• No results found

PROPERTIES OF T – FUZZY SUBSEMIAUTOMATA OVER FINITE GROUPS

N/A
N/A
Protected

Academic year: 2020

Share "PROPERTIES OF T – FUZZY SUBSEMIAUTOMATA OVER FINITE GROUPS"

Copied!
6
0
0

Loading.... (view fulltext now)

Full text

(1)

PROPERTIES OF T – FUZZY

SUBSEMIAUTOMATA OVER FINITE

GROUPS

M. BASHEER AHAMED

University of Tabuk, Kingdom of Saudi Arabia

J. MICHAEL ANNA SPINNELI

Department of Mathematics, Karunya University, Coimbatore, India

Abstract:

In 1967, Wee introduced the concept of fuzzy automata, using Zadeh’s concept of fuzzy sets. A group semiautomaton has been extensively studied by Fong and Clay .This group semiautomaton was fuzzified by Das and he introduced fuzzy semiautomaton over finite group. Sung-jin Cho et al. introduced the notion of T-fuzzy semiautomata, T-fuzzy kernel, and T-fuzzy subsemiautomata over a finite group. In this paper, we give some properties of T-fuzzy subsemiautomata over finite groups with illustrations.

Keywords: T-fuzzy normal subgroup; T-fuzzy kernel; T-fuzzy subsemiautomata

1. Introduction:

Fuzzy automata theory has been developed by many researchers, which was introduced by Wee. [Das (1997)] introduced fuzzy semiautomaton over a finite group. T fuzzy sub group, T fuzzy normal sub group, T fuzzy semi automaton over a finite group is defined by [Kim and Cho (1999)]. Some more results of T fuzzy subsemiautomaton and T fuzzy kernel are proved in [Ahamed and Spinneli (2010)]. In this paper we have proved some properties of T fuzzy subsemiautomaton.

2. Preliminaries:

2.1 Definition

A binary operation T on

[ ]

0,1

is called a t- norm if [Kim and Cho (1999)] (1)

T a

( ,1)

=

a

(2)

T a b

( , )

T a c whenever b

( , )

c

(3)

T a b

( , )

=

T b a

( , )

(4)

T a T b c

( , ( , ))

=

T T a b c for all a b c

( ( , ), )

, ,

[0,1]

The maximum and minimum will be written as

and

respectively. Define T0 on

[0,1]

by

0

( ,1)

0

(1, )

T a

= =

a

T

a

and

T a b

0

( , )

=

0

if a

1

and b

1

for all a b

,

[0,1]

.

Here T always will mean a t- norm on[0,1].A t-norm T on

[0,1]

is said to be

distributive if

T a b

( ,

∨ =

c

)

T a b

( , )

T a c forall a b c

( , )

, ,

[0,1]

.

Throughout this paper, T shall mean a

- distributive t- norm on[0,1] unless otherwise specified. By an abuse

(2)

denote

T a T a T

(

1,

(

2,

(..., (

T a

n1

,

a

n

)....)))

by T a

( ,...

1

a

n

)

where a a

1

,

2,

....

a

n

[0,1]

. The legitimacy of this abuse is ensured by the associativity of T (Definition 2.1(4)).

Note For further discussions we are considering

(

G

,

+

)

as a finite group.

2.2 Definition

A fuzzy subset

λ

of G is called a T – fuzzy subgroup of G if [Kim and Cho (1999)]

(i)

λ

(

x

+

y

)

T

( ( ), ( )),

λ

x

λ

y

(ii)

λ

( )

x

= −

λ

(

x for all x y

)

,

G

.

2.3 Definition A T – fuzzy subgroup

λ

of G is called a T- fuzzy normal subgroup of G if [Kim and Cho (1999)]

λ

(

x

+

y

)

=

λ

(

y

+

x for all x y

)

,

G

.

2.4 Definition

A fuzzy subset µ of

Q

is called a T- fuzzy kernel of a T- fuzzy semiautomaton

M

=

( ,

Q X

, )

τ

[Kim and Cho (1999)]

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

Q

.

(ii)

μ

(

p

− ≥

r

)

T

( (

τ

q

+

k x p

, , ), ( , , ), ( ))

τ

q x r

μ

k

for all

p q r k

, , ,

Q x

,

X

2.5 Definition

A fuzzysubset µ of

Q

is called a T- fuzzy subsemiautomaton of a T-fuzzy semi automaton

M

=

( ,

Q X

, )

τ

if [Kim and Cho (1999)]

(i) µ is a T – fuzzy subgroup

Q

,

(ii)

μ

( )

p

T

( ( , , ), ( ))

τ

q x p

μ

q

forall p q

,

Q x

,

X

.

