Abstract
We consider the fundamental relations β and γ in simple and 0-simple semihypergroups, especially in connection with certain minimal cardinality questions. In particular, we enumerate and exhibit all simple and 0-simple semihypergroups having order 3 where β is not transitive, apart of isomorphisms. Moreover, we show that the least order for which there exists a strongly simple semihypergroup where β is not transitive is 4. Finally, we prove that γ is transitive in all simple semihypergroups, and determine necessary and sufficient conditions for a 0-simple semihypergroup to have γ transitive. The latter results obviously hold also for simple and 0-simple semigroups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anvariyeh S.M., Davvaz B.: Strongly transitive geometric spaces associated to hypermodules. J. Algebra 322, 1340–1359 (2009)
Chaopraknoi S., Triphop N.: Regularity of semihypergroups of infinite matrices. Thai J. Math. 4, 7–11 (2006)
Davvaz B.: Characterizations of sub-semihypergroups by various triangular norms. Czechoslovak Math. J. 55, 923–932 (2005)
Davvaz B.: Applications of the γ*-relation to polygroups. Commun. Algebra 35, 2698–2706 (2007)
De Salvo, M.: K H -hypergroups. Atti Sem. Mat. Fis. Univ. Modena 31, 112–122 (1982, in Italian)
De Salvo, M.; Freni, D.: Cyclic semihypergroups and hypergroups. Atti Sem. Mat. Fis. Univ. Modena 30, 44–59 (1981, in Italian)
De Salvo M., Fasino D., Freni D., Lo Faro G.: Isomorphism classes of the hypergroups of type U on the right of size five. Comp. Math. Appl. 58, 390–402 (2009)
Fasino D., Freni D.: Existence of proper semihypergroups of type U on the right. Discrete Math. 307, 2826–2836 (2007)
Fasino D., Freni D.: Minimal order semihypergroups of type U on the right. Mediterr. J. Math. 5, 295–314 (2008)
Freni D.: r-Hypergroups and their extensions. Atti Società Peloritana Sci. Fis. Mat. Natur. 27, 77–93 (1981)
Freni D.: A new characterization of the derived hypergroup via strongly regular equivalences. Commun. Algebra 30(8), 3977–3989 (2002)
Freni D.: Strongly transitive geometric spaces: applications to hypergroups and semigroups theory. Commun. Algebra 32(3), 969–988 (2004)
Freni D.: Minimal order semihypergroups of type U on the right, II. J. Algebra 340, 77–89 (2011)
Gutan C.: Plongements d’un hypergroupe dans un hypergroupe ayant un élément de plus. Riv. Mat. Pura Appl. 18, 85–92 (1996)
Gutan, C.: Extensions des semigroupes associées à certains semihypergroupes très fins. In: Algebraic Hyperstructures and Applications (Prague, 1996), 43–52. Democritus Univ. Thrace, Alexandroupolis (1997)
Gutan M.: On the transitivity of the relation β in semihypergroups. Rend. Circ. Mat. Palermo (2) 45, 189–200 (1996)
Howie M.: Fundamentals of Semigroup Theory. Oxford University Press, New York (1995)
Jafarabadi H.M., Sarmin N.H., Molaei M.R.: Completely simple and regular semi hypergroups. Bull. Malaysian Math. Sci. Soc. 35, 335–343 (2012)
Koskas H.: Groupoïdes, demi-hypergroupes et hypergroupes. J. Math. Pures Appl. 49, 155–192 (1970)
Vougiouklis T.: The set of hypergroups with operators which are constructed from a set with two elements. Acta Univ. Carolin. Math. Phys. 22, 7–10 (1981)
Vougiouklis T.: Fundamental relations in hyperstructures. Bull. Greek Math. Soc. 42, 113–118 (1999)
Acknowledgments
D. Fasino was supported by PRIN project no. 20083KLJEZ “Problemi di algebra lineare numerica strutturata: analisi, algoritmi e applicazioni”. D. Freni was supported by PRIN project no. 20087P8MN2 “Teoria dei disegni, teoria spettrale dei grafi, teorie combinatorie e loro applicazioni”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Fasino, D., Freni, D. Fundamental relations in simple and 0-simple semihypergroups of small size. Arab. J. Math. 1, 175–190 (2012). https://doi.org/10.1007/s40065-012-0025-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-012-0025-2