Applications of Hyperstructure Theory

Frontmatter

Basic notions and results on Hyperstructure Theory

Abstract

The most important notions and results, obtained on Hyperstructure Theory, are presented here. For more details, see [437].

Piergiulio Corsini, Violeta Leoreanu

Chapter 1. Some topics of Geometry

Abstract

• Several branches of geometry can be treated as certain kinds of hypergroups, known as join spaces. Introduced by W. Prenowitz and studied afterwards by him together with J. Jantosciak, the concept of a join space is “sufficiently general to cover the theories of ordered and partially ordered linear, spherical and projective geometries, as well as abelian groups”.
• If we consider a spherical geometry and identify antipodal points, we obtain a projective geometry. This construction can be described in the context of join spaces as follows:

Let J be the set of points of a spherical join space and for any a ∈ J, let \( \bar a = \{ a,{a^{ - 1}}\} \) .Let \( \bar J = \{ \bar a|a \in J\} \) . We define on \( \bar J \) the following hyperoperation:

$$ \bar ao\bar b = \{ \bar x \in \bar a \cdot \bar b\} $$

, where is the hyperoperation of the spherical join space. Theorem. (see [168]) \( (\bar J,o) \) is a projective join space, such that \( \forall \bar a \in \bar J,\bar ao\bar a = \bar a/\bar a = \{ \bar e,\bar a\} \), where ë is the identity.

Piergiulio Corsini, Violeta Leoreanu

Chapter 2. Graphs and Hypergraphs

Abstract

Since the middle of the last century, Graph Theory has been an important tool in different fields, Iike Geometry, Algebra, Number Theory, Topology, Optimization, Operations Research, Median Algebras and so on. To solve new combinatorial problems, it was necessary to generalize the concept of a Graph.

The notion of a “hypergraph” appeared around 1960 and one of the initial concerns was to extend some classical results of graph theory.

Hypergraph Theory is an useful tool for discrete optimization Problems.

A very good presentation of Graph and Hypergraph Theory is in C. Berge [442] and Harary [448].

In this chapter, we have presented some important connections between Graph, Hypergraph Theory and Hyperstructure Theory.

Piergiulio Corsini, Violeta Leoreanu

Chapter 3. Binary Relations

Abstract

The first connection between a hyperstructure and a binary reIation is implicit in Nieminen [300], who associated a hypergroup with a connected simple graph.

In the same direction, albeit with different hyperoperations associated with graphs, went the papers by Corsini ([74], [79]) and Rosenberg ([326]) and, in the following, by V. Leoreanu and L. Leoreanu ([238]).

Later, Chvalina ([38]) found a correspondence between partially ordered sets and hypergroups. Rosenberg ([326]) generalized Chvalina definition, associating with any binary relation a hypergroupoid.

Rosenberg hypergroup was studied by Corsini ([79]) and then, by Corsini and Leoreanu ([88]), who considered hypergroups associated with union, intersection, product, Cartesian product, direct limit of relations, as we have seen before.

There are stilI open problems on this subject. One of them is to find necessary and sufficient conditions for the hypergroupoids associated with union, intersection, product etc, to be hypergroups. Recently, Spartalis, De SaIvo and Lo Faro have obtained new results on hyperstructures associated with binary relations.

Piergiulio Corsini, Violeta Leoreanu

Chapter 4. Lattices

Abstract

Introduced by Ch.S. Pierce and E. Schröder and independently by R. Dedekind, and afterwards developed by G. Birkhoff, V. Glivenko, K. Menger, J. von Neumann, O. Ore and others, Lattice Theory is a highly topical field, with many applications in mathematics.

Distributive lattices represent the starting point in Lattice Theory; their study is required by more and more frequent situations when distributivity is imposed by applications.

A weaker condition of distributivity is the modularity, introduced by R. Dedekind.

Modularity and distributivity are characterized in this chaper, using hyperstructures, particularly join spaces.

Piergiulio Corsini, Violeta Leoreanu

Chapter 5. Fuzzy sets and rough sets

Abstract

Fuzzy Sets and Hyperstructures introduced by Zadeh, in 1965, and by Marty, in 1934, respectively, are now used in the world both on the theoreticaI point of view and for their many applications. The Rough Sets considered for the first time by Shafer in 1976, have been reintroduced in the international scientific circle by Pawlak, in 1991 especially in connection with Artificial Intelligence. The relations between Rough Sets and Fuzzy Sets have been already considered by Dubois and Prade [137], those between Fuzzy Sets and Hy perstructures by Corsini, Corsini-Leoreanu, Corsini-Tofan, Ameri-Zahedi and others, those between Rough Sets and Hyperstructures by Davvaz. More recently, M. Konstantinidou and A. Kehagias have obtained interesting results on hyperstructures and fuzzy subsets.

