Composition in $EL$-hyperstructures

Year 2019, Volume: 48 Issue: 1, 45 - 58, 01.02.2019

Michal Novak Irina Cristea

Abstract

The link between ordered sets and hyperstructures is one of the classical areas of research in the hyperstructure theory. In this paper we focus on $EL$--hyperstructures, i.e. a class of hyperstructures constructed from quasi-ordered semigroups. In our paper we link this concept to the concept of a \emph{composition hyperring}, a recent hyperstructure generalization of the classical notion of a composition ring.

Keywords

Composition ring, composition hyperring, hyperstructure theory, partially ordered semigroup, quasi-ordered semigroup

References

• I. Adler, Composition rings, Duke Math. J. 29, 607-623, 1962.
• S. M. Anvariyeh and B. Davvaz, Strongly transitive geometric spaces associated to hypermodules, J. Algebra 332, 1340-1359, 2009.
• J. Chvalina, Commutative hypergroups in the sense of Marty and ordered sets, General Algebra and Ordered Sets, Proc. Int. Conf. Olomouc, 19-30, 1994.
• J. Chvalina, Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups (in Czech), Masaryk University, Brno, 1995.
• J. Chvalina and L. Chvalinová, Transposition hypergroups formed by transformation operators on rings of differentiable functions, Ital. J. Pure Appl. Math. 15, 93-106, 2004.
• J. Chvalina, Š. Hošková-Mayerová and A.D. Nezhad, General actions of hyperstructures and some applications, An. St. Univ. Ovidius Constanta, 21 (1), 59-82, 2013.
• A. Connes and C. Consani, The hyperring of adéle classes, J. Number Theory 131 (2), 159-194, 2011.
• P. Corsini, Hyperstructures associated with ordered sets, Bull. Greek Math. Soc. 48, 7-18, 2003.
• P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publishers, Dodrecht, 2003.
• I. Cristea and S. Jancic-Rašovic, Composition hyperrings, An. St. Univ. Ovidius Constanta 21 (2), 81-94, 2013.
• B. Davvaz and V. Leoreanu Fotea, Applications of Hyperring Theory, International Academic Press, Palm Harbor, 2007.
• B. Davvaz, N. Rakhsh-Khorshid and K.P. Shum, Construction of composition $(m,n,k)$-hyperrings, An. St. Univ. Ovidius Constanta 24 (1), 177-188, 2016.
• B. Davvaz and A. Salasi, A realization of hyperrings, Comm. Algebra 34, 4389-4400, 2006.
• A. Deghan Nezhad and B. Davvaz, An Introduction to the Theory of $H_v$-Semilattices, Bull. Malays. Math. Sci. Soc. 32 (3), 375-390, 2009.
• S.H. Ghazavi and S.M. Anvariyeh, $EL$-hyperstructures associated to $n$-ary relations, Soft Comput. 21 (19), 5841-5850, 2017.
• D. Heidari and B. Davvaz, On ordered hyperstructures, U.P.B. Sci. Bull. Series A, 73 (2), 2011.
• K. Iwasava, On linearly ordered groups, J. Math. Soc. 1 (1), 1-9, 1948.
• S. Jancic-Rašovic, About the hyperring of polynomials, Ital. J. Pure Appl. Math. 28, 223-234, 2007.
• M. Krasner, A class of hyperrings and hyperfields, Int. J. Math. Sci. 6 (2), 307-312, 1983.
• M. Krasner, Approximation des corps values complets de characteristique $p\not=0$ par ceux de characteristique 0, Colloque d’Algebre Superieure, CBRM, Bruxelles, 1957.
• Š. Krehlík and M. Novák, From lattices to $H_v$-matrices, An. St. Univ. Ovidius Constanta 24 (3), 209-222, 2016.
• Ch.G. Massouros, On the theory of hyperrings and hyperfields, Algebra i Logika 24 (6), 728-742, 1985.
• G.G. Massouros, The hyperringoid, Multiple Valued Logic 3, 217-234, 1998.
• Ch.G. Massouros and G.G. Massouros, On join hyperrings, Proceedings of the 10th International Congress on Algebraic Hyperstructures and Applications, Brno, Czech Republic, 203-215, 2009.
• Ch.G. Massouros and G. G. Massouros, The transposition axiom in hypercompositional structures, Ratio Mathematica 21, 75-90, 2011.
• S. Mirvakili and B. Davvaz, Applications of the $a^{\ast}$-relation to Krasner hyperrings, J. Algebra 362, 145-156, 2012.
• J.D. Mittas, Sur certaines classes de structures hypercompositionnelles, Proceedings of the Academy of Athens, 48, 298-318, 1973.
• A. Nakasis, Recent results in hyperring and hyperfield theory, Internat. J. of Math. & Math. Sci. 11 (2), 209-220, 1988.
• M. Novák, Some basic properties of $EL$-hyperstructures, European J. Combin. 34, 446-459, 2013.
• M. Novák, Potential of the “Ends lemma" to create ring-like hyperstructures from quasi-ordered (semi)groups, South Bohemia Mathem. Letters 17 (1), 39-50, 2009.
• M. Novák, On $EL$-semihypergroups, European J. Combin. 44 Part B, 274-286, 2015.
• S. Spartalis, A class of hyperrings, Riv. Mat. Pura Appl. 4, 55-64, 1989.
• Th. Vougiouklis, On some representations of hypergroups, Annales scientifiques de l’Université de Clermont-Ferrand 2, Série Mathematiques 26, 21-29, 1990.