A Study on Cayley Graphs of Factorizable Inverse Semi groups

(1)

Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories

Aryabhatta Journal of Mathematics and Informatics

http://www.ijmr.net.in

email id- [email protected]

Page 344 A Study on Cayley Graphs of Factorizable Inverse Semigroups

A. Riyas and K. Geetha Department of Mathematics T. K. M. College of Engineering, Kollam-05

Abstract:

Let

(S

,.)

be a finite semigroup and let T be a non-empty subset of

S

.The graph

Cay

(

S

,

T

)

is defined as the graph whose vertex set is

S

and edges are the pairs

(

x

,

y

)

such that

t

x



y

for some

t



T

_and

y

x



. Such a graph is called the Cayley graph of

S

relative toT. In this paper, we characterize and describe some properties of Cayley graphs of Factorizable inverse Semigroup relative to Green's equivalence classes.

Key Words: Cayley graph, complete graph, Factorizable inverse semigroup, Green's Equivalence classes (L-class, R- class, H-class).

AMS Subject Classification: 05C25, 20M18. 1. Introduction

The Cayley graph of groups was introduced by Arthur Cayley in 1878 and the Cayley graphs of groups have received serious attention since then. The Cayley graphs of semigroups are generalizations of Cayley graphs of groups. The whole section 2.4 of the book [7] is devoted to Cayley graphs of semigroups. In 1964, Bosak [2] and in 1981, Zelinka [9] studied certain graphs over semigroups .Recently in 2006, Kelarev [6] studied on Cayley graphs of inverse semigroups. The concept of Factorizable inverse semigroup has been given Chen and Hsieh in [3].The Green's relations play an important role in the theory of semigroups. In this paper, we study the Cayley graphs of Factorizable inverse semigroups relative to Green's equivalence Classes.

2. Preliminaries

In this section we describe some basic definitions and results in Semigroup theory and Graph theory which are needed in the sequel.

Definition 2.1 A pair

(S

,.)

consisting of a non-empty set

S

and an associative binary operation . on

S

is called a semigroup.

(2)

Aryabhatta Journal of Mathematics and Informatics

http://www.ijmr.net.in

email id- [email protected]

Page 345 Definition 2.3 (c.f. [5]) Let

S

_{be a semigroup. We define}

a

L

b

(

a

,

b



S

)

_{if and only if}a and

b

generates the same principal left ideal, that is, if and only if

S

1

a



S

1

b

. Similarly we define

a

R

b

if and only if a and

b

generates the same principal right ideal, that is, if and only if

a

S

1



b

S

1. We define

b

H

a

if and only if

a

L

b

_and

a

R

b

.

Lemma 2.4 (c. f. [5]) Let

a,

b

be elements of a semigroup

S

. Then

a

L

b

if and only if there exist

1

,

y

S

x



such that

x

a



b

,

y

b



a

_and

a

R

b

_{if and only if} 1

,v S

u  such that

a

u



b

,

b

v



a

.

Notation 2.5 The L-class (R-class, H-class) containing an elementa in a semigroup

S

will be written as a

L (Ra,Ha).

Definition 2.6 A semigroup

S

is called an inverse semigroup if every a in

S

possesses a unique inverse, that is, there exist a unique element

a

1 in

S

such thataa1aa,a1aa1 a1.

Theorem 2.7 ([3]) Let

S

be a semigroup,

G

be a subgroup of

S

and

E



E

(S

)

.Then the following conditions are equivalent

(i)

S



GE

(ii) Le Ge for every

e



E

.

(iii)

R

e



eG

for every

e



E

.

Definition 2.8 A subsemigroup H of a semigroup

S

is said to be factorizable if there exists a subgroup

G

and a subset Eof idempotents of H such that

H



GE



{

ge

/

g



G

,

e



E

}.

