ABSTRACT

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.

Keywords:Fuzzy Systems, Differential Operators, Hyperalgebraic Structures, Multiautomata

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

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.

References