T-Fuzzy Subsemiautomata with Thresholds

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T-FUZZY SUBSEMIAUTOMATA WITH

THRESHOLDS

M.Basheer Ahamed

Department of Mathematics

University of Tabuk

Tabuk-71491

Kingdom of Saudi Arabia.

E-mail: [email protected]

J.Michael Anna Spinneli

Department of Mathematics

Karunya University

Coimbatore-641114

Tamil Nadu, India.

E-mail: [email protected]

Abstract: Rosenfeld introduced fuzzy subgroup in the

year 1971. Bhakat and Das introduced

, , q fuzzy subgroup in 1996. In 2003 Yuan

introduced fuzzy subgroup with thresholds. Bao Qing

Hu introduced T- fuzzy subgroup with thresholds in the

year 2010. In this paper we have defined T-fuzzy kernel

with thresholds and T-fuzzy subsemiautomata with

thresholds and some results concerning them.

Key words: T-fuzzy subgroup with threshold; T-fuzzy

kernel with thresholds and T-fuzzy subsemiautomata with

thresholds.

Mathematics subject Classification 2010: 03E72, 20M35.

I. INTRODUCTION

Zadeh introduced fuzzy sets in 1965.Rosenfield

defined fuzzy subgroup. Anthony and Sherwood

replaced “min” in Rosenfeld axiom by t-norm and

introduced T-Fuzzy subgroup, and Mukherjee

defined fuzzy normal subgroup.Bhakat and Das

introduced , , q fuzzy subgroup in 1996.Kim

.et.al defined T-fuzzy subgroups in 1999.In 2003

Yuan.et.al introduced fuzzy subgroup with thresholds

and implication based fuzzy subgroups .T-fuzzy

subgroups with thresholds was defined by Bao Qing

Hu in 2010.Das introduced fuzzy kernel and fuzzy

subsemiautomata in1996.Kim defined T-fuzzy kernel

and T-fuzzy subsemiautomata.Using Bao Qing Hu

notion of T-fuzzy subgroups with thresholds we have

defined T-fuzzy kernel with thresholds and T-fuzzy

subsemiautomata with thresholds and discuss some

properties of it.

II PRELIMINARIES

Definition 2.1[6]: Let X be a nonempty set. A fuzzy

set A in X is characterized by its membership

function : 0,1 A X and A x is interpreted as

the degree of membership of element x in fuzzy set

A for each x X .

Definition 2.2 [4]: A fuzzy subset of a group G

is a fuzzy subgroup of G if for all , x y G

(i) x y x y

(ii) 1 x x

Definition 2.3[4]: A fuzzy subgroup of G is

called a fuzzy normal subgroup of G if

1 x y x y for all , x y G .

(IJCSIS) International Journal of Computer Science and Information Security,

Vol. 9, No. 8, August 2011

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Definition 2.4[1] A fuzzy subset of G is called a

T – fuzzy subgroup of G if

(i) ( ) ( ( ), ( )), x y T x y

(ii) ( ) ( ) , x x for all x y G .

Definition 2.5[1] A T – fuzzy subgroup of G is

called a T- fuzzy normal subgroup of G if

( ) ( ) , x y y x for all x y G .

Definition 2.6[5]: Let , 0,1 and .Let

A be a fuzzy subset of a group G . Then A is called

a fuzzy subgroup with thresholds of G if for all

, x y G

(i) A x y A x A y

(ii) 1 A x A x

Definition 2.7[2]: Let , 0,1 and < . Let

be a fuzzy subset of a group G . Then is called

a fuzzy normal subgroup with thresholds of G if

1 , , . y x y x x y G

Definition 2.8[3] Let ( , , ) M Q X be a fuzzysemi automaton. Define

* *: [0,1]Q X Q

by* 1

0( , , ) { i fp q

i fp q p q

And

*

1 1 1 1 2 2 1( , ..... , ) ( ( , , ), ( , , )....., ( , , ) /

n n n i p a a q T p a q q a q q a q q Q

where , p q Q1,..... .

na a X When T is applied

M as above, M is called a T-fuzzy semi automaton.

Definition 2.9 [3] A fuzzy subset µ of Q is called a

T- fuzzy kernel of a T- fuzzy semi automaton

( , , ) M Q X

(i) µ is a T- fuzzy normal subgroup of Q .

(ii) ( ) ( ( , , ), ( , , ), ( )) p r T q k x p q x r k

for all , , , , p q r k Q x X

Definition 2.10[3] A fuzzy subset µ of Q is called a

T- fuzzy sub semi automaton of a T-fuzzy semi

automaton ( , , ) M Q X if

(i) µ is a T – fuzzy subgroup Q ,

(ii)

( ) ( ( , , ), ( ))

, ,

p T q x p q forall

p q Q x X

.

Definition 2.11 [2] Let , 0,1 and

.Let be a fuzzy subset of a group G. is

called a T-fuzzy subgroup with thresholds

and of a group G with respect to t-norm norm

and t-conorm c if for all , x y G

(i) c xy x y

(ii) 1

c x x

.

Definition: 2.12[2] Let , 0,1 and .A

T-fuzzy subgroup with thresholds and of a

group G with respect to t-norm norm and t-conorm

c is said to be T-fuzzy normal with thresholds if

1

c y x y x for all

, x y G .

