and
p( x) x 1 the multiplication permutation map
q : is multiplication permutation continuous map and p : the inversion permutation map is permutation continuous map. Example 4.12 Let (4 2 1 5 6 3) be a permutation in symmetric group
S 6 . Then (, 6 ) is permutation topological
space, where {1,2,3,4,5,6} and
6 { , } . Also,
let (, ) be a group with the following table: Table (1)
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
3
4
5
6
1
3
3
4
5
6
1
2
4
4
5
6
1
2
3
5
5
6
1
2
3
4
6
6
1
2
3
4
5
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It is clear that (, 6 ) is an indiscrete permutation
x y is multiplication permutation 1 continuous map and p( x) x is inversion permutation continuous map, x, y . Then (,, 6 ) is a space. Thus q ( x, y )
permutation topological group. Lemma 4.13
Then,
is a permutation continuous and permutation
open map, if (, n ) is (PIS). (b)
is a permutation continuous and permutation open map, if (, n ) is (PSS).
Proof:
(a) Let (, n ) be a (PIS). Then each open trivial
set is
set and hence n { , } . It is clear
( ) and 1 ( ) . Also, () and 1
() (since each permutation map is bijection). Then is a permutation continuous and permutation open map
(b) Let (, n ) be (PSS). Then each proper open set
is
a
singleton.
A permutation topological space (, n ) is called a permutation homogeneous space (PHS), if for any x, y there exists a permutation homeomorphism
: such that ( x) y .
Let : (, n ) (, n ) be a permutation function. (a)
open map). Definition 4.15
This
implies
that,
ˆ & n we have {a} , for
a . 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 some
permutation, but this contradiction. Therefore, for any open set in we consider that
Example 4.16
e be an identity permutation in symmetric group S 9 . Then (, 9 ) is permutation topological Let
space,
where
{1,2,3,4,5,6,7,8,9}
and
9 {{ j}|1 j 9} { , } . It is clear that (, 9 ) is (PSS). Define : as follows: for any x, if t y, x, y , let (t ) y, if t x, t . Therefore t , Otherwise. we get
1 2 1 2
x 1 x x 1 x 1 y x 1
is a permutation in symmetric group
y 1 y y 1 y 1 x y 1
9 9
S 9 and such that
( x) y . Moreover, : (, 9 ) (, 9 ) is a bijection map (since each permutation is bijection). Also, is a permutation continuous and permutation open since
(, 9 ) is (PSS). Then (, 9 ) is a permutation homogeneous. Definition 4.17
A permutation topological group (,, n ) is called a permutation homogeneous topological group, permutation space is a permutation homogeneous. Remark 4.18
if
its
, if Let (,*, n ) be a permutation topological group, and ( ) , if , for some a, b . Also, k , Define : , (r ) kr ( ( r ) rk ), {b}, if {a} r . Then the map r kr (r rk ) is a , if permutation homeomorphism. Also, define : , by similarity we consider that 1 ( ) , if , r , if r 1, {b}, if {a} (r ) n 2 r , if r 1. , r . Then the map for some a, b . Thus
( ) and 1 ( ) 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)- 1 is permutation continuous ( is permutation
r r 1 is a permutation homeomorphism. Lemma 4.19
Every permutation topological group is a permutation homogeneous topological group. Proof:
Let (,, n ) be a permutation topological group, we
need to show that it's permutation space (, n ) is a permutation homogeneous. That means we have to show that x, y there exists a permutation for any
American Journal of Mathematics and Statistics 2017, 7(5): 183-198
homeomorphism : such that ( x) y . Since
Remark 4.23
(,, n ) is permutation topological group. Then there is 1
a permutation homeomorphism such that r r , r and hence for any x, y there exists a permutation homeomorphism such that
r xr 1 y ,
r . Moreover, we consider that this permutation 1 homeomorphism such that x x x y y and y x y 1 y x . Hence (,, n ) is a permutation homogeneous topological group. Lemma 4.20
If
D is a subset of . Then D is an open set if and 1 only if D is an open set.
groups, then 1 2 ... is an Lindelof permutation topological group. Definition 4.24
Let (, n ) be a permutation topological space, and
x . The connected component of x in is the largest connected subset of containing x . Example 4.25
(1 2) be a permutation in symmetric group S 4 . Find connected component of 3 in permutation topological space (, 4 ) . Solution:
t4 {, ,{1,2},{3},{4}}, where {1, 2,3, 4} .
