Applications of Hyperstructure Theory

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.