×

Cyclicity in \(EL\)-hypergroups. (English) Zbl 1423.20070

Summary: In the algebra of single-valued structures, cyclicity is one of the fundamental properties of groups. Therefore, it is natural to study it also in the algebra of multivalued structures (algebraic hyperstructure theory). However, when one considers the nature of generalizing this property, at least two (or rather three) approaches seem natural. Historically, all of these had been introduced and studied by 1990. However, since most of the results had originally been published in journals without proper international impact and later – without the possibility to include proper background and context-synthetized in books, the current way of treating the concept of cyclicity in the algebraic hyperstructure theory is often rather confusing. Therefore, we start our paper with a rather long introduction giving an overview and motivation of existing approaches to the cyclicity in algebraic hyperstructures. In the second part of our paper, we relate these to \(EL\)-hyperstructures, a broad class of algebraic hyperstructures constructed from (pre)ordered (semi)groups, which were defined and started to be studied much later than sources discussed in the introduction were published.

MSC:

20N20 Hypergroups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Marty, F.; Sur une généralisation de la notion de groupe; IV Congrès des Mathématiciens Scandinaves: 1934; ,45-49. · JFM 61.1014.03
[2] Wall, H.S.; Hypergroups; Am. J. Math.: 1934; Volume 59 ,77-98. · Zbl 0016.10302
[3] De Salvo, M.; Freni, D.; Sugli ipergruppi ciclici e completi; Matematiche (Catania): 1980; Volume 35 ,211-226. · Zbl 0571.20071
[4] Corsini, P.; Sur les semi-hypergroupes complètes et les groupoides; Atti Soc. Pel. Sci. Fis. Mat. Nat.: 1980; Volume 26 ,391-398. · Zbl 0585.20038
[5] Corsini, P.; Romeo, G.; ; Hypergroupes complètes et T-groupoids: Taormina, Italy 1978; ,129-146. · Zbl 0603.20060
[6] De Salvo, M.; Sugli ipergruppi completi finiti; Riv. Mat. Univ. Parma: 1982; Volume 8 ,269-280. · Zbl 0587.20039
[7] Freni, D.; Ipergruppi ciclici e torsione negli ipergruppi; Matematiche (Catania): 1980; Volume 35 ,270-286. · Zbl 0527.20055
[8] De Salvo, M.; Freni, D.; Semi-ipergruppi e ipergruppi ciclici; Atti Sem. Mat. Fis. Univ. Modena: 1981; Volume 30 ,44-59. · Zbl 0567.20051
[9] Freni, D.; Una nota su gli ipergruppoidi ciclici; Ratio Mathematica: 1995; Volume 9 ,101-111.
[10] Vougiouklis, T.; Cyclicity in a special class of hypergroups; Acta Univ. Carolinae Math. Phys.: 1981; Volume 22 ,3-6. · Zbl 0495.20042
[11] Konguetsof, L.; Vougiouklis, T.; Kessoglides, M.; Spartalis, S.; On cyclic hypergroups with period; Acta Univ. Carolinae Math. Phys.: 1987; Volume 28 ,3-7. · Zbl 0658.20045
[12] Vougioulis, T.; Isomorphisms on P-hypergroups and cyclicity; Ars Combinatoria: 1990; Volume 29A ,241-245. · Zbl 0744.20064
[13] De Salvo, M.; Freni, D.; Ipergruppi finitamente generati; Riv. Mat. Univ. Parma: 1986; Volume 12 ,177-186. · Zbl 0646.20062
[14] De Salvo, M.; Su le potenze ad esponente intero in un ipergruppo e gli r-ipergruppi; Riv. Mat. Univ. Parma: 1985; Volume 4 ,409-421. · Zbl 0647.20074