Proof:
Hence (, 4 ) is a permutation topological space, let {L1
1
Since the map r r is a permutation homeomorphism. Then the proof is obvious. Lemma 4.21
Let (,*, n ) be a permutation topological group, and
k . Then D is an open set if and only if kD ( Dk ) is an open set. Proof:
Since the map r kr ( r rk ) is a permutation homeomorphism. Then the proof is obvious. Theorem 4.22 A permutation topological group is an Lindelof permutation topological group. Proof: Let (,, t n ) be permutation topological group where and ( ) (1, 2 ,...,c( ) ) , then for each
1 i c( )
we
have
i {b , b ,...,bi } i 1
i 2
D {i }iI
i j
iI
{ i }iI is a collection of permutation topological
Let
Let (,*, n ) be a permutation topological group, and
Sn ,
191
i
the
proper
set
is a countable set, and for each base
for permutation space where
open
j
i
k 1
k 1
we have
bkj sup{ bki | i I } , but j
is a
countable set (each finite set is a countable), (see Runde, 2005), so D 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 (, , tn ) is an Lindelof permutation topological group.
{3}, L2 {1,3}, L3 {2,3}, L4 {3, 4}, L5 {1, 2,3}, L6 {1,3, 4}, L7 {2,3, 4}, L8 } be the family of all
subsets of which are contain point 3 . Then we consider that each one of the permutation subspaces{ {(, t
(, 4 ) is two open are
L2
i
)}i73 , (, t4
L8
L2 ) , (, t3 ) } of
decomposed, for all (2 i 8) into
8 sets {1, 2} and {3} and hence {Li }i 2
disconnected, where
t 4 Li t3
4 L
Li
for
all
{1, 2,3} ,
3i 8 ,
t4
L8
t4
{, ,{1, 2},{3}} . Further only (, t3
L1
L
t4 i
and
) is
connected where t L1 {, ,{1},{2},{3}} . Hence 3 L1 {3} is connected component of 3 in permutation topological space
(, 4 ) . In another side,
is not connected component of all its points and then is not connected. Definition 4.26 A permutation topological group (,, t n ) is
connected topological group iff is connected component of all its points. Example: 4.27 Let (,, t 6 ) be a permutation topological group in example (4.12). Then is connected component of all its points and hence the permutation topological group (,, t 6 ) is connected
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Extension Permutation Spaces with Separation Axioms in Topological Groups
topological group. Lemma 4.28
Definition 4.29
Let be a permutation in symmetric group S n . Then
connected component of 1 in permutation
the
topological space (, t n ) is , if c( ) n .
c( ) n , then every proper open set in (, t n ) is a singleton (i.e, B t n , t satisfies B {t} ). Moreover, for any L {1, b1 ,, bk } subset of , we have L B
k L, if bi 1 t i 1 k . {t}, if t bi 1 i 1 {1}, if L {1} {t} , if L &{t} are disjo int
Now, we looking for the largest
subset L of
L
contains 1 with permutation subspace (, tm
connected. Thus we first discuses L
) is
L with
n
(, t L), if L , n bi 1 . Here (, tm ) i 1 (, t n ), if L . If L . Then {L, ,{1},{2}, {n}} is a k
collection of all non-empty open
L {a} {a}
,
L sets and such that
{a}, if a b {b} {a} {b}, if a b
,
{a} {a} for any 1 a, b n and L L . Also, If L . Then {,{1},{2}, {n}} is a
collection of all non-empty open
{a}, if a b {b} {a} {b}, if a b
L sets and such that
, {a} {a}
for
any
1 a, b n . Then, cannot be L decomposed
into two open
L sets. Because there exist no two open
L
sets are disjoint
set in such that x , but y .
