APPLICATIONS OF A GROUP IN GENERAL FUZZY AUTOMATA. Communicated by S. Alikhani

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1 Algebraic Structures and Their Applications Vol. 4 No. 2 ( 2017 ) pp APPLICATIONS OF A GROUP IN GENERAL FUZZY AUTOMATA M. HORRY Communicated by S. Alikhani Abstract. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2) be a general fuzzy automaton and the set of its states be a group. The aim of this paper is the study of applications of a group in a general fuzzy automaton. For this purpose, we define the concepts of fuzzy normal kernel of a general fuzzy automaton, fuzzy kernel of a general fuzzy automaton, adjustable, multiplicative. Then we obtain the relationships between them. 1. Introduction The concept of fuzzy automata was introduced by Wee in 1967 [11]. Let Σ be a set. A word in Σ is the product of a finite sequence of elements in Σ. Λ will denote the empty word and Σ the set of all words on Σ. The length l(x) of the word x Σ is the number of its letters, so l(λ) = 0. For a nonempty set X, P (X) will denote the set of all fuzzy sets on X and P (X) will denote the set of all subsets on X. A deterministic finite-state automaton is a five-tuple denoted as A = (Q, Σ, f, T, s), where Q is a finite set of states, Σ is a finite set of input symbols, the total function f from Q Σ into DOI: MSC(2010): Primary:18B20 Keywords: (General) Fuzzy automata, group, normal subgroup, fuzzy subgroup, fuzzy normal subgroup. Received: 10 November 2017, Accepted: 17 July 2018 Corresponding author c 2017 Yazd University. 57

2 58 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Q is the state transition, T is a subset of Q of accepting states and s Q is the initial state. A word x = x 1 x 2... x n Σ is said to be accepted by A if there exist states q 0, q 1,..., q n satisfying (1) q 0 = s (2) f(q i 1, x i ) = q i for i = 1, 2,..., n, (3)q n T. The empty word is accepted by A if and only if s T. A nondeterministic finite-state automaton is a five-tuple denoted as A = (Q, Σ, f, T, s), where Q is a finite set of states, Σ is a finite set of input symbols, the partial function f from Q Σ into P (Q) is the state transition, T is a subset of Q of accepting states and s Q is the initial state. A fuzzy finite-state automaton (FFA) is a six-tuple F = (Q, Σ, R, Z, δ, ω), where Q is a finite set of states, Σ is a finite set of input symbols, R is the initial state of F, Z is a finite set of output symbols, δ : Q Σ Q [0, 1] is the fuzzy transition function which is used to map a state (current state) into another state (next state) upon an input symbol, attributing a value in the interval [0, 1] and ω : Q Z is the output function. Associated with each fuzzy transition, there is a membership value in [0, 1] called the weight of the transition. The transition from state q i (current state) to state q j (next state) upon input a k is denoted by δ(q i, a k, q j ). We use this notation to refer both to a transition and its weight. Whenever δ(q i, a k, q j ) is used as a value, it refers to the weight of the transition. Otherwise, it specifies the transition itself. The set of all transitions of F will be denoted by. The above definition is generally accepted as a formal definition of a fuzzy finite-state automaton [4, 5, 6, 7, 8, 9]. In 2004, M. Doostfatemeh and S.C. Kremer extended the notion of fuzzy automata and introduced the notion of general fuzzy automata [1]. In this paper, we define the concepts of fuzzy normal kernel of a general fuzzy automaton, fuzzy kernel of a general fuzzy automaton, adjustable, multiplicative and obtain the relationships between them. Definition 1.1. [1] A general fuzzy automaton (GFA) is an eight-tuple machine F = (Q, Σ, R, Z, δ, ω, F 1, F 2 ), where (i) Q is a finite set of states, Q = {q 1, q 2,..., q n }, (ii) Σ is a finite set of input symbols, Σ = {a 1, a 2,..., a m }, (iii) R is the set of fuzzy start states, R P (Q), (iv) Z is a finite set of output symbols, Z = {b 1, b 2,..., b k }, (v) ω : Q Z is the output function, (vi) δ : (Q [0, 1]) Σ Q [0, 1] is the augmented transition function,

