α -Stable local necks of fuzzy automata

In this paper we introduce α -stable local necks, α -stable monogenically directable, α -stable monogenically strongly directable, α -stable monogenically trap directable, α -stable uniformly monogenically directable, α -stable uniformly monogenically strongly directable, α -stable uniformly monogenically trap-directable fuzzy automata. We have shown that α -stable local necks of fuzzy automaton exists then it is α -stable subautomaton. Further we prove a some equivalent conditions on fuzzy automaton.


Introduction
Fuzzy set was introduced by L. A. Zadeh in 1965 [8].The fuzzy set is a simple mathematical tool for representing the inevitability of vagueness, uncertainty, and imprecision in everyday life.W.G. Wee extended the fuzzy idea to automata in 1967 [7].Later, numerous academics adapted the fuzzy notion to a wide range of domains, and it has a wide range of applications.J.N. Mordeson and D. S. Malikgave a detailed account of fuzzy automata and languages in their book [6].
T. Petkovic et al. [1] discussed directable automata, monogenically directable, generalized directable using necks.T. Petkovic et al. [3] introduce and studied trapdirectable, trapped automata and other related automata.Also, we refer the survey paper Directable automata and their generalizations were investigated by S. Bogdanovic et al [2].Fur-ther the necks and local necks of fuzzy automata were studied and discussed in [4,5].In this paper we introduce α-stable local necks, α-stable monogenically directable, α-stable monogenically strongly directable, α-stable monogenically trap directable, α-stable uniformly monogenically directable, αstable uniformly monogenically strongly directable, α-stable uniformly monogenically trap-directable fuzzy automata.We have shown that α-stable local necks of fuzzy automata exists then it is α-stable subautomata.Further we prove a some equivalent conditions on fuzzy automaton.Let S = (D, I, ψ) be a fuzzy automaton.S is said to be strongly connected if for every d i , d j ∈ D, there exists t ∈ I * such that ψ * (d i , t, d j ) > 0. Equivalently, S is strongly connected if it has no proper subautomaton.

Preliminaries
Definition 2.5.[4] Let S = (D, I, ψ) be a fuzzy automaton.A state d j ∈ D is called a neck of S if there exists t ∈ I * such that ψ * (d i , t, d j ) > 0 for every d i ∈ D. In that case d j is also called t-neck of S and the word t is called a directing word of S. If S has a directing word, then we say that S is a directable fuzzy automaton.Definition 2.6.[5] Let S = (D, I, ψ) be a fuzzy automaton.If d i ∈ Q is called local neck of S, if it is neck of some directable subautomaton of S. The set of all local necks of S is denoted by LN(S).
Let S = (D, I, ψ) be a fuzzy automaton.If S is said to be α-stable fuzzy automaton then Let S = (D, I, ψ) be a fuzzy automaton and let d i ∈ D. The α-stable subautomaton of S generated by d i is denoted by d i .It is given by Let S = (D, I, ψ) be a fuzzy automaton.A state d i ∈ D is called α-stable local neck of S if it is α-stable neck of some α-stable directable subautomaton of S. The set of all α-stable local necks of S is denoted by αSLN(S).Definition 3.5.
Let S = (D, I, ψ) be a fuzzy automaton.S is called α-stable monogenically trap-directable if every monogenic subautomaton of S has a single α-stable neck.
Definition 3.8.Let S = (D, I, ψ) be a fuzzy automaton.If t ∈ I * is α-stable common directing word of S if t is a αstable directing word of every monogenic subautomaton of S. The set all α-stable common directing words of S will be denoted by αSCDW (S).In other words, αSCDW (S) = ∩ d i ∈ D αSDW ( d i ).Definition 3.9.
Let S = (D, I, ψ) be a fuzzy automaton.S is called α-stable uniformly monogenically directable fuzzy automaton if every monogenic subautomaton of S is α-stable directable and have atleast one β -weak common directing word.Definition 3.10.
Let S = (D, I, ψ) be a fuzzy automaton.S is called α-stable uniformly monogenically strongly directable fuzzy automaton if every monogenic subautomaton of S is strongly α-stable directable and have atleast one α-stable common directing word.Definition 3.11.
Let S = (D, I, ψ) be a fuzzy automaton.S is called α-stable uniformly monogenically trap directable fuzzy automaton if every monogenic subautomaton of S has a single α-stable neck and have atleast one α-stable common directing word.(ii) d i is a strongly α-stable directable fuzzy automaton;

Properties of α-Stable Local Necks of Fuzzy Automata
(iii) for every t ∈ I * , there exists t ∈ I * such that ψ * (d i , t t, d i ) ≥ α > 0.

Proof. (i) ⇒ (ii)
Let d i be a α-stable local neck of S. Then there exists a α-stable directable subautomaton S of S such that d i ∈ αSN(S ).Thus αSN(S ) is a strongly α-stable directable fuzzy automaton.Also, d i ⊆ αSN(S ), and αSN(S ) is strongly connected, then d i = αSN(S ).Therefore, d i is a

Definition 2 . 3 .
[4] Let S = (D, I, ψ) be a fuzzy automaton.Let D ⊆ D. Let ψ is the restriction of ψ and let S = (D , I, ψ ).The fuzzy automaton S is called a subautomaton of S if (i) ψ : D × I × D → [0, 1] and (ii) For any d i ∈ D and ψ (d i ,t, d j ) > 0 for some t ∈ I * , then d j ∈ D .Definition 2.4.[6] exists, then it is called the α-stable least subautomaton of S containing d i .Definition 3.3.Let S = (D, I, ψ) be a fuzzy automaton.For any non-empty D ⊆ D, the α-stable subautomaton of S generated by D is denoted by D and is given by D = {{ d j | ψ * (d i , t, d j ) ≥ α} > 0, d i ∈ D , t ∈ I * }.It is called the α-stable least subautomaton of S containing D .The α-stable least subautomaton of a fuzzy automaton S if it exists is called the α-stable kernel of S. Definition 3.4.

Theorem 4 . 1 .
Let S = (D, I, ψ) be a fuzzy automaton and d i ∈ D. Then the following conditions are equivalent: (i) d i is a α-stable local neck;