HYPERGROUPS AND THEIR APPLICATIONS
Abstract
IN THIS DISSERTATION, THE HYPERGROUPS IN GENERAL, AS WELL AS SOME SPECIAL CATEGORIES OF HYPERGROUPS (JOIN, HOMIOGENE, STRONGLY HOMIOGENE) ARE STUDIED. BY THE DEVELOPED THEORY, CONCLUSIONS ARE DERIVED IN THE LINEAR SPACES, AS WELL AS IN CERTAIN GEOMETRIES. THE SEMI-SUB-HYPERGROUPS AND THE CLOSED SUB-HYPERGROUPS TOGETHER WITH SOME SPECIAL CATEGORIES OF ELEMENTS OF A HYPERGROUP (SUCH AS THE FUNDAMENTAL, DEPENDANT, CORRELATED ETC) PLAY A SIGNIFICANT ROLE IN THE WHOLE STUDY.THEOREMS ON THESE ELEMENTS, DESCRIBING THEIR PROPERTIES (MAINLY GENETIC) IN THE HYPERGROUP, ARE INTRODUCED. MOREOVER THE PROPERTIES WITH WHICH THE ELEMENTS OF SOME SPECIAL HYPERGROUPS ARE LED TO THE INTRODUCTION AND THE STUDY OF THE HOMIOGENE AND STRONGLY HOMIOGENE HYPERGROUPS. BESIDES IT WAS PROVED THAT IN EVERY LINEAR SPACE, CAN BE DEFINED. THIS ALLOWS US TO GET THEOREMS OF THE LINEAR SPACES SUCH AS THE THEOREMS OF KAKUTANI, STONE, HELLY, RANDON, CARATHEODORY, STEINITZ, WHICH DERIVE AS COROLLARIES OF MUCH MORE GENERAL ...
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