**Applied Mathematics
**Vol.5 No.9(2014), Article ID:46147,6 pages
DOI:10.4236/am.2014.59130

F-Multiautomata on Join Spaces Induced by Differential Operators

Rajab Ali Borzooei^{1}, Hamid Reza Varasteh^{2}, Abbas Hasankhani^{3}

^{1}Department of Mathematics, Shahid Beheshti University, Tehran, Iran

^{2}Department of Mathematics, Payame Noor University, Tehran, Iran

^{3}Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman,
Iran

Email: borzooei@sbu.ac.ir, varastehhamid@gmail.com, abhasan@mail.uk.ac.ir

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 15 February 2014; revised 22 March 2014; accepted 1 April 2014

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

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.

References

- Corsini, P. and Leoreanu, V. (2003) Applications of Hyperstructure Theory. Kluwer Academic Publishers, Dordrecht. http://dx.doi.org/10.1007/978-1-4757-3714-1
- Corsini, P. (1993) Prolegomena of Hypergroup Theory. Aviani Editore, Tricesimo.
- Davvaz, B. and Leoreanu, V. (2008) Hyperring Theory and Applications. International Academic Press, USA.
- Chvalina, J. and Rakova, P. (2008) Join Spaces of Smooth Functions and Their Actions on Transposition Hypergroups of Second Order Linear Differential Operators. Journal of Applied Mathematics, 55-63.
- Chvalina, J. and Hoskova, S. (2007) Action of Hypergroups of the First Order Partial Differential Operators. 6th International Conference Aplimat, Department of Mathematics, FME Slovak University of Technology, Bratislava, 177- 184.
- Chvalina, J. (2008) Infinite Multiautomata with Phase Hypergroups of Various Operators. Proceedings of the 10th International Congress on Algebraic Hyperstructures and Applications, University of Defence, Brno, 57-69.
- Chvalina, J. and Chvalina, L. (2009) Action of Join Spaces of Continuous Functions on the Underlying Hypergroups of 2-Dimentional Linear Spaces of Functions. Journal of Applied Mathematics, 2, 24-34.
- Hoskov’a, S. and Chvalina, J. (2008) Multiautomata Formed by First Order Partial Differential Operators. Journal of Applied Mathematics, 1, 423-430.
- Zadeh, L.A. (1965) Fuzzy Sets. Inform and Control, 8, 338-353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X
- Wee, W.G. (1967) On Generalization of Adaptive Algorithm and Application of the Fuzzy Sets Concept to Pattern Classification. Ph.D. Dissertation, Purdue University, West Lafayette.
- Mordeson, J. and Malik, D. (2002) Fuzzy Automata and Languages: Theory and Applications. ACRC Press Company, Boca Raton. http://dx.doi.org/10.1201/9781420035643
- Prenowitz, W. (1943) Projective Geometries as Multigroups. The American Journal of Mathematics, 65, 235-256. http://dx.doi.org/10.2307/2371812
- Prenowitz, W. and Jantosciak, W. (1972) Geometries and Join Spaces. Journal für die Reine und Angewandte Mathematik, 257, 100-128.
- Chvalina, J. (1995) Functional Graphs, Quasi Ordered Sets and Commutative Hypergroups. Masaryk University, Brno.
- Novák, M. (2013) Some Basic Properties of EL-Hyperstructures. European Journal of Combinatorics, 34, 446-459. http://dx.doi.org/10.1016/j.ejc.2012.09.005
- Hoskova, S. and Chvalina, J. (2008) Discrete Transformation Hypergroups and Transformation Hypergroups with Phase Tolerance Space. Discrete Mathematics, 308, 4133-4143. http://dx.doi.org/10.1016/j.disc.2007.08.005