3 Alg. Struc. Appl. Vol. 4 No. 2 (2017) (vii) F 1 : [0, 1] [0, 1] [0, 1] is the membership assignment function, (viii) F 2 : [0, 1] [0, 1] is called the multi-membership resolution function. We note that the function F 1 (µ, δ) has two parameters µ and δ, where µ is the membership value of a predecessor and δ is the weight of a transition and δ : Q Σ Q [0, 1] is the fuzzy transition function which is used to map a state (current state) into another state (next state) upon an input symbol, attributing a value in the interval [0, 1]. In this definition, the process that takes place upon the transition from state q i to q j on input a k is represented as: µ t+1 (q j ) = δ((q i, µ t (q i )), a k, q j ) = F 1 (µ t (q i ), δ(q i, a k, q j )). This means that t {0, 1, 2,, n} and the membership value (mv) of the state q j at time t + 1 is computed by function F 1 using both the membership value of q i at time t and the weight of the transition. The usual options for the function F 1 (µ, δ) are max{µ, δ}, min{µ, δ} and (µ + δ)/2. The multi-membership resolution function resolves the multi-membership active states and assigns a single membership value to them. and Let Q act (t i ) be the fuzzy set of all active states at time t i, i 0. We have Q act (t 0 ) = R Q act (t i ) = {(q, µ t i (q)) : q Q act (t i 1 ), a Σ, δ(q, a, q) }, i 1. Since Q act (t i ) is a fuzzy set, in order to show that a state q belongs to Q act (t i ) and T is a subset of Q act (t i ), we should write: q Domain(Q act (t i )) and T Domain(Q act (t i )). Hereafter, we simply denote them as: q Q act (t i ) and T Q act (t i ). The combination of the operations of functions F 1 and F 2 on a multi-membership state q j leads to the multi-membership resolution algorithm. Algorithm 1. [1] (Multi-membership resolution) If there are several simultaneous transitions to the active state q j at time t + 1, the following algorithm will assign a unified membership value to it: (1) Each transition weight δ(q i, a k, q j ) together with µ t (q i ), will be processed by the membership assignment function F 1, and will produce a membership value. Call this v i. v i = δ((q i, µ t (q i )), a k, q j ) = F 1 (µ t (q i ), δ(q i, a k, q j )). (2) These membership values are not necessarily equal. Hence, they need to be processed by the multi-membership resolution function F 2. (3) The result produced by F 2 will be assigned as the instantaneous membership value of the active state q j, µ t+1 (q j ) = F n 2 [v i ] = F n 2 [F 1 (µ t (q i ), δ(q i, a k, q j ))]. i=1 i=1

4 60 Alg. Struc. Appl. Vol. 4 No. 2 (2017) where n is the number of simultaneous transitions to the active state q j at time t + 1. δ(q i, a k, q j ) is the weight of a transition from q i to q j upon input a k. µ t (q i ) is the membership value of q i at time t. µ t+1 (q j ) is the final membership value of q j at time t + 1. Definition 1.2. [12] Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton. define max-min general fuzzy automata as F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) such that : We δ : Q act Σ Q [0, 1] where Q act = {Q act (t 0 ), Q act (t 1 ), Q act (t 2 ),... } and for all i 0, δ ((q, µ t i 1, q = p, (q)), Λ, p) = 0, otherwise Also, if the input at time t i be u i, where u i Σ, 1 i n, then δ ((q, µ t i 1 (q)), u i, p) = δ((q, µ t i 1 (q)), u i, p), δ ((q, µ t i 1 (q)), u i u i+1, p) = ( δ((q, µ t i 1 (q)), u i, q ) δ((q, µ t i (q )), u i+1, p)), q Q act(t i ) and recursively δ ((q, µ t 0 (q)), u 1 u 2... u n, p) = { δ((q, µ t 0 (q)), u 1, p 1 ) δ((p 1, µ t 1 (p 1 )), u 2, p 2 )... δ((p n 1, µ t n 1 (p n 1 )), u n, p) p 1 Q act (t 1 ), p 2 Q act (t 2 ),..., p n 1 Q act (t n 1 )}. If q Q act (t i ), we should write q belongs to an element of Q act. Hereafter, we simply denote it as: q Q act. Definition 1.3. [5] A fuzzy subset µ of X is a function of X into the closed interval [0,1]. The support of µ is defined to be the set, Supp(µ) = {x X µ(x) > 0} Definition 1.4. [5] Let λ and µ be fuzzy subsets of G. The product λ µ of λ and µ is defined by (λ µ)(x) = {λ(y) µ(z) y, z G, x = y z} for all x G. We let F P (X) denote the fuzzy power set of X. So F P (X) is the set of all fuzzy subsets of X. F P (X) is a semigroup with respect to the product of fuzzy subsets of X.