III MAIN RESULTS

Definition3.1: Let , 0,1 and < . Let

( , , )S Q X be a fuzzy semiautomaton over a

finite group. A fuzzy set of Q is called a T-

fuzzy kernel of ( , , )S Q X with thresholds

, with respect to t-norm and t-co norm c

(i) if is a T-fuzzy normal subgroup of

Q with thresholds ,



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(ii)

1 , ,

, ,

c p r q k x p

q x r k

for

all , , , p q r k Q

Definition3.2: Let , 0,1 and < .

Let

, ,S Q X be a fuzzy semiautomaton over a

finite Group. A fuzzy subset of Q is called a T-

fuzzy subsemiautomaton of S with thresholds with

respect to t-norm and t-co norm c if the

following conditions hold

(i) is a T-fuzzy subgroup of Q with

thresholds

(ii) , ,c p q x p q for all , p q Q and x X .

Definition3.3: Let and be fuzzy subsets of G

with thresholds , . The product is defined

by

/

,c

y z

x y z G suchthat

x y z

.

Note: Let , ,S Q X be a fuzzy semiautomaton

over a finite group in the remaining of the results.

The element ' 'e be the identity of , .Q

Proposition3.4:

Let , 0,1 and . Let be a T-fuzzy

kernel of , ,S Q X with thresholds , .

Then is a T-fuzzy subsemiautomaton of S with

thresholds , if and only if

, ,c p e x p e for all

, p Q x X .

Proof: We have

1

c c p p r r

Since every fuzzy subgroup with thresholds is a

Rosenfield’s subgroup

1

c p r r

1

c p r r

, , , ,e q x p e x r q r

By definition

, , , ,q x p e x r e q

By given condition

, , , ,q x p e x r q ,

Since e q

, ,q x p q . □

Proposition 3.5: Let , 0,1 and .

Let be a T-fuzzy kernel of , ,S Q X with

thresholds , and be a T- fuzzy

subsemiautomaton of S with thresholds , .

Then is a T-fuzzy subsemiautomaton of S with

thresholds , .

Proof: Since is a T -fuzzy normal subgroup withthresholds and is a T- fuzzy subgroup with

thresholds of Q , it follows that is a T-fuzzy

sub group of Q with thresholds ,




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1

c c p p r r

1 p r r

1

c p r r

1

c c p r r

, , , ,

, ,

a b x p a x r b

a x r a

, ,a b x p b a

1

, ,

/

, ,

a b x p b

p a

a b Q a b q

/

, ,, ,

b aq x p

a b q a b Q

, ,q x p q

, ,q x p q

, ,q x p q

□

Proposition 3.6: Let , 0,1 and .If

and are T-fuzzy kernels of , ,S Q X with

thresholds , ,then T-fuzzy kernels of

, ,S Q X with thresholds , .

Proof: Since and are T-fuzzy normal subgroups

of Q with thresholds then is also a T-fuzzy

normal subgroup of Q with thresholds.

1

1 1

c

c

p r

p q q r

1 1 p q q r (By definition)

1 1

c p q q r

1 1

c c p q q r

, , , ,

, , , ,

a b c x p a b x q c

a b x q a x r b

(By definition)

= , ,a b c x p c b

1

, ,

, ,

/

, ,

c

q b c x p

q x r c p r

b

b c Q b c k

, , , ,, ,

c bq k x p q x r

b c k b c Q




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, , , , cq k x p q x r k

, , , ,q k x p q x r k

□

REFERENCES

[1] Anthony J. M. and Sherwood, H., A characterization of fuzzy

groups, Fuzzy Sets and System, 7, (1982) pp 297-305.

[2] Bao Qing Hu, Fuzzy groups and T- fuzzy Groups with

Thresholds, Advances in fuzzy mathematics Vol. 5, No.1,

(2010) pp. 17-29.

[3] Cho, S.J., Kim, J.G., Tae Kim, S. (1999). “T- fuzzy

semiautomata over finite groups”. Fuzzy sets and systems 108, pp.

341-351

[4] P. Das, On some properties of fuzzy semiautomaton over afinite group, Information Sciences 101, (1997) pp.71-84

[5] Yuan, X., Zhang, C., and Ren, Y., Generalized fuzzy groups

and many valued implications, Fuzzy Sets and Systems, 138,

(2003) pp. 205-211

[6] Zadeh, L. A., Fuzzy sets, Information and Control, 8, (1965)

pp. 338-353.

AUTHORS PROFILE

Basheer Ahamed. M Received his Ph,D., Degree in Mathematics fromBharathidasan University, Tiruchirappalli, Tamilnadu in2005. He is working in Deparment of Mathematics,University of Tabuk, Kingdom of Saudi Arabia. Hisresearch areas are Fuzzy Mathematics and DiscreteMathematics.

J. Michael Anna Spinneli

Received her M.Sc and M.Phil degrees in mathematicsfrom Manonmanium Sundaranar University.Tirunelveli,

Tamilnadu. Now she is working as an Assistant Professorof mathematics in Karunya University, Coimbatore, India.She is doing research on Fuzzy Automata.




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T-Fuzzy Subsemiautomata with Thresholds