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**Hyperstructures and their representations.**
*(English)*
Zbl 0828.20076

Hadronic Press Monographs in Mathematics. Palm Harbor, FL: Hadronic Press. vi, 180 p. (1994).

This book deals with algebraic hyperstructures verifying some weak axioms in comparison with the axioms of the classical hyperstructures [studied for instance by P. Corsini, Prolegomena of hypergroup theory (Aviani Editore, Tricesimo, 1993; Zbl 0785.20032)]. For instance, the commutativity of a hypergroupoid \((xy = yx\), for every \(x\), \(y\) in \(H\)) is replaced by the weak commutativity \((xy \cap yx \neq \emptyset\), for every \(x\), \(y\) in \(H\)) and the associativity is replaced by the weak associativity \(((xy) z \cap x (yz) \neq \emptyset\), for every \(x\), \(y\), \(z\) in \(H\)). These hyperstructures have direct applications in various branches of mathematics like combinatorics, group theory, probability and statistics, cryptography, computer science.

In the first chapter (Basic definitions) the most important concepts of the book are introduced and many suggestive examples are presented. In the second chapter, the \(H_v\)-groups, according to the author’s terminology, i.e. the weakly associative quasihypergroups, are studied. The third chapter takes aim at \(H_v\)-rings, which are hyperstructures \((R, +, \cdot)\) such that \((R, +)\) is a \(H_v\)-group, “\(\cdot\)” is weakly associative, and which also verify the condition of weak distributivity \((x (y + z) \cap (xy + xz) \neq \emptyset\) and \((x + y)z \cap (xz +yz) \neq \emptyset\), for all \(x\), \(y\), \(z\) in \(H\)). The following two chapters (Uniting elements and \(H_v\)-hypergroupoid algebra) give methods to construct hyperstructures like those studied in the first part of the book. The purpose of the last four chapters (Generalized permutations, Representations by generalized permutations, \(H_v\)-matrix representations, Reducibility and characters) is to study the hyperstructure representations by generalized permutations and hypermatrices. The book also contains three appendices, namely: Classical hyperstructures, Very thin hyperstructures and \(P\)-hyperstructures.

The exposition is clear and many open problems in the field are presented.

In the first chapter (Basic definitions) the most important concepts of the book are introduced and many suggestive examples are presented. In the second chapter, the \(H_v\)-groups, according to the author’s terminology, i.e. the weakly associative quasihypergroups, are studied. The third chapter takes aim at \(H_v\)-rings, which are hyperstructures \((R, +, \cdot)\) such that \((R, +)\) is a \(H_v\)-group, “\(\cdot\)” is weakly associative, and which also verify the condition of weak distributivity \((x (y + z) \cap (xy + xz) \neq \emptyset\) and \((x + y)z \cap (xz +yz) \neq \emptyset\), for all \(x\), \(y\), \(z\) in \(H\)). The following two chapters (Uniting elements and \(H_v\)-hypergroupoid algebra) give methods to construct hyperstructures like those studied in the first part of the book. The purpose of the last four chapters (Generalized permutations, Representations by generalized permutations, \(H_v\)-matrix representations, Reducibility and characters) is to study the hyperstructure representations by generalized permutations and hypermatrices. The book also contains three appendices, namely: Classical hyperstructures, Very thin hyperstructures and \(P\)-hyperstructures.

The exposition is clear and many open problems in the field are presented.

Reviewer: M.Guţan (Aubière)

### MSC:

20N20 | Hypergroups |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

16Y99 | Generalizations |

08A05 | Structure theory of algebraic structures |