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Some mathematical disciplines can be presented and developed in the context of other disciplines, for instance Boolean algebras, that Stone has converted in a branch of ring theory, projective geome tries, characterized by Birkhoff as lattices of a special type, projec tive, descriptive and spherical geometries, represented by Prenowitz, as multigroups, linear geometries and convex sets presented by Jan tosciak and Prenowitz as join spaces. As Prenowitz and Jantosciak did for geometries, in this book we present and study several ma thematical disciplines that use the Hyperstructure Theory. Since the beginning, the Hyperstructure Theory and particu larly the Hypergroup Theory, had applications to several domains. Marty, who introduced hypergroups in 1934, applied them to groups, algebraic functions and rational fractions. New applications to groups were also found among others by Eaton, Ore, Krasner, Utumi, Drbohlav, Harrison, Roth, Mockor, Sureau and Haddad. Connections with other subjects of classical pure Mathematics have been determined and studied: • Fields by Krasner, Stratigopoulos and Massouros Ch. • Lattices by Mittas, Comer, Konstantinidou, Serafimidis, Leoreanu and Calugareanu • Rings by Nakano, Kemprasit, Yuwaree • Quasigroups and Groupoids by Koskas, Corsini, Kepka, Drbohlav, Nemec • Semigroups by Kepka, Drbohlav, Nemec, Yuwaree, Kempra sit, Punkla, Leoreanu • Ordered Structures by Prenowitz, Corsini, Chvalina IX x • Combinatorics by Comer, Tallini, Migliorato, De Salvo, Scafati, Gionfriddo, Scorzoni • Vector Spaces by Mittas • Topology by Mittas , Konstantinidou • Ternary Algebras by Bandelt and Hedlikova.
Introduction. Basic notions and results on Hyperstructure Theory. 1: Some topics of Geometry. 1. Descriptive geometries and join spaces. 2. Spherical geometries and join spaces. 3. Projective geometries and join spaces. 4. Multivalued loops and projective geometries. 2: Graphs and Hypergraphs. 1. Generalized graphs and hypergroups. 2. Chromatic quasi-canonical hypergroups. 3. Hypergroups induced by paths of a direct graph. 4. Hypergraphs and hypergroups. 5. On the hypergroup HGamma associated with a hypergraph Gamma. 6. Other hyperstructures associated with hypergraphs. 3: Binary Relations. 1. Quasi-order hypergroups. 2. Hypergroups associated with binary relations. 3. Hypergroups associated with union, intersection, direct product, direct limit of relations. 4. Relation beta in semihypergroups. 4: Lattices. 1. Distributive lattices and join spaces. 2. Lattice ordered join space. 3. Modular lattices and join spaces. 4. Direct limit and inverse limit of join spaces associated with lattices. 5. Hyperlattices and join spaces. 5: Fuzzy sets and rough sets. 2. Direct limit and inverse limit of join spaces associated with fuzzy subsets. 3. Rough sets, fuzzy subsets and join spaces. 4. Direct limits and inverse limits of join spaces associated with rough sets. 5. Hyperstructures and Factor Spaces. 6. Hypergroups induced by a fuzzy subset. Fuzzy hypergroups. 7. Fuzzy subhypermodules over fuzzy hyperrings. 8. On Chinese hyperstructures. 6: Automata. 1. Language theory and hyperstructures. 2. Automata and hyperstructures. 3. Automata and quasi-order hypergroups. 7: Cryptography. 1. Algebraic cryptography and hypergroupoids. 2. Cryptographic interpretation of some hyperstructures. 8: Codes. 1. Steiner hypergroupoids and Steiner systems. 2. Some basic notions about codes. 3. Steiner hypergroups and codes. 9: Median algebras, Relation algebras, C-algebras. 1. Median algebras and join spaces. 2. Relation algebras and quasi-canonical hypergroups. 3. C-algebras and quasi-canonical hypergroups. 10: Artificial Intelligence. 1. Generalized intervals. Connections with quasi-canonical hypergroups. 2. Weak representations of interval algebras. 11: Probabilities. Bibliography.