# PROPERTIES OF T – FUZZY SUBSEMIAUTOMATA OVER FINITE GROUPS

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### 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)

(2)

(3)

(4)

### [0,1]

The maximum and minimum will be written as

and

### ∧

respectively. Define T0 on

by

0

0

and

0

### [0,1]

.

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

is said to be

distributive if

### [0,1]

.

Throughout this paper, T shall mean a

### ∨

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

(2)

denote

1,

2,

n1

n

1

n

1

2,

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

### )

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)

(ii)

### 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)]

### G

.

2.4 Definition

A fuzzy subset µ of

### Q

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

### τ

[Kim and Cho (1999)]

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

.

(ii)

for all

### X

2.5 Definition

A fuzzysubset µ of

### Q

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

### τ

if [Kim and Cho (1999)]

(i) µ is a T – fuzzy subgroup

,

(ii)

### X

.

2.6 Definition [Sessa. (1984)]

Let

and

### μ

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

and

is defined by

for all

### G

2.7Remark

(3)

2.8 Example[Mordeson and Nair (1992)]

Let

### }

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

### =

.Clearly A is a fuzzy subgroup of G.

2.9 Example [Bhakat and Kumar (2000)]

Let

### }

be the Klein four groups. Define the fuzzy set

of G defined by

.Clearly

### λ

is not a fuzzy group of G.

2.10 Definition [Das (1997)] Let T =

### )

be a fuzzy semiautomaton over a finite group

. An element

0

### X

is called an e- input if

0

### 0

.

3. Main Results:

Definition 3.1.

A fuzzy semiautomaton

### )

over a finite group

### ∗

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

0

### X

having the following properties.

(1)

0

(2)

1

2−1

0

1

2−1

1

0

1

2

0

2

1

2

1 2

Proposition 3.2.

Let

### )

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

0

.Let

### λ

be a T – fuzzy subsemiautomaton of

, and

### ν

be a T- fuzzy kernel of

.If

### ν

is a T- fuzzy normal subgroup of Q then

### ν

is a T- fuzzy kernel of

.

Proof.

Given

### )

is T- multiplicative

.

0

0

=

0

0

0

−1

0

0

0

−1

0

0

(4)

' 1

1

1

' 1

1

0 0 0

− −

− −

## )

1

' 1 1

0 0 0

1

0 0 0

− − − −

=

0

' 1

0

1

0

0

− −

Since

### ν

is a T fuzzy kernel of

, for all

and

supp

,

−1

0

0

0

1

0

0

=

Hence

### ν

is a T fuzzy kernel of

### )

. □

Example 3.3.

Consider a T multiplicative fuzzy semiautomaton

where

and

.

### )

is a group defined by

### p

is the identity element.

1 1 1

Let

### X

be the e- input since

### 0

T norm is defined by

(5)

Since

,

,

,

Since

,

,

,

Since

,

,

,

Since

,

,

,

Since

,

,

,

Since

,

,

,

Let

### λ

be a fuzzy subset of

defined by

,

,

,

Clearly

### λ

is a T fuzzy subgroup of

### Q

and also a T fuzzy subsemiautomaton 0f

.

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

defined by

,

,

,

also

### ν

is T fuzzy normal subgroup of

.Clearly

### ν

is T fuzzy kernel of

Proposition 3.4

Let

### λ

be a T fuzzy subsemiautomaton of S, and let

### ν

be a T fuzzy kernel of S, such that

. Then

### ∗

is a T fuzzy subsemiautomaton of S,

### ν

is a T fuzzy kernel of

and

### ∩

is a T fuzzy kernel of

### λ

.

Proof

By proposition 3.3 of [1]

### ∗

is T fuzzy subsemiautomaton of S.

and

### ν

is a T fuzzy normal subgroup of

,since

.

### λ

is a T fuzzy kernel of S. It follows that

### λ

is a T fuzzy kernel of

. We know that

so

### ∩

is a T fuzzy normal subgroup of

. Now for

supp (

),

1

1

Thus

### ∩

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

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