Rajab Ali Borzooei, Hamid Reza Varasteh, Abbas Hasankhani

Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman, Iran.

Department of Mathematics, Payame Noor University, Tehran, Iran.

Department of Mathematics, Shahid Beheshti University, Tehran, Iran.

**DOI: **10.4236/am.2014.59130
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Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman, Iran.

Department of Mathematics, Payame Noor University, Tehran, Iran.

Department of Mathematics, Shahid Beheshti University, Tehran, Iran.

In this paper, we introduce the notion of fuzzy multiautomata and we investigate the hyperstructures induced by the linear second-order differential operators which can be used for construction of fuzzy multiautomata serving as a theoretical background for modeling of processes.

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Borzooei, R. , Varasteh, H. and Hasankhani, A. (2014) *F*-Multiautomata on Join Spaces Induced by Differential Operators. *Applied Mathematics*, **5**, 1386-1391. doi: 10.4236/am.2014.59130.

1. Introduction

Hyperstructure theory was born in 1934 when Marty defined hypergroups as a generalization of groups. This theory has been studied in the following decades and nowadays by many mathematicians. The hypergroup theory both extends some well-known group results and introduces new topics, thus leading to a wide variety of applications, as well as to a broadening of the investigation fields. There are applications of algebraic hyperstructures to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, combinatorics, codes, artificial intelligence, and probabilistic. A comprehensive review of the theory of hyperstructures appears in [1] -[3] .

Further, since the beginning of the first decade of this century relationships between ordinary linear differential operators and the hypergroup theory have been studied [4] -[8] .

Zadeh [9] introduced the theory of fuzzy sets and, soon after, Wee [10] introduced the concept of fuzzy automata. Automata have a long history both in theory and application and are the prime examples of general computational systems over discrete spaces. Fuzzy automata not only provide a systematic approach for handling uncertainty in such systems, but also can be used in continuous spaces [11] . In this paper, we introduce F-multiautomaton, without output function, where the transition function or next state function satisfies so called Fuzzy Generalized Mixed Condition (FGMC).These -multiautomata are systems that can be used for the transmission of information of certain type. Then we construct -multiautomata of commutative hypergroups and join spaces created from second order linear differential operators.

2. Preliminaries

Let J be an open interval of real numbers, and be the group of all continuous functions from J to interval. In what follows we denote that named differential operators of second order. And define. Recall some basic notions of the hypergroup theory. A hypergroupoid is a pair where and is a binary hyperoperation on H. (Here denotes the system of all nonempty subsets of (H)). If holds for all then is called a semihypergroup. If moreover, the reproduction axiom (, for any element) is satisfied, then the pair is called a hypergroup. Join spaces are playing an important role in theories of various mathematical structures and their applications. The concept of a join space has been introduced by Prenowitz [12] and used by him and afterwards together with James Jantoisciak to reconstruct several branches of geometry. In order to define a join space, we need the following notation: If are elements of a hypergroupoid then we denote and we intend the set.

Definition 2.1 [12] [13] A commutative hypergroup is called a join space (or commutative transposition hypergroup) if the following condition holds for all elements of:

By a quasi-ordered (semi)group we mean a triple where is a (semi) group and binary relation is a quasi ordering (i.e. is reflexive and transitive) on the set G such that, for any triple with the property also and hold.

The following lemma is called Ends-Lemma that is proved on [14] [15] .

Lemma 2.2 Let be a quasi-ordered semigroup. Deﬁne a hyperoperation

For all pairs of elements. Then is a semihypergroup which is commutative if the semigroup is commutative. If moreover, is a group, then is a transposition hypergroup. Therefore, if is a commutative group, then is a join space.

Proposition 2.3 For any pair of diﬀerential operators deﬁne a binary operation as below:

and define a quasi-ordered relation as following:

Then is a commutative ordered group with the unit element □

Now we apply the simple construction of a hypergroup from Lemma 2.2 into this considered concrete case of diﬀerential operators:

For arbitrary pair of operators we put:

Then we obtain the following Corollary from Lemma 2. 2 immediately:

Corollary 2.4 For each, if

Then is a commutative hypergroup and a join space.

