Hypercompositional Algebra, Computer Science and Geometry

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.

Regular Parameter Elements and Regular Local Hyperrings

“…In this article, the free and cyclic hypermodules are studied and several examples are provided such as the one obtained as a quotient of a P-module over a unitary ring P. Recently, Bordbar and his collaborators in [16][17][18] have introduced the length and the support of hypermodules and studied some properties of them, that are used also in this paper. Moreover, the connection between the hyperrings/hyperfields theory and geometry is very clearly explained by Massouros in [19]. This paper is a pylon in the current literature on algebraic hypercompositional algebra, because it describes the development of this theory (with a lot of examples and explanation of the terminology) from the first definition of hypergroup proposed by F. Marty to the hypergroups endowed with more axioms and used now a days.…”

“…The importance of these hyperrings and hyperfields in Krasner's studies is very clear explained by G. Massouros and Ch. Massouros in [19], as well as their different names given by some authors [7][8][9][10][11], with the risk of creating confusions. Therefore, in order to keep the original terminology, we recommend to read the papers of Nakasis [21] and Massouros [15,22,23].…”

On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations

“…He named this new hypergroup join space. Prenowitz was followed by others, such as Jantosciak [62,63], Barlotti and Strambach [64], Freni [65,66], Massouros [67][68][69][70][71], Dramalidis [72], etc. For the sake of terminology unification, the commutative hypergroups which satisfy the transposition axiom are called join hypergroups.…”

Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators

“…For further reading on automata theory and its links to the theory of hypercompositional structures (also known as algebraic hyperstructures), see, e.g., [24][25][26]. Furthermore, for clarification and evolution of terminology, see [8].…”