On exactness and homology of hyperchain complexes

Abstract

In this paper we define the concepts of a “strict morphism” as a suitable map between geometric spaces (as a generalization of hyperstructures), and its kernel. Then we introduce the concepts of a “quotient geometric space”, an “exact sequence”, an “exact hypersequence”, a “homology” and a “hyperhomology” of geometric spaces. Finally we generalize some famous theorems from groups and modules in this direction, such as the “isomorphism theorem”, the “five short lemma”, the “snake lemma”, the “exact triangle”, etc., by some conditions that are naturally valid for groups.