L
sets. Also, for each z
T1 if for any two distinct points x, y , there are
y 1 and y 2 , x 2 . Example 4.31
Let (, 9 ) be a permutation topological space in example (4.16), where {1,2,3,4, 5,6,7,8,9} and
9 {{ j} | 1 j 9} { , }
. It is clear that
(, 9 ) is (PSS) and hence each singleton set is an open set. Then, for any two distinct points x, y ,
there are two open
sets 1 {x} , 2 { y} in
such that x 1 , y 1 …(1) and y 2 ,
x 2 …(2). Hence from (1) we get (, 9 ) is T0 .
Also, from (1) and (2) we have (, 9 ) is T1 . Lemma 4.32 Let be a permutation in symmetric group S n . Then
c ( ) n if and only if (, t n ) is T1 space. Proof: Assume c( ) n , then by lemma(4.4) we have
(, n ) is a (PSS) and hence any singleton set is open set. Hence for any two distinct points x, y ,
there are two open
sets 1 {x} , 2 { y} in
such that x 1 , y 1 and y 2 , x 2 . Conversely, suppose that (, t n ) is T1 space and c( ) n . Hence c ( ) k n , for some k n (since 1 c ( ) n ), and hence 12 k . k
However,
ˆ . Therefore, is the largest L connected subset of containing 1 . Then the connected component of 1 in permutation topological space (, t n ) is .
i i
i 1
sets 1 , 2 in such that x 1 ,
two open
L {1, b1,, bz } subset of with n bi 1 we have
open
A permutation topological space (, t n ) is said to be
Since
T0 if for any two distinct points x, y , there is an
Definition 4.30
Proof:
A permutation topological space (, t n ) is said to be
i n , i 1
where ( ) (1 , 2 ,, k ) ,
, (1 i k ) . Then
i i 1
for some
(1 i k ) . This implies that i contains more one element. That means there are two distinct elements
x, y i . However, (, t n ) is T1 space, then
American Journal of Mathematics and Statistics 2017, 7(5): 183-198
y 1 and
x 1 ,
sets 1 , 2 in such that
there are two open
y 2 , x 2 . Thus
x supp(1 ) supp(i ) , y supp( 2 )
supp(i ) . But this contradiction since the cycles for
sets are disjoint and hence we consider
any pair of open that
supp(1 )
supp( )
Remarks 5.3 (1) For any permutation there
supp(2 )
is
in symmetric group S n ,
(, E (t n ))
extension
permutation
topological space (EPTS). ˆ is open (closed) set in (, t n ) , (2) If A
A
then
is
open
(closed)
E ( ) set in
(, E (t n )) . However, the converse is not true in
,
193
general. ˆ are similar (3) Any pair of subsets A, B
supp( ) . Then c( ) n .
(disjoint) sets in (, t n ) if and only if they are similar (disjoint) E ( ) sets in (, E (t n )) .
5. Extension Permutation Topological Space (EPTS) Suppose that (, tn ) is a permutation topological space.
ˆ are disjoint (4) Any pair of subsets A, B sets if and only if their complements are disjoin sets or disjoin E ( ) sets.
Now, we define new set by E (tn ) { A B | A, B tn } . Here we used the normal union ( ) between open sets
similar E ( ) sets in (, E (t n )) , then their
to generate the new topology E (t n ) on with two operations and (see definition 3.3). In another side,
tn { A
B | A, B E (tn )} .