5 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Definition 1.5. [5] Let (G, ) be a group. A fuzzy subset λ of G is called a fuzzy subgroup of G if the following properties hold: i) λ(x y) λ(x) λ(y) ii) λ(x) = λ(x 1 ) for all x, y G. Definition 1.6. [5] A fuzzy subgroup λ of G is called a fuzzy normal subgroup of G if λ(x y x 1 ) λ(y) for all x, y G. Definition 1.7. [5] Let λ and µ be fuzzy subsets of G and λ µ. Then λ is called a fuzzy normal subgroup of µ if λ(x y x 1 ) λ(x) λ(y) for all x, y G. 2. Applications of a group in general fuzzy automata Definition 2.1. Let F 1 = (Q 1, Σ 1, R 1, Z, ω, δ 1, F 1, F 2 ) and F 2 = (Q 2, Σ 2, R 2, Z, ω, δ 2, F 1, F 2 ) be general fuzzy automata and (Q 1, ) and (Q 2, ) be groups. A pair of functions (f, g), where f : Q 1 Q 2, g : Σ 1 Σ 2, is called a homomorphism from F 1 into F 2 if the following conditions hold: i) f is a group homomorphism, ii) δ 1 ((p, µ t (p)), x, q) δ 2 ((f(p), µ t (f(p))), g(x), f(q)), p, q Q, x Σ 1. Definition 2.2. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and let λ be a fuzzy subset of Q and (Q, ) be a group. λ is called fuzzy normal kernel of F if the following conditions hold: i) λ is a fuzzy normal subgroup of Q, ii) λ(p r 1 ) δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) λ(k) for all p, q, k, r Q, x Σ. Theorem 2.3. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a max-min general fuzzy automaton and let λ is a fuzzy normal subgroup of Q and (Q, ) be a group. Then λ is a fuzzy normal kernel of F if and only if λ(p r 1 ) δ ((q k, µ t (q k)), x, p) δ ((q, µ t (q)), x, r) λ(k) for all p, q, k, r Q, x Σ.

6 62 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Proof. Let λ be a fuzzy normal kernel of F. We prove the theorem by induction on x = n. Let n = 0. Then x = Λ. If p = q k, r = q, then since λ is a fuzzy normal subgroup, we have δ ((q k, µ t (q k)), x, p) δ ((q, µ t (q)), x, r) λ(k) = λ(k) λ(q k q 1 ) = λ(p r 1 ) If p q k or r q, then δ ((q k, µ t (q k)), x, p) δ ((q, µ t (q)), x, r) λ(k) = 0 λ(p r 1 ) Thus the result holds for n = 0. Suppose that the result holds for all x Σ such that x = n 1. Let x = ya, y Σ, y = n 1, n > 0. Then we have δ ((q k, µ t (q k)), x, p) δ ((q, µ t (q)), x, r) λ(k) = δ ((q k, µ t (q k)), ya, p) δ ((q, µ t (q)), ya, r) λ(k) = ( u Q δ ((q k, µ t (q k)), y, u) δ((u, µ t (u)), a, p)) ( v Q δ ((q, µ t (q)), y, v) δ((v, µ t (v)), a, r)) λ(k) = u Q δ v Q ((q k, µ t (q k)), y, u) δ((u, µ t (u)), a, p) δ ((q, µ t (q)), y, v) δ((v, µ t (v)), a, r) λ(k) u Q v Q λ(u v 1 ) δ((u, µ t (u)), a, p) δ((v, µ t (v)), a, r) u Q v Q λ(v 1 u) δ((v v 1 u, µ t (v v 1 u)), a, p) δ((v, µ t (v)), a, r) = u Q v Q λ(k ) δ((v k, µ t (v k )), a, p) δ((v, µ t (v)), a, r) λ(p r 1 ) The converse is trivial. Definition 2.4. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and let λ be a fuzzy subset of Q and (Q, ) be a group. λ is called fuzzy kernel of F if the following conditions hold: i)λ is a fuzzy subgroup of Q, ii)λ(p) δ((q, µ t (q)), x, p) λ(q) for all p, q Q, x Σ. Theorem 2.5. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a max-min general fuzzy automaton and let λ is a fuzzy subgroup of Q and (Q, ) be a group. Then λ is a fuzzy kernel of F if and only if λ(p) δ ((q, µ t (q)), x, p) λ(q) for all p, q Q, x Σ. Proof. The proof is similar to that of Theorem 2.3.