Lemma 2.9 ([3]) If the semigroup

S

is factorizable as

S



GE

,then (i)

S



EG

(ii)

S

has an identity 1 which is the identity of

G

(iii)

G



H

1, the H-class containing 1

(iv) For every

x



S

there is a unique

e



E

such that xge for some

g



G

and

E



E

(S

)

, the set of all idempotents of

S

.

(3)

Aryabhatta Journal of Mathematics and Informatics

http://www.ijmr.net.in

email id- [email protected]

Page 346 Definition 2.11 A graph

G

*is a pair

(

V

,

E

)

where

V

is a non-empty set whose elements are called vertices of

G

*

and Eis a subset of

V



V

whose elements are called edges of

G

*

.

The vertex set of a graph

G

* is denoted by

V

(

G

*

)

and edge set is denoted by E(G*).

Definition 2.12 A subgraph H* (U,F) of a graph

G

*=

(

V

,

E

)

is said to be vertex induced subgraph if

F consists of all the edges of

G

* joining pairs of vertices of

U

.

Definition 2.13 A graph of order nwith all possible edges is called a complete graph of order n and is denoted by

K

_n

.

Definition 2.14 Let

S

_{be a finite semigroup and let}

T

be a non-empty subset of

S

. The Cayley graph

)

,

(

S

T

Cay

of

S

relative to

T

is defined as the graph with vertex set

S

and edge set

}

,

:

)

,

{(

x

y

t

x



y

for

some

t



T

x



y

. 3. Main Results

Throughout this section

S

is considered as a Factorizable inverse semigroup on a finite set X and

G

be the group elements, the subgroup of

S

. Let

N



K



S

,

where

N

be the set of all nilpotent elements with nilpotency level 2 and K be the set of all charts or partial symmetries of the form

_











j

i

,where

i

,

j



X

.

Lemma 3.1 Let

S

be a factorizable inverse semigroup on a finite setX with

x

,

y



S

and

g



G

.Then

y

xR

if and only if yxg.

Proof. Suppose that

xR

y

. Then x,yRefor some

e



E

.

Therefore, by Theorem2.7, there exist

G

g

₁

,

₂



such that

x



eg

₁and

y



eg

₂

.

From

x



eg

₁, we have 11 

xg

e and so

y



eg

₂implies

3 2

1 1 2

1

1 ) ( )

(xg g x g g xg

y     for some g₃ g₁1g₂G.

Conversely suppose that yxg for some

g



G

. Therefore

x



yg

1

.

Hence

xR

y

.

Thus

xR

y

if and only if yxg.

(4)

Aryabhatta Journal of Mathematics and Informatics

http://www.ijmr.net.in

email id- [email protected]

Page 347 Proof. Let

x

,

y



K

with

x



y

. Suppose that there is an edge between x andyin

Cay

(

K

,

N

).

Then there exist

g

₁

,

g

₂



G

such that

y



g

₁

x

and

x



g

₂

y

.

Then by lemma 2.4, we have

xLy

.

Conversely, suppose that

x

,

y



K

with

xLy

.

Then

x

,

y



L

efor some

e



E

.

Then by Theorem 2.7, there exist

g

₁

,

g

₂



G

such that

x



g

₁

e

and

y



g

₂

e

.

Then e g₁1x andyg₂g₁1x.That is

x g

y ₃ for some g₃ g₂g₁1G.Since

x



K

, we have

x



k

₁

x

for some ₁

K

.

i

k

_















Thus

x g

y 3 impliesy(g3k1)x. Suppose

(

)

0

2 1

3

k



g

(where 0-empty mapping). Then

g

3

k

1



k

1for some

K

k

₁



,it contradicts our assumption that

xLy

.

Thus g₃k₁N.Hence there is an edge from x toyin

).

,

(

K

N

Cay

Also, we havexk₁1g₃1y=k₁g₃1y.Therefore there is an edge from y to x. Hence there is an edge between x and y in

Cay

(

K

,

N

).

Proposition 3.3 Let

S

be a factorizable inverse semigroup on a finite set X and

L

ebe any Lclass containing e , where

e



E

(K

).