Let (, 5 ) be a permutation topological space in
Let (, tn ) be a permutation topological space. Then is
called
an
Extension
Permutation
ˆ is called an Topological Space (EPTS), and each A Extension Permutation set and denoted by E ( ) set. Example 5.2 Let (1 3)(2 5) be a permutation in symmetric group space,
S 5 . Hence (, 5 ) is a permutation topological t5 {, ,{1,3},{2,5},{4}}
where
and
{1,2,3,4,5} . Thus (, E (t n )) is (EPTS), where
E (t5 ) {, ,{1,3},{2,5},{4},{1,2,3,5},{1,3,4},{2,4,5}} . Moreover, , ,{2,4,5},{1,3,4},{1,2,3,5},{4},{2,5}, and {1,3} are all closed E ( ) subset of , for example {1, 2,3,5} and {2, 4,5} are similar E ( ) sets and ˆ {2, 4,5} (since 4 {2,4,5} ). Further, {1, 2,3,5} {4} and {1,3} are disjoint E ( ) sets, thus neither
ˆ {1,3} nor {1,3} ˆ {4} . In another side, {4} ({1, 2,3,5}) {1, 2,3,5} o
,
({2,4,5}) {2,4,5} o
,
{1,2,3,5} {1,2,3,5} , {2,4,5} {2,4,5} , ({4}) {4} , o
{4} {4} , ({1,3}) {1,3} and {1,3} {1,3} . o
complements it is not necessary to be similar sets or similar E ( ) sets. Example 5.4
Definition 5.1
(, E (t n ))
ˆ are similar sets in (, t n ) or (5) If A, B
example (5.2), where t 5 {, , {1,3}, {2,5},{4}} and
{1,2,3,4,5} . Thus (, E (t n )) is (EPTS), where
E (t5 ) {, ,{1,3},{2,5},{4},
{1,2,3,5}, {1,3,4},
{2, 4,5}} . Let A {1,2,3}, B {1,5}, D {2,3,5}, ˆ . Then A, B and their complements C {1,2,3,4} are similar sets in (, t n ) and similar E ( ) sets in (, E (tn )) , However, C, D are similar sets in
(, t n ) and similar E ( ) sets in (, E (t n )) , but their complements are neither similar sets nor similar E ( ) sets. Lemma 5.5 Let (, t n ) be a permutation topological space. Then
(, t n ) is (EPTS) if c( ) 1 . Proof: Let (, t n ) be a permutation topological
space and
c( ) 1 . Then (, t n ) is (PIS) by lemma (3.5). This implies
E (tn ) { A
that
tn { , }
.
However,
B | A, B tn } , thus E (tn ) { , } tn .
Then (, tn ) is (EPTS).
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be E ( ) T1 if for any two distinct points x, y ,
Lemma 5.6 Let : be a permutation map. Then, (a) : (, E ( n )) (, E ( n )) is
a
there are two open E ( ) sets
permutation
: (, n ) (, n ) continuous, if permutation continuous map. (b) : (, E ( n )) (, E ( n ))
is
is a permutation
open, if : (, n ) (, n ) is permutation
open map. (c) : (, E ( n )) (, E ( n )) homeomorphism, if
is a permutation
: (, n ) (, n ) is
permutation homeomorphism. Proof: (a) Suppose
that
: (, n ) (, n )
is
a
B C A E (t n ) , for some B, C t n , but
and
(, n ) map,
thus
hence 1 ( B)
in such
2
that x , y and y , x . 1 2 1 2 Definition 5.9 A permutation topological space (, E (tn )) is said to
be E ( ) T2 if for any two distinct points x, y , there are two open disjoint E ( ) sets
1
,
2
in
such that x 1 and y 2 . Definition 5.10 Let (, E (t n )) be (EPS) and (,) be a group. Then
permutation continuous map. Let A E (t n ) , then
continuous
,
we say that (,, E ( n )) is an Extension Permutation
: (, n )
1
is
a
permutation
1
1
( B), (C ) tn
1 (C ) E (tn ) .