7 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Definition 2.6. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and (Q, ) be a group. F is called adjustable if δ((p q, µ t (p q)), x, p) δ((p, µ t (p)), x, k) for all p, q, r, k Q, x Σ. Theorem 2.7. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable general fuzzy automaton, (Q, ) be a group and let λ be a fuzzy normal kernel of F and ν be a fuzzy kernel of F. Then λ ν is a fuzzy kernel of F. Proof. Since λ is a fuzzy normal subgroup of Q and ν is a fuzzy subgroup of Q, it follows that λ ν is a fuzzy subgroup of Q and λ ν = ν λ. Since p = (p r 1 ) r, then we have (λ ν)(p) λ(p r 1 ) ν(r) ( δ((a b, µ t (a b)), x, p) δ((a, µ t (a)), x, r) λ(b)) ( δ((a, µ t (a)), x, r) ν(a)) for all a, b, p Q, x Σ. Since F is adjustable, so we have δ((a b, µ t (a b)), x, p) δ((a, µ t (a)), x, r) Then we have ( δ((a b, µ t (a b)), x, p) δ((a, µ t (a)), x, r) λ(b)) ( δ((a, µ t (a)), x, r) ν(a)) Thus for all p, q Q, x Σ, = δ((a b, µ t (a b)), x, p) λ(b) ν(a) (λ ν)(p) { δ((a b, µ t (a b)), x, p) λ(b) ν(a) a, b Q, a b = q} = δ((q, µ t (q)), x, p) ( {λ(b) ν(a) a, b Q, a b = q}) = δ((q, µ t (q)), x, p) (ν λ)(q) = δ((q, µ t (q)), x, p) (λ ν)(q) Hence λ ν is a fuzzy kernel of F. Theorem 2.8. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable general fuzzy automaton and (Q, ) be a group. If λ and ν are fuzzy normal kernels of F, then λ ν is a fuzzy normal kernel of F.

8 64 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Proof. Since λ and ν are fuzzy normal subgroups of Q, it follows that λ ν is a fuzzy normal subgroup of Q and λ ν = ν λ. Since p r 1 = (p q 1 ) (q r 1 ), then we have (λ ν)(p r 1 ) λ(p q 1 ) ν(q r 1 ) ( δ((a b c, µ t (a b c)), x, p) δ((a b, µ t (a b)), x, q) λ(c)) ( δ((a b, µ t (a b)), x, q) δ((a, µ t (a)), x, r) ν(b)) for all a, b, c, p, r Q, x Σ. Since F is adjustable, so we have δ((a b c, µ t (a b c)), x, p) δ((a b, µ t (a b)), x, q) Then we have δ((a b c, µ t (a b c)), x, p) δ((a b, µ t (a b)), x, q) λ(c) δ((a b, µ t (a b)), x, q) δ((a, µ t (a)), x, r) ν(b) = δ((a b c, µ t (a b c)), x, p) δ((a, µ t (a)), x, r) λ(c) ν(b) Thus for all p, q, r, k Q, x Σ, we have (λ ν)(p r 1 ) { δ((a b c, µ t (a b c)), x, p) δ((a, µ t (a)), x, r) λ(c) ν(b) b, c Q, b c = k} = δ((a k, µ t (a k)), x, p) δ((a, µ t (a)), x, r) ( {λ(c) ν(b) b, c Q, b c = k}) = δ((a k, µ t (a k)), x, p) δ((a, µ t (a)), x, r) (ν λ)(k) = δ((a k, µ t (a k)), x, p) δ((a, µ t (a)), x, r) (λ ν)(k) Hence λ ν is a fuzzy normal kernel of F. Theorem 2.9. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton, (Q, ) be a group and let λ be a fuzzy normal kernel of F with Supp(λ) = Q. Then there exists a general fuzzy automaton say H and there exists a homomorphism from F to H.

