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Hypercompositional Algebra, Computer Science and Geometry

Author

Listed:
  • Gerasimos Massouros

    (School of Social Sciences, Hellenic Open University, Aristotelous 18, GR 26335 Patra, Greece)

  • Christos Massouros

    (Core Department, Euripus Campus, National and Kapodistrian University of Athens, Psahna, GR 34400 Euboia, Greece)

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.

Suggested Citation

  • Gerasimos Massouros & Christos Massouros, 2020. "Hypercompositional Algebra, Computer Science and Geometry," Mathematics, MDPI, vol. 8(8), pages 1-33, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1338-:d:397392
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    References listed on IDEAS

    as
    1. Ameri, R. & Zahedi, M.M., 1997. "Hypergroup and join space induced by a fuzzy subset," Pure Mathematics and Applications, Department of Mathematics, Corvinus University of Budapest, vol. 8(2-4), pages 155-168.
    2. Leoreanu, V., 2000. "Direct limit and inverse limit of join spaces associated with fuzzy sets," Pure Mathematics and Applications, Department of Mathematics, Corvinus University of Budapest, vol. 11(3), pages 509-516.
    3. Anastase Nakassis, 1988. "Recent results in hyperring and hyperfield theory," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 11, pages 1-12, January.
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    Citations

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    Cited by:

    1. Gerasimos G. Massouros & Christos G. Massouros, 2022. "State Machines and Hypergroups," Mathematics, MDPI, vol. 10(14), pages 1-25, July.
    2. Christos G. Massouros & Gerasimos G. Massouros, 2023. "On the Borderline of Fields and Hyperfields," Mathematics, MDPI, vol. 11(6), pages 1-35, March.
    3. Mario De Salvo & Dario Fasino & Domenico Freni & Giovanni Lo Faro, 2022. "Commutativity and Completeness Degrees of Weakly Complete Hypergroups," Mathematics, MDPI, vol. 10(6), pages 1-17, March.

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