Since
1 ( B C ) 1 ( B) 1 (C ) . Then this implies
Topological Group (EPTG) if
p( x) x
1
the
q ( x, y ) x y and
multiplication
permutation
map
q : is multiplication permutation continuous map and p : the inversion permutation map is permutation continuous map. Lemma 5.11 If (, , tn ) is (PTG), then (, , E ( n )) is (EPTG). Proof:
Suppose that (, , tn ) is (PTG). Then there are two
(, E ( n )) is a permutation continuous map.
permutation continuous maps q : (, tn ) (, tn ) (, tn )
1
that ( A) E (tn ) . Hence : (, E ( n )) (b) Assume : (, n ) (, n ) is a permutation
and
p : (, tn ) (, tn ) such that q ( x, y ) x y
A E (t n ) . Then
and
p( x) x 1 . By lemma (5.6) we have q : (, E (tn ))
continuous map and let
B C A E (tn ) , for some B, C t n , but
: (, n ) (, n ) is a permutation open map, thus ( B), (C ) tn and hence ( B)
E (tn ) . Further, since ( B
(C )
C ) ( B )
(C ) .
Then this implies that ( A) E (tn ) . Hence
: (, E ( n )) (, E ( n ))
is a permutation
open map. (c) By (a) and (b), it is clear the proof is obvious. Definition 5.7 A permutation topological space (, E (tn )) is said to be E ( ) T0 if for any two distinct points x, y , there is an open E ( ) set
in such that
(, E (tn )) (, E (tn )) and p : (, E (tn )) (, E (tn )) are permutation continuous maps and hence (, , E ( n )) is (EPTG). Definition 5.12
Let (, E (t n )) be (EPTS). Then (, E (t n )) is called an Indiscrete Extension Permutation Topological Space (IEPTS) if and only if each open E ( ) set is trivial
E ( ) set. Definition 5.13 Let (, E (t n )) be (EPTS). Then (, E (t n )) is called a discrete Extension Permutation Topological Space (DEPTS) if and only if each E ( ) subset in is open E ( ) set. Remark 5.14 Let (,, E ( n )) be (EPTG), then (, , E ( n )) is
x , but y .
E ( ) T1 , E ( ) T2 )] is [res. (, E (t n )) E( ) T0
said to be E ( ) T0 [res.
Definition 5.8
A permutation topological space (, E (t n )) is said to
group
iff
American Journal of Mathematics and Statistics 2017, 7(5): 183-198
E ( ) T1 , E ( ) T2 )] space. Also, (,, E ( n )) is
said to (DEPTG) [(IEPTG)] iff (, E (t n )) is (DEPTS) [(IEPTS)]. Lemma 5.15
Let (, E (t n )) be a permutation topological space. Then (, E (t n )) is (DEPTS) if and only if c( ) n .
Proof: Suppose that (, E (tn )) is a (DEPTS). Then for each
ˆ and hence is an open E ( ) x we have {x}
set [since (, E (tn )) is a (DEPTS)]. That means there are two open
sets A and B such that {x} A B ,
but {x}
is singleton and this implies that either
{x} A B {x} A B or or {x} B A . That means each open singleton E ( ) set is open set. Then (, t n ) is a (PSS). Thus by lemma (4.4) we have c( ) n . Conversely, if c( ) n . Then by lemma (4.4) we have (, t n ) is a (PSS). Thus each singleton is an open set and hence an open E ( ) set. For any E ( ) subset ˆ A . Therefore, A of and x A we have x {x} x is an interior point of A , thus x A o and hence ˆ A o , but in general Ao ˆ A . Thus Ao A and A hence A E (t n ) . That means any E ( ) subset of is open E ( ) set. Hence (, E (t n )) is (DEPTS).
