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.