The various branches of Mathematics are not separated between themselves. On the contrary, they interact and extend into each other’s sometimes seemingly different and unrelated areas and help them advance. In this sense, the Hypercompositional Algebra’s path has crossed, among others, with the paths of the theory of Formal Languages, Automata and Geometry. This paper presents the course of development from the hypergroup, as it was initially defined in 1934 by F. Marty to the hypergroups which are endowed with more axioms and allow the proof of Theorems and Propositions that generalize Kleen’s Theorem, determine the order and the grade of the states of an automaton, minimize it and describe its operation. The same hypergroups lie underneath Geometry and they produce results which give as Corollaries well known named Theorems in Geometry, like Helly’s Theorem, Kakutani’s Lemma, Stone’s Theorem, Radon’s Theorem, Caratheodory’s Theorem and Steinitz’s Theorem. This paper also highlights the close relationship between the hyperfields and the hypermodules to geometries, like projective geometries and spherical geometries.

中文翻译：

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超组合代数，计算机科学与几何

数学的各个分支之间没有分隔。相反，它们相互作用并延伸到彼此有时看似不同且无关的领域，并帮助他们前进。从这个意义上讲，超组合代数的道路与形式语言，自动机和几何学的道路相交叉。本文介绍了超群的发展历程，它是由F. Marty于1934年最初定义的，是超群所赋予的更多公理，并允许定理和命题的证明推广Kleen定理，确定阶次和等级自动机的状态，将其最小化并描述其操作。相同的超群位于“几何”下，它们产生的结果得出推论，即众所周知的“几何中的定理”，例如Helly定理，Kakutani引理，Stone定理，Radon定理，Caratheodory定理和Steinitz定理。本文还强调了超场和超模与几何之间的紧密关系，例如射影几何和球形几何。