2.6 Definition [Sessa. (1984)]

Let

λ

and

μ

be T- fuzzy subsets of G, then the sum of

λ

and

μ

is defined by

{

}

(

λ μ

+

)( )

x

= ∨

T

( ( ), ( )) / ,

λ

y

μ

x

y z

G such that x

= +

y

z

for all

x

G

2.7Remark

(3)

2.8 Example[Mordeson and Nair (1992)]

Let

G

=

{

e a b c

, , ,

}

be the Klein four group. Define the fuzzy subset A of G by

( )

1,

( )

1,

( )

3 / 4,

( )

3 / 4

A e

=

A a

=

A b

=

A c

=

.Clearly A is a fuzzy subgroup of G.

2.9 Example [Bhakat and Kumar (2000)]

Let

G

=

{

e a b c

, , ,

}

be the Klein four groups. Define the fuzzy set

λ

of G defined by

( )

e

0.6,

( )

a

0.7,

( )

b

0.4,

( )

c

0.4

λ

=

λ

=

λ

=

λ

=

.Clearly

λ

is not a fuzzy group of G.

2.10 Definition [Das (1997)] Let T =

(

Q X

,

,

δ

)

be a fuzzy semiautomaton over a finite group

( )

G

,

. An element

x

0

X

is called an e- input if

δ

(

e x e

,

0

,

)

>

0

.

3. Main Results:

Definition 3.1.

A fuzzy semiautomaton

T

=

(

Q X

,

,

δ

)

over a finite group

( )

G

,

is called T- multiplicative if there exists an e – input

x

0

X

having the following properties.

(1)

δ

(

q x p r

, ,

∗ ≥

)

T

(

δ

(

q x p

,

0

,

) (

,

δ

e x r

, ,

)

)

p q r

, ,

Q x

,

X

(2)

δ

(

p

1

p

2−1

,

x q

0

,

1

q

2−1

)

T

(

δ

(

p x q

1

,

0

,

1

) (

,

δ

p x q

2

,

0

,

2

)

)

p p q q

1

,

2

, ,

1 2

Q

Proposition 3.2.

Let

T

=

(

Q X

,

,

δ

)

be a T – multiplicative fuzzysemiautomon over a finite group with e – input

x

0

X

.Let

λ

be a T – fuzzy subsemiautomaton of

T

=

(

Q X

,

,

δ

)

, and

ν

be a T- fuzzy kernel of

λ

.If

ν

is a T- fuzzy normal subgroup of Q then

ν

is a T- fuzzy kernel of

T

=

(

Q X

,

,

δ

)

.

Proof.

Given

T

=

(

Q X

,

,

δ

)

is T- multiplicative

p q r k

, , ,

Q and x

X

.

(

) (

)

(

, ,

,

, ,

)

(

(

, ,

) (

,

, ,

)

)

T

δ

q k x p

δ

q x r

=

T

δ

q k e x p e

∗ ∗

δ

q e x r e

T T

(

(

δ

(

q k x p

,

0

,

) (

,

δ

e x e

, ,

)

)

,

T

(

δ

(

q x r

,

0

,

) (

,

δ

e x e

, ,

)

)

)

=

T

(

δ

(

q k x p

,

0

,

) (

,

δ

e x e

, ,

) (

,

δ

q x r

,

0

,

)

)

T T

(

(

δ

(

q x p

,

0

,

)

,

δ

(

k

−1

,

x e

0

,

)

)

,

δ

(

e x e

, ,

) (

,

δ

q x r

,

0

,

)

)

T

(

δ

(

q x p

,

0

,

)

,

δ

(

k

−1

,

x e

0

,

)

,

δ

(

e x e

, ,

) (

,

δ

q x r

,

0

,

)

)

(4)

(

) (

)

(

)

(

(

(

' 1

)

1

)

(

1

)

(

)

(

(

' 1

)

1

)

)

0 0 0

, ,

,

, ,

,

,

,

,

,

,

, ,

,

,

,

T

δ

q k x p

δ

q x r

=

T

δ

b

∗ ∗

b q

− −

x p

δ

k

x e

δ

e x e

δ

b

∗ ∗

b q

− −

x r

(

)