9 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Proof. Now λ is a fuzzy normal subgroup of Q. Define λ p : Q [0, 1] by λ p (q) = λ(p q 1 ) for all q Q. Let E = {λ p p Q}. Then E is a group with respect to the binary operation λ p λ q = λ p q for all λ p, λ q E. Define Q/λ : E [0, 1] by (Q/λ)(λ p ) = λ(p) for all λ p E. Thus Q/λ is a fuzzy subgroup of E. Let G = Supp(Q/λ). Define σ : (G [0, 1]) Σ G [0, 1] by σ((λ p, µ t (λ p )), x, λ q ) = { δ((a, µ t (a)), x, b) a, b Q, λ a = λ p, λ b = λ q } for all λ p, λ q G. Define f : Q G by f(q) = λ q for all q Q and let g : Σ Σ be idendity map. Then we have f(p q) = λ p q = λ p λ q = f(p) f(q) for all p, q Q. Thus (f, g) is a homomorphism from F to H = (G, Σ, R, Z, ω, σ, F 1, F 2 ). Definition Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and let λ be a fuzzy kernel of F and (Q, ) be a group. A fuzzy set ν of Q is called a fuzzy normal kernel of λ if the following conditions hold: i) ν λ and ν is a fuzzy normal subgroup of λ, ii) ν(p r 1 ) δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) ν(k) for all p, k, r Q, q Supp(λ), x Σ. Definition Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and (Q, ) be a group. F is called multiplicative if having the following properties: i) there exists the element x 0 of Σ such that δ((e, µ t (e)), x 0, e) > 0 ii) δ((q, µ t (q)), x, p r) = δ((q, µ t (q)), x 0, p) δ((e, µ t (e)), x, r), iii) δ((p 1 p 1 2, µt (p 1 p 1 2 )), x 0, q 1 q 1 2 ) = δ((p 1, µ t (p 1 )), x 0, q 1 ) δ((p 2, µ t (p 2 )), x 0, q 2 ) for all p, q, r, p 1, p 2, q 1, q 2 Q, x Σ. Theorem Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable multiplicative general fuzzy automaton and (Q, ) be an abelian group. Let λ be a fuzzy kernel of F, ν be a fuzzy normal kernel of λ and Supp(λ) Φ. If ν is a fuzzy normal subgroup of Q, then ν is a fuzzy normal kernel of F. Proof. Since F is multiplicative, for all p, q, r, k Q, x Σ, we have δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) = δ((q k, µ t (q k)), x, p e) δ((q, µ t (q)), x, r e) = ( δ((q k, µ t (q k)), x 0, p) δ((e, µ t (e)), x, e)) ( δ((q, µ t (q)), x 0, r) δ((e, µ t (e)), x, e)) = ( δ((q k, µ t (q k)), x 0, p e) δ((e, µ t (e)), x, e)) ( δ((q, µ t (q)), x 0, r) δ((e, µ t (e)), x, e))

10 66 Alg. Struc. Appl. Vol. 4 No. 2 (2017) = δ((q, µ t (q)), x 0, p) δ((k 1, µ t (k 1 )), x 0, e) δ((e, µ t (e)), x, e) δ((q, µ t (q)), x 0, r) Now for any q Q, b Supp(λ), q = b (q 1 b) 1. Thus we have δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) = δ((b (q 1 b) 1 k, µ t (b (q 1 b) 1 k)), x, p e) δ((e q, µ t (e q)), x, r e) = δ((b, µ t (b)), x 0, p) δ((q 1 b, µ t (q 1 b)), x 0, e) δ((k 1, µ t (k 1 )), x 0, e) δ((e, µ t (e)), x, e) δ((q 1, µ t (q 1 )), x 0, r) = δ((b, µ t (b)), x 0, p) δ((k 1, µ t (k 1 )), x 0, e) δ((e, µ t (e)), x, e) ( δ((q 1 b, µ t (q 1 b)), x 0, e) δ((q 1, µ t (q 1 )), x 0, r)) = δ((b, µ t (b)), x 0, p) δ((k 1, µ t (k 1 )), x 0, e) δ((e, µ t (e)), x, e) δ((b, µ t (b)), x 0, r) Since ν is a fuzzy normal kernel of λ, for all p, q, r Q, b Supp(λ), x Σ, we have ν(p r 1 ) δ((b k, µ t (b k)), x, p) δ((b, µ t (b)), x, r) ν(k) = δ((b k, µ t (b k)), x, p e) δ((b, µ t (b)), x, r e) ν(k) = δ((b, µ t (b)), x 0, p) δ((k 1, µ t (k 1 )), x 0, e) δ((b, µ t (b)), x 0, r) δ((e, µ t (e)), x, e) ν(k) = δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) ν(k) Hence ν is a fuzzy normal kernel of F. Theorem Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable multiplicative general fuzzy automaton, (Q, ) be a group, λ be a fuzzy subset of Q and Supp(λ) Φ. Then the following statements are equivalent: i) λ is a fuzzy normal kernel of F ii) λ is a fuzzy normal subgroup of Q and λ(q) δ((p, µ t (p)), x 0, q) λ(p), for all p, q Q. Proof. (i ii) Since λ is a fuzzy normal kernel of F, we have: λ(q) = λ(q e 1 ) δ((e p, µ t (e p)), x 0, q) δ((e, µ t (e)), x 0, e) λ(p) = δ((p, µ t (p)), x 0, q) λ(p) (Since F is adjustable, for all p, q Q, we have δ((p, µ t (p)), x 0, q) = δ((e p, µ t (e p)), x 0, q) δ((e, µ t (e)), x 0, e) )