Piergiulio Corsini, Violeta Leoreanu

Chapter 6. Automata

Abstract

The definition of an automaton, we shall present here, has its origins in a paper of Kleene (1956). The title “Representation of events in nerve sets and finite automata” of Kleene’s paper gives an idea of its motivation.

The concept of automaton had led to important results, both in mathematics and in theoretical computer science.

Automata are in fact very familiar objects, in the shape of coin machines.

The last twenty years have developed a body of research known ùnder the names of Automaton Theory and Formal Language Theory.

We mention Biology between the fields which have significant connections with Automaton Theory.

Here, we have presented the connections of Automaton Theory and Language Theory with another field, known as Hyperstructure Theory.

Using tools and methods of Hyperstructure Theory, G. G. Massouros gave a new proof of the famous Kleene’s Theorem, which states that:

„A subset of the set of words M* is acceptable from an automaton ℳ if and only if it is defined by a regular expression.”

Piergiulio Corsini, Violeta Leoreanu

Chapter 7. Cryptography

Abstract

For ages, cryptography has been used in military and diplomatic communication, in order to make the meaning of transmitted messages incomprehensible to unauthorized users.

As Francis Bacon said, ”The art of ciphering, half for relative an art of deciphering, by supposition unprofitable, but as things are, of great use”. Lately, W. Diffie and M. Hellman [126] point out the new directions in Cryptography.

In this chapter, we have presented some hyperstructures derived from generalized designs and some cryptographic interpretations on hyperstructures. As being a science in a continuous development, ciphering can still be improved, using a relative new theory, that one of Hyperstructure Theory.

Piergiulio Corsini, Violeta Leoreanu

Chapter 8. Codes

Abstract

In general, Code Theory and more precisely Error—Correcting Code Theory is one branch of applied mathematics, which massively uses algebraic methods and results.

Through a channel, recall that Error—Correcting Code Theory is essential for all types of communications (for instance, telephonic communications, radio communications and so on).

Among the most remarkable codes, we recall Hamming codes, QR-codes, which are important classes of cyclic codes.

We present below a connection between Steiner hypergroups and linear codes. We think that the study of this connection deserves to be studied in depth. For more details on Error—Correcting Codes, see [452], [454] and [457].

Piergiulio Corsini, Violeta Leoreanu

Chapter 9. Median algebras, Relation algebras, C-algebras

Abstract

• For the first time, median algebras appeared in the late fourties. A.A. Grau [148] characterized Boolean algebras in terms of median operation and complementation, G. Birkhoff and S.A. Kiss [25] discusses the median operation for distributive lattices. The concept of abstract median algebra was introduced by S.P. Avann [12] and later M. Scholander [356], [357], [358] and S.P. Avann [13] performed a detailed study of median algebras. More recently, J. Nieminen [301], E. Evans [139], H.M. Mulder A. Schrijver [297], J.R. Isbell [165], H. Werner [424] worked on this subject.
• We shall see that quasi-canonical hypergroups can be characterized as the atomic structures of complete atomic integral relation algebras (§2). Moreover, the Tarski complex- algebra construction gives a fulI embedding of quasi-canonical hypergroups into relation algebras. Therefore, certain combinatorial properties of quasi-canonical hypergroups transfer to relation aI gebras. Using this process, results of Monk [295], [296] or McKenzie [263], [453], about relation algebras (or cylindric algebras) turn out to be just interpretations of quasi-canonical hypergroup results.
• Let us remember some remarkable C-algebras: the adjancency algebras of association schemes [441], Salgebras over finite groups [31], and centralizer algebras of homogeneous coherent configurations [449].

Piergiulio Corsini, Violeta Leoreanu

Chapter 10. Artificial Intelligence

Abstract

Weak representations of an interval algebra are the objects of interest in the Artificial Intelligence.

Let us give some words about the Mathematicians who worked on this subject.

Allen [3] defined the calculus of time intervals and Ladkin and Maddux [220] showed the interpretation of the calculus of time intervals, in terms of representations of a par ticular relation algebra, in the sense of Tarski [178]. They proved that there is, up to an isomorphism, a unique countable representation of this algebra.

Ligozat [241] generalized the calculus of time intervals to a calculus of n-intervals and presented this generalization expressed in terms of relation algebras As.

Defining canonical functors between the category of weak representations of A _n and those of A ₁ , Ligozat [241] extended the results obtained by Ladkin [219].

Finally, it can be seen that the set of (p, q)-positions can be endowed with natural operations which give rise to a family of quasi—canonical hypergroupoids.

Piergiulio Corsini, Violeta Leoreanu

Chapter 11. Probabilities

Abstract

Using a particular non—standard algebraic hyperstructure, A. Maturo [251] proved that the problems on the coherent assessments of probability and their solutions can be expressed in a very useful and simple form.

Thus, new algorithms to control the coherence can be introduced in this new algebraic context.

Piergiulio Corsini, Violeta Leoreanu

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