(

(

)

)

(

)

(

)

(

) (

)

(

(

)

(

)

)

1

' 1 1

0 0 0

1

0 0 0

,

,

,

,

,

,

,

,

,

, ,

,

,

,

,

,

,

,

,

,

T

b x p

b q

x e

k

x e

T

e x e

b x r T

b x r

b q

x r

δ

δ

δ

δ

δ

δ

δ

− − − −

=

T

(

δ

(

b x p

,

0

,

)

,

δ

(

b q

' 1

,

x e

0

,

) (

,

δ

k

1

,

x e

0

,

)

,

δ

(

e x e

, ,

) (

,

δ

b x r

,

0

,

)

)

− −

Since

ν

is a T fuzzy kernel of

λ

, for all

p q r k

, , ,

Q

and

b

supp

( )

λ

,

x

X

ν

(

p r

−1

)

T

(

δ

(

b k x p

, ,

) (

,

δ

b x r

, ,

) ( )

,

ν

k

)

T

(

δ

(

b k x p

,

0

,

) (

,

δ

b x r

,

0

,

) (

,

δ

e x e

, ,

) ( )

,

ν

k

)

T

(

δ

(

b x p

,

0

,

)

,

δ

(

k

1

,

x e

0

,

)

,

δ

(

b x r

,

0

,

) (

,

δ

e x e

, ,

) ( )

,

ν

k

)

=

T

(

δ

(

q k x p

, ,

) (

,

δ

q x r

, ,

) ( )

,

ν

k

)

Hence

ν

is a T fuzzy kernel of

T

=

(

Q X

,

,

δ

)

. □

Example 3.3.

Consider a T multiplicative fuzzy semiautomaton

T

=

(

Q X

,

,

δ

)

where

Q

=

{

p q r k

, , ,

}

and

X

=

{

a b c

, ,

}

.

(

Q

,

+

)

is a group defined by

,

,

,

p

+ =

p

p p

+ =

q

q p

+ =

r

r p

+ =

k

k

,

,

,

q

+ =

p

q q

+ =

q

p q

+ =

r

k q

+ =

k

r

,

,

,

r

+ =

p

r r

+ =

q

k r

+ =

r

q r

+ =

k

p

,

,

,

k

+ =

p

k k

+ =

q

r k

+ =

r

p k

+ =

k

q

p

is the identity element.

1 1 1

,

,

q

=

q r

=

k k

=

r

Let

a

X

be the e- input since

δ

(

p a p

, ,

)

>

0

T norm is defined by

T a b

( )

,

=

ab

(5)

(

q b k r

, ,

)

T

(

(

q a k

, ,

) (

,

p b r

, ,

)

)

δ

∗ ≥

δ

δ

Since

δ

(

q b k r

, ,

∗ =

)

0

,

δ

(

q a k

, ,

)

=

0

,

δ

(

p b r

, ,

)

=

0

,

T

( )

0, 0

=

0

(

q c k r

, ,

)

T

(

(

q a k

, ,

) (

,

p c r

, ,

)

)

δ

∗ ≥

δ

δ

Since

δ

(

q c k r

, ,

∗ =

)

0.4

,

δ

(

q a k

, ,

)

=

0

,

δ

(

p c r

, ,

)

=

0

,

T

(

0, 0.1

)

=

0

(

r b q k

, ,

)

T

(

(

r a q

, ,

) (

,

p b k

, ,

)

)

δ

∗ ≥

δ

δ

Since

δ

(

r b q k

, ,

∗ =

)

0

,

δ

(

r a q

, ,

)

=

0

,

δ

(

p b k

, ,

)

=

0

,

T

( )

0, 0

=

0

(

r c q k

, ,

)

T

(

(

r a q

, ,

) (

,

p c k

, ,

)

)

δ

∗ ≥

δ

δ

Since

δ

(

r c q k

, ,

∗ =

)

0.3

,

δ

(

r a q

, ,

)

=

0

,

δ

(

p c k

, ,

)

=

0

,

T

( )

0, 0

=

0

(

k b q r

, ,

)

T

(

(

k a q

, ,

) (

,

p b r

, ,

)

)

δ

∗ ≥

δ

δ

Since

δ

(

k b q r

, ,

∗ =

)

0.2

,

δ

(

k a q

, ,

)

