Abstract
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.
Highlights
This paper is written in the context of the special issue “Hypercompositional Algebra and Applications” in “Mathematics” and it aims to shed light on two areas where the HypercompositionalAlgebra has expanded and has interacted with them: Computer Science and Geometry.Hypercompositional Algebra is a branch of Abstract Algebra which appeared in the 1930s via the introduction of the hypergroup.It is interesting that the group and the hypergroup are two algebraic structures which satisfy exactly the same axioms, i.e., the associativity and the reproductivity, but they differ in the law of synthesis
It is interesting that the groups and the hypergroups are two algebraic structures which satisfy exactly the same axioms, i.e., the associativity and the reproductivity, but they differ in the law of synthesis
We have presented the connection of the Hypercompositional Algebra to the Formal Languages and Automata theory as well as its close relationship to Geometry
Summary
Hypergroup this has hypergroup not c-elements. c-elements. Hypercompositional should expected, Algebra this and relation appears between be expected, this relation between be this relation appears. It is really of exceptional interest that the axioms of the hypergroup are directly related to certain ( x,0 )introduce geometries andand spherical geometries. It is is really really of exceptional exceptional interest that the axioms of the hypergroup. It is really are of directly exceptional related interest to certain that the axioms of th. More examples ofἐπί hyperringoids can be found in σθω πó παντóς σημε oυ π π ν σημε εεὐθεῖαν θε αν γραμμ ν γαγε “Hιτ oids can of be hyperringoids found in [61,113,114,115,116].ἀ [61,113,114,115,116].ἀ xamples can be found in ἐἐπί ὐand ῖ oν ῖwhich underneath
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