Synopses & Reviews
This book presents some of the numerous applications of hyperstructures, especially those that were found and studied in the last fifteen years. There are applications to the following subjects: 1) geometry; 2) hypergraphs; 3) binary relations; 4) lattices; 5) fuzzy sets and rough sets; 6) automata; 7) cryptography; 8) median algebras, relation algebras; 9) combinatorics; 10) codes; 11) artificial intelligence; 12) probabilities. Audience: Graduate students and researchers.
Table of Contents
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.