However, every open
195
set is open E ( ) set. Then
(, E (tn )) is E ( ) T1 . Conversely, if (, E (tn )) is E ( ) T1 . Then for any
two distinct points x, y , there are two open E ( )
sets A , B in such that x A , y A and y B ,
x B . Moreover, there are open set A1 , A2 , B1 and B2 such that x A A1 A2 , y A A1 A2 , y B B1 B2 and x B B1 B2 . Thus
x A2 ) & ( y A1 and y A2 ) & ( y B1 or y B2 ) & ( x B1 and x B2 ). Hence, ( x A1 or
there are four cases cover all probabilities which are holed as following: (1) x A1 and y B1
y B2 (3) x A2 and y B1 (4) x A2 and y B2 (2) x A1 and
y A2 ) & ( x B1 and x B2 ). Then (, n ) is T1 space. However, ( y A1 and
Lemma 5.18 Let 1 be an identity element in extension permutation topological group
(, , E ( n )) , then
(, , E ( n )) is a E ( ) T1 topological group if and only if {1} is open E ( ) set. Proof: Let (,, E ( n )) be a E ( ) T1 group, then by
Lemma 5.16 (, E (t n )) is (DEPTS) if and only if (, t n ) is (PSS).
Proof: Let (, E (t n )) be (DEPTS). Then by lemma (5.15) we
have c( ) n and hence by lemma (4.4) we have
(, t n ) is (PSS).
Conversely, if (, t n ) is (PSS), then by lemma (5.15) and Lemma (4.4) we have (, E (t n )) is (DEPTS). Lemma 5.17 Every permutation topological space
(, n ) is
T1 if and only if its extension (, E (t n )) is E ( ) T1 . Proof:
Suppose that (, n ) is T1 . Then for any two distinct points x, y , there are two open
sets A ,
B in such that x A , y A and y B , x B .
lemma (5.17) we have (, n ) is a T1 and hence
c( ) n [by lemma (4.32)]. Then (, n ) is a (PSS) [by lemma (4.4)]. Hence any singleton set is open set. Then {1} is open set and hence {1} is open E ( ) set [ since each open set is open E ( ) set]. Conversely, suppose that {1} is open E ( ) set. That means there are two open sets A and B such that {1} A B , but {1} is singleton and this implies that {1} A B {1} A B either or or {1} B A . That means each open singleton E ( ) set is open set. Then by Lemma (4.21) we have {k } is open set for any k , because k{1} {k 1} {k} is open set. Hence (, n ) is (PSS) and hence c( ) n [by lemma (4.4)]. Therefore
(, t n ) is T1 space [by lemma (4.32)]. Then
(, E (t n )) is E ( ) T1 [by lemma(5.17)].
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space. The collection of E ( ) sets { Ai }iI is said
Lemma 5.19
Let (, , E (tn )) be extension topological group. Then
(, , E (tn ))
E ( ) T1 ,
is
E( ) T0 .
if
(, , E (tn ))
is
Let (,, E (t n )) be a E( ) T0 topological group, then for any two distinct points x, y , there is open
E ( ) set A in such that x A and y A . Define : , ( r ) kr , r . Then the map kx
1
is
a
permutation
and
{x 1 t | t A}
homeomorphism.
1
Dx A ,
then
Put
( A) D
E ( ) set in
and 1 x x D . Then yD is open E ( ) set in such that y y 1 yD . Thus A yD (since y A ). D1 D2 , and A A1 A2 where Let yD is open
1
A1 , A2 , D1 , D2 n . Now, if x yD . This implies that
x Di and x A j , for some 1 i, j 2 . Thus x
supp( Aj ) supp(Di ) . But this contradiction since the cycles for any pair of open sets are disjoint and hence we consider that x yD . Then (, , E (tn )) is a E ( ) T1 topological group. If (, , E ( n )) is (DEPTG), then (, , E ( n )) is
E ( ) T2 group.
(,, E ( n ))
is
(DEPTG).
