A Brief Survey on the two Different Approaches of Fundamental Equivalence Relations on Hyperstructures

Nikolaos Antampoufis, Sarka Hosková-Mayerová


Fundamental structures are the main tools in the study of hyperstructures. Fundamental equivalence relations link hyperstructure theory to the  theory of corresponding classical structures. They also introduce new hyperstructure classes.The present paper is a brief reference to the two different approaches to the notion of the fundamental relation in hyperstructures. The first one belongs to Koskas, who introduced the β ∗ - relation in hyperstructures and the second approach to Vougiouklis, who gave the name fundamental to the resulting quotient sets. The two approaches, the necessary definitions and the theorems for the introduction of the fundamental equivalence relation in hyperstructures, are presented.


Fundamental equivalence relations; strongly regular relation; hyperstructures; Hv - structures.

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N. Antampoufis, s2−H v (h/v)-structures and s2-hyperstructures, Int. J. of Alg. Hyperstructures and Appl. (IJAHA), 2(1), (2016), 163–174.

N. Antampoufis, s1-H v -groups, s1-hypergroups and the ∂ - operation, Proc. of 10 th AHA Congress 2008, Brno, Czech Republic, (2009), 99–112.

N. Antampoufis, Contribution to the study of hyperstructures with applications in Compulsory Education, Doctoral thesis, Ed. telia+pavla, Xanthi, Greece, 2008.

N. Antampoufis, T. Vougiouklis and A. Dramalidis, Geometrical and Circle Hyperoperations in Urban Applications, Ratio Sociologica 4(2)(2011), 53–66.

P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, 1993.

P. Corsini and Th. Vougiouklis, From Groupoids to Groups through Hypergroups, Renticcoti Mat. S. VII, 9 (1989), 173–181.

P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publishers, 2002.

J. Chvalina and S. Hoskova-Mayerova, General ω-hyperstructures and certain applications of those, Ratio Mathematica, 23, (2012), 3–20.

B. Davvaz, Semihypergroup Theory, Elsevier, Academic Press, 2016.

A. Dramalidis, Geometical H v -strucures, Stucture elements of Hyperstructures, Proceedings (Eds. N. Lygeros and Th. Vougiouklis), Spanidis Press, Alexandoupolis, Greece, (2005), 41–52.

D. Freni, Hypergroupoids and fundamental relations, Proceedings of AHA, Ed. M. Stefanescu, Hadronic Press, (1994), 81–92.

D. Freni, A note on the core of a hypergroup and the transitive closure β ∗ of β, Riv. Mat. Pura Appl. 8, (1991), 153–156.

M. Gutan, Properties of hyperproducts and the relation β in Quasihypergroups, Ratio Mathematica 12, (1997), 19–34.

S. Hoskova-Mayerova, Quasi-Order Hypergroups determinated by T-Hypergroups, Ratio Mathematica, 32, (2017), 37–44

S. Hoskova-Mayerova and A. Maturo, Algebraic Hyperstructures and Social Relations, Ital. J. Pure Appl. Math. 39, (2018), in print.

M. Koskas, Groupoids, demi-hypergroupes et hypergroupes, J. Math. Pures Appl., 49(9), (1970), 155–192.

F. Marty, Sur une gènèralisation de la notion de groupe, Huitieme Congrès Mathématiciens Scandinaves, Stockholm, (1934), 45–49.

A. Maturo, Propability, utility and Hyperstructures, Proceedings of the 8 th International Congress on AHA , Samothraki, Greece,(2003), 203–214.

R. Migliorato, Fundamental relation on non-associative Hypergroupoids, Ital. J. Pure Appl. Math., 6 (1999), 147–160.

M. Novák, EL-hyperstructures: an overview, Ratio Mathematica, 23 (2012), 65–80.

O. Ore, Structures and group theory I, Duke Math. J., 3, (1937), 149–174.

Th. Vougiouklis, An H v -interview, i.e. weak (interviewer N. Lygeros), Structure Elements of Hyperstructures, Proceedings,(Eds. N.Lygeros and Th. Vougiouklis), Spanidis Press, (2005), Alexandroupolis, Greece, 5–15.

Th. Vougiouklis, The h/v - structures, J. Discrete Math. Sci. Cryptogr., 6, (2003), 235–243.

Th. Vougiouklis, H v -groups defined on the same set, Discrete Mathematics 155, (1996), 259–265.

Th. Vougiouklis, Hyperstructures and their representations, Monographs, Hadronic Press, USA, 1994.

Th. Vougiouklis, The fundamental relation in hyperrings. The general hyperfield, 4 th AHA Cong. Xanthi, World Scientific (1991), 209–217.

Th. Vougiouklis, Groups in hypergroups, Annals Discrete Math., 37, (1988), 459–468.

T. Vougiouklis, S. Vougiouklis, Helix-Hopes on Finite Hyperfields, Ratio Mathematica, 31 (2016), 65–78.

DOI: http://dx.doi.org/10.23755/rm.v33i0.388


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