×

Construction of \(n\)-ary \((H,G)\)-hypergroups. (English) Zbl 1268.20067

Let \((H,f)\) be an \(n\)-ary hypergroup and \((G,h)\) be an \(n\)-ary group derived from some binary group. Let \(K\) by the union of a family of non-empty disjoint subsets \(A_g\) of \(H\) uniquely indexed by elements of \(G\) in the way that \(A_e=H\). If the hyper-operation \(f\) is extended to \(K\) in the way that \(f(x_1^n)=A_{h(g_1^n)}\) for all \((x_1^n)\in A_{g_1}\times\cdots\times A_{g_n}\neq H^n\) the structure obtained is an \(n\)-ary hypergroup called an \(n\)-ary \((H,G)\)-hypergroup. Properties and the fundamental relation of such hypergroups are described.

MSC:

20N20 Hypergroups
20N15 \(n\)-ary systems \((n\ge 3)\)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. M. Anvariyeh and B. Davvaz, Strongly transitive geometric spaces associated to hypermodules. J. Algebra 322 (2009), 1340-1359. · Zbl 1185.16049 · doi:10.1016/j.jalgebra.2009.05.014
[2] S. M. Anvariyeh and B. Davvaz, On the heart of hypermodules. Math. Scand. 106 (2010), 39-49. · Zbl 1193.16037
[3] S. M. Anvariyeh, S. Mirvakili, and B. Davvaz, Fundamental relation on \eth m; nÞ-ary hypermodules over \eth m; nÞ-ary hyperrings. Ars Combin. 94 (2010), 273-288. · Zbl 1227.16031
[4] S. D. Comer, Polygroups derived from cogroups. J. Algebra 89 (1984), 397-405. · Zbl 0543.20059 · doi:10.1016/0021-8693(84)90225-4
[5] P. Corsini, Prolegomena of hypergroup theory. Supplement to Riv. Mat. Pura Appl., Aviani Editore, Tricesimo 1993. Supplement to Riv. Mat. Pura Appl. · Zbl 0785.20032
[6] P. Corsini and V. Leoreanu, Applications of hyperstructure theory. Adv. Math. 5, Kluwer Academic Publishers, Dordrecht 2003. · Zbl 1027.20051
[7] I. Cristea and M. S\?tefa\?nescu, Binary relations and reduced hypergroups. Discrete Math. 308 (2008), 3537-3544. · Zbl 1148.20051 · doi:10.1016/j.disc.2007.07.011
[8] B. Davvaz, Approximations in n-ary algebraic systems. Soft Comput. 12 (2008), 409-418. · Zbl 1131.08002 · doi:10.1007/s00500-007-0174-y
[9] B. Davvaz, W. A. Dudek, and S. Mirvakili, Neutral elements, fundamental relations and n-ary hypersemigroups. Internat. J. Algebra Comput. 19 (2009), 567-583. · Zbl 1185.20061 · doi:10.1142/S0218196709005226
[10] B. Davvaz and M. Karimian, On the gÃ-complete hypergroups. European J. Combin. n 28 (2007), 86-93. · Zbl 1117.20053 · doi:10.1016/j.ejc.2004.09.007
[11] B. Davvaz and V. Leoreanu-Fotea, Hyperring theory and applications. International Academic Press, USA, 2007. · Zbl 1204.16033
[12] B. Davvaz and T. Vougiouklis, n-ary hypergroups. Iran. J. Sci. Technol. Trans. A Sci. 30 (2006), 165-174, 243.
[13] M. De Salvo, \eth H; GÞ-hypergroups. Riv. Mat. Univ. Parma (4) 10 (1984), 207-216. · Zbl 0599.20115
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.