Quasi-Order Hypergroups and T-Hypergroups

Šárka Hošková-Mayerová


Quasi-order hypergroups were introduced by J. Chvalina in 90s of the last
century. They form a subclass of the class of all hypergroups, i.e. structures
with one associative hyperoperation fulfilling the reproduction axiom. In this
paper a theorem which allows an easy description of all quasi-order hyper-
groups is mentioned and some results concerning the relation of quasi-order
and upper quasi-order hypergroups are given. Furthermore the transformation
hypergroups acting on tolerance spaces are defined and an example of them is


Quasi-order hypergroup; order hypergroup; tolerance relation; transformation semihypergroup; transformation hypergroup

Full Text:



I. Chajda, Algebraic Theory of Tolerance Relations, The Palacky University Olomouc, Czech Republic (1991).

J. Chvalina, Commutative hypergroups in the sence of Marthy, Proceeding of the Summer School (1994), 19–30, Horní Lipová, Czech Republic.

J. Chvalina, Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups, MU Brno, (1995), (in Czech).

J. Chvalina, L. Chvalinová, State hypergroup of automata, Acta Mat. et. Inf. Univ. Ostraviensis 4 (1996), 105–120.

J. Chvalina, Š. Hošková, Abelization of proximal H ν -rings using graphs of good homomorphisms and diagonals of direct squares of hyperstructures, Internat. Congress on AHA 8 (Samothraki, Greece 2002), 147–158.

P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, Tricesimo, (1993).

P. Corsini, Hypergraphs and hypergroups, Algebra Universalis, 35, (1996), 548–555.

P. Corsini, V. Leoreanu, Hypergroups and Binary Relations , Algebra Universalis, 43, (2000), 321–330.

P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publishers, Dordrecht, Hardbound, (2003).

Š. Hošková, Abelization of weakly associative hyperstructures and their proximal modifications, Ph.D. thesis, Masaryk University Brno (2003), 73p.

Š. Hošková, Upper order hypergroups as a reflective subcategory of subquasi-order hypergroups, Italian Journal of Pure and Applied Math., No.20, 215–222, (2006).

Š. Hošková, Representation of quasi-order hypergroups, Global Journal of Pure and Applied Mathematics (GJPAM), Vol.1 No. 2, 173–176, India, (2005).

Š. Hošková, J. Chvalina, The unique square root condition for quasi-order hypergroups and the corresponding reflector for the category of all order-hypergroups, Proc. of 3 th International Conference Aplimat 2004, Bratislava, 471–476, Slovakia.

Š. Hošková, J. Chvalina, Transformation tolerance hypergroup, Thai Mathematical Journal, Volume 4, No.1, (2006), in print.

Marty, F.: Sur une généralisation de la notion de groupe, Huiti` eme Congr. math. Scan. (1934), Stockholm, 45–49.

Vougiouklis, T.: Hyperstructures and Their Representations, Hadronic Press Monographs in Mathematics, Palm Harbor Florida 1994.

DOI: http://dx.doi.org/10.23755/rm.v32i0.333


  • There are currently no refbacks.

Copyright (c) 2017 Šárka Hošková

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.