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
:
![](https://www.scirp.org/html/htmlimages\6-7402125x\5646fb85-8534-4dac-b0d4-ece9a5e63349.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402125x\13e5d18d-0a14-457c-951c-186dfd09ee53.png)
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:
![](https://www.scirp.org/html/htmlimages\6-7402125x\93da167b-d7c7-4949-8a0a-f1429faee7a5.png)
and define a quasi-ordered relation as following:
![](https://www.scirp.org/html/htmlimages\6-7402125x\8305a075-14dd-486f-ada9-614dbf35d34e.png)
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:
![](https://www.scirp.org/html/htmlimages\6-7402125x\4ba02e9a-c3bd-4fc1-9c17-72072870bb77.png)
Then we obtain the following Corollary from Lemma 2. 2 immediately:
Corollary 2.4 For each
, if
![](https://www.scirp.org/html/htmlimages\6-7402125x\e123f51b-9d74-437f-805e-32bddf681c50.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402125x\55dcc38e-224c-4b5d-bba6-1ae75555da7e.png)
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} |
![](https://www.scirp.org/html/htmlimages\6-7402125x\eecdf5f0-f008-4f4a-bf44-f4c78473e08a.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\53d8d824-56df-4166-8fab-3351952c87fc.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\0f5f2a9d-aa77-49ba-a24b-3900769ce0a2.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\c05f13d6-2d4d-4218-8a75-f3605c3adace.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\eb890771-a628-4f5f-8546-e7bf8ff50f8b.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\a5b00b34-9cab-4f99-9c38-94b2864383d3.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\1cbd3a80-217d-47a6-b366-40ef4734d544.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\a82a14fc-6c6b-41cc-a665-1cfc20e2015b.png)
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 ![](https://www.scirp.org/html/htmlimages\6-7402125x\26f15f31-2de4-49ff-a6cf-05a5cd679f4c.png)
Figure 1. The
-multiautomaton of Example 3.5.
![](https://www.scirp.org/html/htmlimages\6-7402125x\7578a3d3-f0a4-43f1-ab71-5a0c682e965e.png)
And define:
![](https://www.scirp.org/html/htmlimages\6-7402125x\03af2c98-439f-4d68-84fc-fd2f8a2b54b1.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\c5c1c54e-2ed2-41c9-9906-da225c9f5568.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\907a4124-7c5b-478d-a78c-33abd309c8b0.png)
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![](https://www.scirp.org/html/htmlimages\6-7402125x\882bdc55-3bf2-473b-880e-8a189ae8a231.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\37c63b18-556f-4634-a72a-e762f17ddf3e.png)
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:
![](https://www.scirp.org/html/htmlimages\6-7402125x\8d77a6a9-ebe9-4808-a976-ccf5421fc6e1.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\c7774b1f-d0f9-462c-8cd5-f7c9226d7931.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\da6e683d-05de-47ec-a709-420d485fe43b.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402125x\cbf9c322-d47e-4e6c-912b-ce9245df8272.png)
for all,
and
.
Then
𝒿![](https://www.scirp.org/html/htmlimages\6-7402125x\b6deee10-83b6-4304-b908-ca9475e546dd.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\fe0a5293-0feb-433e-868f-25372c0d67c1.png)
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
:
![](https://www.scirp.org/html/htmlimages\6-7402125x\b9d2d37a-1cee-42ee-9c49-63b3031ceb62.png)
And define:
![](https://www.scirp.org/html/htmlimages\6-7402125x\9562cddd-5c25-4af9-9200-50462e268ece.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\374b6683-8809-4d76-8497-c22804b6d160.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\5ccc5d80-4ae7-4790-aab8-d457f363c87b.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402125x\0385e105-c45c-4ce6-9437-6df716a22e3b.png)
And
, for all
and
.
Then
![](https://www.scirp.org/html/htmlimages\6-7402125x\4e003bb9-a517-491f-81c4-53ed1bb7a065.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\f96655ca-9537-485a-a04f-d033f05873da.png)
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:
![](https://www.scirp.org/html/htmlimages\6-7402125x\2b5108f4-f4cb-4511-9c81-48f8b52e8330.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\61c3a1ae-5a4c-443a-ace2-f6ad8edf2be8.png)
![](https://www.scirp.org/html/htmlimages\6-7402125x\d7d7e671-5708-48bf-b75b-45f23dd2d6d5.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402125x\851e4c15-40fb-4a0f-8632-9f2b3eb3061b.png)
for all
and
.
Then
𝓂
![](https://www.scirp.org/html/htmlimages\6-7402125x\c662e4d0-4552-4883-ad4f-97a2e6dbc9f0.png)
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.