11 Alg. Struc. Appl. Vol. 4 No. 2 (2017) (ii i) Since F is multiplicative, for all p, q, r, k Q, x Σ, we have δ((q k, µ t (q k)), x, p r) δ((q, µ t (q)), x, p) λ(k) = δ((q k, µ t (q k)), x, p r e) δ((q, µ t (q)), x, p e) λ(k) = ( δ((q k, µ t (q k)), x 0, p r) δ((e, µ t (e)), x, e)) ( δ((q, µ t (q)), x 0, p) δ((e, µ t (e)), x, e)) λ(k) = δ((q, µ t (q)), x 0, p) δ((k 1, µ t (k 1 )), x 0, r 1 ) δ((e, µ t (e)), x, e)) λ(k) = δ((q, µ t (q)), x 0, p) δ((e k 1, µ t (e k 1 )), x 0, e r 1 ) δ((e, µ t (e)), x, e)) λ(k) = δ((q, µ t (q)), x 0, p) ( δ((e, µ t (e)), x 0, e) δ((k, µ t (k)), x 0, r)) δ((e, µ t (e)), x, e)) λ(k) δ((k, µ t (k)), x 0, r) λ(k) λ(r) λ(p r p 1 ) Now let p 1, p 2, q, k Q, x Σ. Since Q is a group, then there exists a unique element r Q such that p 1 = p 2 r. Thus λ(p 1 p 1 2 ) = λ(p 2 r p 1 2 ) δ((q k, µ t (q k)), x, p 2 r) δ((q, µ t (q)), x, p 2 ) λ(k) Hence λ is a fuzzy normal kernel of F. = δ((q k, µ t (q k)), x, p 1 ) δ((q, µ t (q)), x, p 2 ) λ(k) Theorem Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable multiplicative general fuzzy automaton and (Q, ) be a group. Then there exists a semigroup homomorphism f : Q F P (Q) and there exists a function g : Σ F P (Q) such that δ((q, µ t (q)), x, p) = f(q)(p) g(x)(e), for all p, q Q, x Σ.

12 68 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Proof. Define f : Q F P (Q) by f(q) = µ q for all q Q, where µ q : Q [0, 1] is defined by µ q (p) = δ((q, µ t (q)), x 0, p) for all p Q. Since F is multiplicative, for all p 1, p 2, q 1, q 2 Q, we have µ q1 q 2 (p 1 p 2 ) = δ((q 1 q 2, µ t (q 1 q 2 )), x 0, p 1 p 2 ) = δ((q 1, µ t (q 1 )), x 0, p 1 ) δ((q 1 2, µt (q 1 2 )), x 0, p 1 2 ) = δ((q 1, µ t (q 1 )), x 0, p 1 ) δ((q 2, µ t (q 2 )), x 0, p 2 ) Also we have = µ q1 (p 1 ) µ q2 (p 2 ) (µ q1 µ q2 )(p) = {µ q1 (q) µ q2 (r) q, r Q, p = q r} = µ q1 q 2 (p) for all p Q. Hence f(q 1 q 2 ) = f(q 1 ) f(q 2 ) For x Σ, define the fuzzy subset λ x of Q by λ x (q) = δ((e, µ t (e)), x, q) for all q Q. Define g : Σ F P (Q) by g(x) = λ x for all x Σ. Since F is multiplicative, for all p, q Q, x Σ, we have f(q)(p) g(x)(e) = µ q (p) λ x (e) = δ((q, µ t (q)), x 0, p) δ((e, µ t (e)), x, e) = δ((q, µ t (q)), x, p) 3. Conclusion In this paper, we defined the concepts of fuzzy normal kernel of a general fuzzy automaton, fuzzy kernel of a general fuzzy automaton, adjustable, multiplicative. Then we obtained the relationships between them. 4. Acknowledgments The author wish to sincerely thank the referees for several useful comments.

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