[15] Antampoufis, N.; Hošková-Mayerová, Š.; A brief survey on the two different approaches of fundamental equivalence relations on hyperstructures; Ratio Math.: 2017; Volume 33 ,47-60.
[16] Koskas, M.; Groupoids, demi-hypergroupes et hypergroupes; J. Math. Pure Appl.: 1970; Volume 48 ,155-192. · Zbl 0194.02201
[17] Freni, D.; A note on the core of a hypergroup and the transitive closure β⋆ of β; Riv. Mat. Pura Appl.: 1991; Volume 8 ,153-156. · Zbl 0780.20047
[18] Corsini, P.; ; Prolegomena of Hypergroup Theory: Tricesimo, Italy 1993; . · Zbl 0785.20032
[19] Corsini, P.; Leoreanu, V.; ; Applications of Hyperstructure Theory: Dodrecht, The Netherlands 2003; . · Zbl 1172.03324
[20] Al Tahan, M.; Davvaz, B.; On some properties of single power cyclic hypergroups and regular relations; J. Algebra Appl.: 2017; Volume 16 ,1750214. · Zbl 1394.20042
[21] Al Tahan, M.; Davvaz, B.; On a special single-power cyclic hypergroup and its automorphisms; Discrete Math. Algorithm. Appl.: 2016; Volume 8 ,1650059. · Zbl 1359.20041
[22] Karimian, M.; Davvaz, B.; On the γ-cyclic hypergroups; Commun. Algebra: 2006; Volume 34 ,4579-4589. · Zbl 1115.20062
[23] Freni, D.; A new characterization of the derived hypergroup via strongly regular equivalencies; Commun. Algebra: 2002; Volume 30 ,3977-3989. · Zbl 1021.20045
[24] Chvalina, J.; Commutative hypergroups in the sense of Marty and ordered sets; General Algebra and Ordered Sets, Proceedings of the International Conference Olomouc: Olomouc, Czech Republic 1994; ,19-30. · Zbl 0827.20085
[25] Chvalina, J.; ; Functional Graphs, Quasi-Ordered Sets and Commutative Hypergroups: Brno, Czech Republic 1995; .
[26] Massouros, C.G.; On path hypercompositions in graphs and automata; MATEC Web Conf.: 2016; Volume 41 ,05003.
[27] Novák, M.; On EL-semihypergroups; Eur. J. Combin.: 2015; Volume 44 ,274-286. · Zbl 1309.06008
[28] Novák, M.; Some basic properties of EL-hyperstructures; Eur. J. Combin.: 2013; Volume 34 ,446-459. · Zbl 1258.06012
[29] Novák, M.; Cristea, I.; Composition in EL-hyperstructures; Hacet. J. Math. Stat.: 2018; . · Zbl 1488.20099
[30] Křehlík, Š.; Novák, M.; From lattices to Hv-matrices; An. Şt. Univ. Ovidius Constanţa: 2016; Volume 24 ,209-222. · Zbl 1399.15041
[31] Novák, M.; Křehlík, Š.; EL-hyperstructures revisited; Soft Comput.: 2018; Volume 22 ,7269-7280. · Zbl 1401.06031
[32] Chvalina, J.; Křehlík, Š.; Novák, M.; Cartesian composition and the problem of generalising the MAC condition to quasi-multiautomata; An. Şt. Univ. Ovidius Constanţa: 2016; Volume 24 ,79-100. · Zbl 1399.68071
[33] Pickett, H.E.; Homomorphisms and subalgebras of multialgebras; Pac. J. Math.: 1967; Volume 21 ,327-342. · Zbl 0149.26101
[34] Novák, M.; EL-semihypergroups in which the quasi-ordering is not antisymmetric; Mathematics, Information Technologies and Applied Sciences 2017: Post-Conference Proceedings of Extended Versions of Selected Papers: Brno, Czech Republic 2017; ,183-192.
[35] Jantosciak, J.; Reduced hypergroups; Algebraic Hyperstructures and Applications, Proceedings of the Fourth International Congress, Xanthi, Greece, 1990: Singapore 1991; ,119-122. · Zbl 0791.20087
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.