Definition 2.5 [16] Let be a non-empty set, be a (semi) hypergroup and be a mapping such that, for all, and:

(2.1)

Then is called a discrete transformation (semi)hypergroup or an action of the (semi)hypergroup H on the set X. The mapping is usually said to be simply an action.

Remark 2.6 The condition (2.1) used above is called Generalized Mixed Associativity Condition, shortly GMAC.

Definition 2.7 [6] [7] (Quasi)multiautomaton without output is a triad, where is a (semi)hypergroup, S is a non-empty set, and is a transition map satisfying GMAC condition. The set S is called the state set of the (quasi)multiautomaton M, the structure is called a input (semi)- hypergroup of the (quasi)multiautomaton M and is called a transition function. Elements of the set S are called states and the elements of the set H are called input symbols.

3. (-Multi Automata

Definition 3.1 A fuzzy transformation (semi)hypergroup (or a fuzzy action) of (semi)hypergroup H on S is a triple where is a non-empty set, is a (semi)hypergroup, and is a fuzzy subset of such that, for all and:

(3.2)

Remark 3.2 The condition (3.2) used above is called Fuzzy Generalized Mixed Condition, shortly FGMC.

Definition 3.3 -(quasi) multiautomaton without outputs is a triad, where is a (semi)hyper-group, is a non-empty set and is a fuzzy transition map satisfying FGMC condition.

Set S is called the state set and the hyperstructure is called the input (semi)hypergroup of the - (quasi)multiautomaton and is called fuzzy transition function. Elements of the set are called states and the elements of the set are called input symbols.

Definition 3.4 -(quasi)multiautomaton is said to be abelian (or commutative) if

Example 3.5 Suppose that Let hyperoperation on H and fuzzy transition function are defined as follows:

* |
a |
B |

a |
{a} |
{a,b} |

b |
{a,b} |
{b} |

And for all other ordered triples we define. Then is a commutative - multiautomaton (Figure 1).

4. (-Multi Automata on Join Spaces Induced by Differential Operators

Proposition 4.1: Let where, for all

Figure 1. The -multiautomaton of Example 3.5.

And define:

Then is a commutative -multiautomaton.

Proof: By Lemma 2.2 the hypergroupoid is a join space. Now, we prove this structure is satisfying FGMC property. Let

𝒾

and

, for all and.

Then_{}

𝒾

Clearly 𝒾 (since we can take or for each). Then FGMC property holds. Hence is a -multiautomaton. In addition, since, for all then is commutative. □

Proposition 4.2: Let where hyperoperation was defined in proposition 4.1.

And define:

Then is a commutative -multiautomaton.

Proof: By Lemma 2.2 the hypergroupoid is a join space. Now, we prove this structure is satisfying FGMC property. Let

𝒿

and

for all, and.

Then

𝒿_{}

Since for all then 𝒿. Hence FGMC property holds. Therefore is a -multiautomaton. In addition, It is clear that is commutative.

Proposition 4.3: Let where, for all:

And define:

Then is a commutative -multiautomaton.

Proof: According to Corollary 2.4 is a join space. Now we check the FGMC property for this structure. Let

And

, for all and.

Then_{}

Since for all then. Hence is a -multiautomaton. It is clear that is commutative. □

Proposition 4.4: Let, where hyperoperation * was defined in proposition 3.4.

And define:

Then is a commutative -multiautomaton.

Proof: According to Corollary 2.4 is a join space. Now, we prove this structure is satisfying FGMC property. Let

𝓂

for all and.

Then

𝓂

Since and, for all then 𝓂. Hence is a multiautomaton. It is clear that is commutative.

5. Conclusion

In this research, we introduced -multistructures which can be used for construction of -multiautomata serving as a theoretical background for modeling of processes. Then we obtain some -multiautomata of linear second-order differential operators. In future work, we can introduce -multiautomaton with output and concrete interpretations of these structures can be studied.

Conflicts of Interest

The authors declare no conflicts of interest.

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