Then the graph induced by

L

e of

Cay

(

K

,

N

)

is a complete graph

K

n, where n |L_e|.

Proof. Let

x

,

y



L

e with

x



y

. Then we have

x

Ly

. Hence, by Proposition 3.2, there is an edge between x and y in

Cay

(

K

,

N

).

Hence there is an edge between x and y in the graph induced by

e

L

of

Cay

(

K

,

N

).

Therefore the graph induced by

L

e of

Cay

(

K

,

N

)

is a complete graph

K

nwhere .

| |L_e n

Theorem 3.4 Let

S

be a factorizable inverse semigroup on a finite set X and

L

ebe any Lclass containing e , where

e



E

(K

).

Then

Cay

(

K

,

N

)

is the disjoint union of induced subgraphs with vertex set

L

e of

Cay

(

K

,

N

).

Proof. Let

e

,

f



E

(

K

).

For

e



f

, we have Le Lf 



,since they are distinct Lclasses. Hence the induced subgraphs with vertex set

L

e and Lf of

Cay

(

K

,

N

)

are disjoint whenever eand

f

are distinct.

Let us consider

Cay

(

L

_e

,

N

)

be the graph induced by the vertex set of

L

eof

Cay

(

K

,

N

).

Now;

V

(

Cay

(

K

,

N

))

= K = eE(K) Le= eE(K)V(Cay(Le,N)). Let

x

,

y



K

with

x



y

. Suppose that there is an edge from x to y in

Cay

(

K

,

N

).

Then by Proposition 3.2 ,we have

x

Ly

.

Then

e

L

y

(5)

Aryabhatta Journal of Mathematics and Informatics

http://www.ijmr.net.in

email id- [email protected]

Page 348 to y in

Cay

(

L

e

,

N

).

Hence there is an edge from x to y in eE(K)Cay(Le,N).Therefore



))

,

(

Cay

K

N

E

E(_e__E₍_K₎Cay(L_e,N)).On the other hand , suppose that there is an edge from x to

y in the disjoint union of

Cay

(

L

_e

,

N

).

Since the union is disjoint and each

Cay

(

L

_e

,

N

)

is complete, it follows that anx and y belongs to

an

L

e for some

e



E

(K

).

Hence

x

Ly

.

Therefore by Proposition 3.2, there is an edge from x to y in

).

,

(

K

N

Cay

Hence E(e_E(K)Cay(Le,N))

E

(

Cay

(

K

,

N

)).

Thus

E

(

Cay

(

K

,

N

))



)).

, ( ( ₍ ₎Cay L N

E _e__E_K _e Hence

Cay

(

K

,

N

)

_{is the disjoint union of}

Cay

(

L

_e

,

N

).

Theorem 3.5 Let

S

be a factorizable inverse semigroup on a finite set X and factorizable as

S



GE

.If e

R is any R-class of

S

for

e



E

, then for

x

,

y



S

with

x



y

there is an edge from x to y in )

, (S R_e

Cay if and only if there exist an

r



R

e such that

r



x

Ly

for every rRe Lr.

Proof. Let

x

,

y



S

with

x



y

.Suppose that there is an edge from xto y inCay(S,R_e), where Re is any R-class in

S

. Then there exist an

r



R

esuch that rx y.

Case (1): Let

e



1

. Then

Cay

(

S

,

R

₁

)

=

Cay

(

S

,

G

)

and so rx yfor some

r



R

eimplies gx yfor some

g



G

.

Since

g



G

, we have

g



g

₁

g

₂for some

g

₁

,

g

₂



G

.

Thus gx yimplies

g

₁

(

g

₂

x

)



y

and so g₂xg₁1y.Hence

g

₂

x

Ly

for every

g

₂



G

.

Conversely suppose that

g

₂

x

Ly

for every

g

₂



G

.

Then g2x,yLefor some

e



E

. Therefore by Theorem 2.7, there exist

g



and

g





G

such that

g

₂

x



g



e

and

y



g



e

.

