**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]

by0

### ( ,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 abusedenote

*T a T a T*

### (

_{1,}

### (

_{2,}

### (..., (

*T a*

_{n}_{−}

_{1}

### ,

*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 **

**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}

### ,

### )

### )

### (

### ) (

### )

### (

### )

## (

### (

### (

_{' 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*

### (

*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

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