Then
(, E ( n )) is (DEPTS). Let x y be any two distinct points in . Then, either ( x y ) or ( y x ). y, x ( y x) . Let A {x, y x} , and B {y} . Hence A {x, n x} , ˆ are two open E ( ) sets [since B {y} if
decomposition of we also say that has been E ( ) decomposed into the E ( ) sets of . Assume the extension permutation topological space (, E (tn )) has been E ( ) decomposed into two open E ( ) sets A
and B . In this form the permutation space is called E ( ) disconnected. Moreover, and its topology
E (tn ) are both said to be E ( ) connected if cannot be E ( ) decomposed into two open E ( ) sets. Lemma 5.22 Let (, tn ) be permutation topological space. Then
(, tn ) is
connected, if its extension space (, E (tn )) is E ( ) connected. Proof: Suppose that (, E (tn )) is E ( ) connected. Then
cannot be E ( ) decomposed into two open E ( ) sets. That means for any pair of non empty open E ( ) sets A, B we have A B and hence for A, B tn
x y
(, E ( n )) is (DEPTS)]. Also, x y x y . Then there are two open disjoint E ( ) sets A , B in such that x A and y B . Also, if y x we have A {x} , and B { y , x y} are two open disjoint E ( ) sets in such that x A and y B . Hence (,, E ( n )) is E ( ) T2 group. Definition 5.21: ( E ( ) Connectedness) Let (, E (tn )) be extension permutation topological
we have
A B [since
tn E (tn )] . Thus cannot be decomposed into two open
Proof:
Thus,
iI
any
Lemma 3.20
Assume
Ai and if the members Ai of are all nonempty and disjoint E ( ) sets. Then is called E ( )
Proof:
r kr
to be a E ( ) decomposition of the set {1,2,..., n} if
sets. Then (, t n ) is connected.
Definition 5.23 An extension
permutation
topological
group
(, , E (tn )) is called E ( ) connected topological
group, if (, E (t n )) is E ( ) connected. Lemma 5.24
If (, E (t n )) is (DEPTS), then (, E (t n )) is
E ( ) disconnected space.
Proof: Assume (, E (t n )) is (DEPTS). Then there are two open disjoint E ( ) sets {n} and {1, n 1} , where
ˆ and {n},{1, n 1} E (tn ) [since {n}, {1, n 1} (, E (t n )) is (DEPTS)], {n} {1, n 1} and {n} {1, n 1} . Thus is E ( ) decomposed into two open E ( ) sets and hence (, E (tn )) is E ( ) disconnected space. Lemma 5.25 If {1} is open E ( ) set, where 1 is an identity element in extension permutation topological group
American Journal of Mathematics and Statistics 2017, 7(5): 183-198
(, , E (tn )) , then (, , E (tn )) disconnected topological group. Proof:
E ( )
is
Assume {1} is open E ( ) set. Then by lemma (5.18)
we get (,, E (t n )) is a E ( ) T1 topological group
and hence by (5.17) we have (, n ) is T1 . This implies
c( ) n
that
[by
lemma
(4.32)].
197
( E ( ) T2 group) and others are introduced. Assume
(, n ) is permutation space and (,, f ) , where f 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 (, n ,, f ) is permutation topological d-algebra (resp. permutation topological BCK-algebra, permutation topological BCL-algebra).?
Then
(, E (t n )) is (DEPTS) [by lemma (5.15)]. Hence (, E (t n )) is E ( ) disconnected space [by lemma (5.24)]. Example 5.26 Let
e be an identity permutation in symmetric
group
S 9 . Then (, E( 9 )) is (DEPTS), where
ˆ } . Also, let {1,2,3,4,5,6,7,8,9} and E (tn ) {D | D (,) be a group with the following table:
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1
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9
1
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9
1
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1
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5
6
7
8
Thus q ( x, y ) x y , and
x d in an Alternating Group for all n & H C , journal of the Association of Arab n
P( x) x 1 , x, y
are permutation continuous maps. Then (, , E ( 9 )) is
E( ) T0 group, E ( ) T1 group, group and E ( ) disconnected group.
Universities for Basic and Applied Sciences, 16, (2014), 38–45. http://dx.doi.org/10.1016/j.jaubas.2013.10.003.
E ( ) T2
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, E ( ) -connected group,
E ( ) -disconnected topological group, (EPTG), (IEPTG), (DEPTG), ( E( ) T0 group), ( E ( ) T1 group),
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Extension Permutation Spaces with Separation Axioms in Topological Groups
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