Then e g 1g₂x

) (  

 and

x g x g g g

y 2 3

1

)

(  



 

for some ( ) 2 .

1 1

3 g g g G

g     Thus there exist an edge from xto y in

).

,

(

S

G

Cay

Case (2): Let

e



1 .

Suppose that there is an edge from xto yin Cay(S,R_e).Then there exist an

r



R

e such that yrx. Let rR_e L_r.Then

r





R

eand

r





L

r. Since

r



,

r



L

r

,

we have

r

L

r



.

Then

e

L

r

,





for some

e



E

.Then by Theorem 2.7, there exist

g

₁

,

g

₂



G

such that

r



g

₁

e

and

r





g

₂

e

.

Then e g1 1r



 and r g2g1 1r.





 That is rg₃r for some g3 g2g1 1G.

 Therefore . ) ( )

(g₃r x g₃ rx g₃y x

r    Thus

r



x

Ly

for every rR_e L_r.

Conversely suppose that

r



x

Ly

for every rR_e L_r.Then

r



x

,

y



L

e for some

e



E

.

Therefore by Theorem 2.7, there exist

g

₁

,

g

₂



G

such that

r



x



g

₁

e

and

y



g

₂

e

.

Theneg1 1(rx)



and yg2g1 1(rx).



That is

y



g

₃

(

r



x

)

for some g3 g2g1 1G.



(6)

Aryabhatta Journal of Mathematics and Informatics

http://www.ijmr.net.in

email id- [email protected]

Page 349 r

e L

R

r  ,we have

r





g

₁

r

and

r



g

₂

r



for someg₂,g₃G.We can choose an element g₃G

such that rg₃r and

r



g

₃

r



.

Suppose not, it contradicts

r





L

_r

.

Thus

k





r



R

eand so there exist an edge from xto y inCay(S,R_e).

Example 3.6 Let

S



{

e

,

g

,

(

1 ,

1 ),

(

1 ,

2 ),

(

2 ,

1 ),

(

2 ,

2 )}

be a set with multiplication defined by

. e

g

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

e e

g

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

g

_g _e _{(2,1) (2,2)} _(1,1) _(1,2) (1,1) (1,1) (1,2) (1,1) (1,2) 0 0 (1,2) (1,2) (1,1) 0 0 (1,1) (1,2) (2,1) (2,1) (2,2) (2,1) (2,2) 0 0 (2,2) (2,2) (2,1) 0 0 (2,1) (2,2)

Then

S

is a factorizable inverse semigroup on a finite set

X



{

1 ,

2 }

with group of units

G



{

e

,

g

}.

Again the LClasses of

S

are

L

₁



{(

1 ,

1 ),

(

2 ,

1 )}

and

L

₂



{(

1 ,

2 ),

(

2 ,

2 )}

and the R Classes of

S

are

)}

2 ,

1 (

),

1 ,

1 {(

1



R

and

R

₂



{(

2 ,

1 ),

(

2 ,

2 )}

respectively. References

[1] Biggs, N. Algebraic Graph Theory, Cambridge University, United Kingdom, (1994).

[2] Bosac, J. Theory of Graphs and application, The graphs of semigroups, Academic Press, Newyork,(1964),119-125.

[3] Chen S.Y., and Hsieh S.C., Factorizable Inverse Semigroups, Semigroup forum, 8(1974), 283-297.

[4] Harary., Graph theory, Narosa publishing House, New Delhi,(1987).

[5] Howie J.M., An Introduction to semigroup theory, Academic press, New York (1976).

[6] Kelarev A.V., On Cayley graphs of inverse semigroups, Semigroup forum,72(2006),411-418.

[7] Kelarev A.V., Graph algebras and automata, Dekker, NewYork (2003).

(7)

Aryabhatta Journal of Mathematics and Informatics

http://www.ijmr.net.in

email id- [email protected]

Page 350 of the International Conference in Algebra,(2010),566-596.