=

0

,

δ

(

p b r

, ,

)

=

0

,

T

( )

0, 0

=

0

(

k c q r

, ,

)

T

(

(

k a q

, ,

) (

,

p c r

, ,

)

)

δ

∗ ≥

δ

δ

Since

δ

(

k c q r

, ,

∗ =

)

0.2

,

δ

(

k a q

, ,

)

=

0

,

δ

(

p c r

, ,

)

=

0.1

,

T

(

0, 0.1

)

=

0

Let

λ

be a fuzzy subset of

Q

defined by

λ

( )

p

=

0.4

,

λ

( )

q

=

0.1

,

λ

( )

r

=

0.25

,

λ

( )

r

=

0.25

Clearly

λ

is a T fuzzy subgroup of

Q

and also a T fuzzy subsemiautomaton 0f

Q

.

p q

r k

c/0.1

a/0.1

a/0.1 b/0.1

c/0.3

a/0.3 b/0.2

c/0.4 b/0.3

(6)

Let

ν

be a fuzzy subset of

Q

defined by

ν

( )

p

=

0.6

,

ν

( )

q

=

0.2

,

ν

( )

r

=

0.1

,

ν

( )

k

=

0.1

also

ν

is T fuzzy normal subgroup of

Q

.Clearly

ν

is T fuzzy kernel of

T

=

(

Q X

,

,

δ

)

Proposition 3.4

Let

λ

be a T fuzzy subsemiautomaton of S, and let

ν

be a T fuzzy kernel of S, such that

λ

( ) ( )

e

=

ν

e

. Then

λ ν

is a T fuzzy subsemiautomaton of S,

ν

is a T fuzzy kernel of

λ ν

and

λ ν

is a T fuzzy kernel of

λ

.

Proof

By proposition 3.3 of [1]

λ ν

is T fuzzy subsemiautomaton of S.

ν λ ν

⊆ ∗

and

ν

is a T fuzzy normal subgroup of

λ ν

,since

λ

( ) ( )

e

=

ν

e

.

λ

is a T fuzzy kernel of S. It follows that

λ

is a T fuzzy kernel of

λ ν

. We know that

λ ν λ

∩ ⊆

so

λ ν

is a T fuzzy normal subgroup of

λ

. Now for

, ,

p r k

Q and q

supp (

λ

),

x

X

(

)

(

1

)

(

(

) (

) ( )

(

1

)

( )

)

, ,

,

, ,

,

,

, ,

,

p r

T

q k x p

q x r

k

k x p r

k

λ ν

μ

μ

ν

μ

λ

=

T

(

μ

(

q k x p

, ,

) (

,

μ

q x r

, ,

) (

,

λ ν

)( )

k

)

Thus

λ ν

is a T fuzzy kernel of

λ

. □

References

[1] Basheer Ahamed, M., Michael Anna Spinneli, J. (2010). “Some Results of T-Fuzzy Subsemiautomata over Finite Groups”. International Journal of Computer and Network Security. 2, pp. 48- 51

[2] Bhakat, Sandeep Kumar. (2000). “(, , Vq)-fuzzy normal, quasinormal and maximal subgroups” . Fuzzy Sets and Systems. 112, pp. 299-312

References

Related documents

Mastery goal orientation (MGO) Scale and Teaching Self-Efficacy (TSE) Scale were used to know the level of Mastery goal orientation and the level of Teaching self- efficacy of

In conclusion, the findings of this review of reviews suggest that first generation migrants including refugees/ asylum seekers are at increased risk of mental ill-health,

the baseline and test shelters in grid zones throughout the case study countries. Differences between the test and baseline cases were calculated to reveal the fuel savings

There is also the difficulty that Brown and Levinson never really attempted to prove, in any methodologically rigorous fashion, the connection between either type of politeness and

Determination of the factors affecting duration of hospitalization in patients with chronic obstructive pulmonary disease (COPD) in Iran.. Seyed Ali Javad Mousavi,

In the model that included loneliness, living arrangements, social networks outside the household, socio-demographics, and health status indicators (corresponding to Model 3 above)

From the results of our study, we concur that the initial expenditure for the da Vinci® Surgical Systems is substan- tial, it is associated with the potential cost benefits of re-

When comparing all patients with complete data amenable for risk strati fi cation, following adjustment for age group, ethnicity, year of diagnosis and tumour characteristics (model