Abelizations of weakly associative hyperstructures based on their direct squares. (English) Zbl 1110.20304

Summary: The paper contains a simple construction of some types of commutative hyperstructures as hypergroupoids, weakly associative semi-hypergroupoids, hypergroups, quasi-hypergroups and weakly associative hyperrings from non-commutative hyperstructures. It is proved that the used construction induces reflectors on suitable categories of the mentioned hyperstructures.


20N20 Hypergroups
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
16Y99 Generalizations
Full Text: EuDML


[1] Adámek J.: Mathematical Structures and Categories. SNTL Praha 1982)
[2] Adámek J., Koubek V., Reiterrhan J.: Základy obecne topologie. SNTL Praha 1977.
[3] Beránek J., Chvalina J.: From groups of linear functions to non-commutative transposition hypergroups. Dept. Math. Report Series 7, Univ. of South Bohemia, České Budějovice (1999), 1-10.
[4] Čech E.: Topological Spaces. revised by Z. Frolík and M. Katětov, Academia, Praha 1966. · Zbl 0141.39401
[5] Corsini P.: Prolegomena of Hypergroup Theory. Aviani Editore, Tricesimo, 1993. · Zbl 0785.20032
[6] Darafsheh M. R., Davvaz B.: \(H_ v\)-rings of fractions. Italian Journal of Pure and App. Math., N. 5, (1999), 25-34. · Zbl 0980.16042
[7] Freni D.: Hypergroupoids and fundamental relations. Algebraic Hyperstructures and Applications, Proceedings of the Fifth International Congress, Iasi, Romania 1993, Palm Harbor, Hadronic Press (1994), 81-92. · Zbl 0842.20056
[8] Chvalina J.: Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups. Masarykova Universita, Brno 1995)
[9] Chvalina J., Chvalinová L.: Multistructures determined by differential rings. Arch. Math., Brno (2000), T. 36, CDDE 2001 issue, 429-434. · Zbl 1090.12500
[10] Chvalina J., Chvalinová L.: Transposition hypergroups formed by transformation operators on rings of differentiable functions. Italian Journ. of Pure and Appl. Mathematics, N. 17, (2003), 13 pp. In print. · Zbl 1184.47019
[11] Chvalina J., Hošková S.: Abelization of quasi-hypergroups as reflexion. Second Conf. Math. and Physics at Technical Universities, Military Academy Brno, Proceedings of Contributions, MA Brno (2001), 47-53
[12] Herrlich H., Strecker G.: Topologische Reflexionen and Coreflexionen. Lecture Notes on Math. 78, 1968. · Zbl 0182.25302
[13] Herrlich H., Strecker G.: Category Theory. Allyn and Bacon, Boston 1973. · Zbl 0265.18001
[14] Ježek J., Kepka T.: Bijective Reflexions and Coreflexions of Commutative Unars. Acta Univ. Carol. - Math. Phys., Vol. 37, No 1, (1996), 31-40. · Zbl 0899.08005
[15] Kannan V.: Reflexive and coreflexive categories in topology. Math. Ann. 195 (1972), 168-174. · Zbl 0215.09701
[16] Kuroš A. G.: Teorija grupp. Izdatelstvo ”Nauka”, Moskva 1967.
[17] Migliorato R.: Some topics on the feebly associative hypergroupoids. Algebraic Hyperstructures and Applications, Proceedings of the Fifth International Congress, Iasi, Romania 1993, Palm Harbor, Hadronic Press (1994), 67-79. · Zbl 0842.20060
[18] Migliorato R., Gentile G.: Feebly associative hypergroupoids. Proceedings of the International Conference on Finite Geometries, Perugia, Italy 1992, (1993), 259-268. · Zbl 0802.20057
[19] Skula L.: On a reflexive subcategory of the category of all topological spaces. Trans. Amer. Math. Soc. 142 (1969), 37-41. · Zbl 0185.50401
[20] Spartalis S.: On reversible \(H_ v\)-groups, Algebraic Hyperstructures and Applications. Proc. Fifth Intefnat. Congress on AHA Iasi, Romania 1993. Hadronic Press, Inc. Palm Harbor (1994), 163-170.
[21] Spartalis S.: On the number of \(H_ v\)-rings with P-hypergroupoids. Discrete Math. 155, No. 1-3 (1966), 225-231.
[22] Spartalis S.: Quotients of the P - \(H_ v\)-rings. Proceedings of the International Workshop, Molise, Italy 1995, Palm Harbor, Hadronic Press, Series on New Frontiers in Advanced Mathematics (1996), 167-176. · Zbl 0883.16040
[23] Spartalis S., Vougiouklis T.: The fundamental relations on \(H_ v\)-rings. Rivistadi Matematica Pura ed Appl., 14 (1994), 7-20. · Zbl 0803.16043
[24] Tallini G.: Geometric hypergroups and line spaces. Acta Univ. Carol. Math. Phys, 25, 1 (1984), 69-73. · Zbl 0566.51016
[25] Tallini G: General multivalued algebraic structures and geometric space. Algebraic Hyperstructures and Applications, J. Comb. Inf. Syst. Sci. 20, No. 14 (1995), 229-235.
[26] Vougiouklis T.: Generalization of \(H_ v\)-hypergroups. Rend. Circ. Math. Palermo, Ser.II, 36 (1987), 114-121. · Zbl 0652.20069
[27] Vougiouklis T.: Groups in hypergroups. Annals Discrete Math., 37 (1988), 459-468. · Zbl 0652.20071
[28] Vougiouklis T.: Isomorphism on \(H_ v\)-hyperstructures and cyclicity. Ars Combinatoria, V. 29 A, (1990), 242-245.
[29] Vougiouklis T.: The fundamental relation in hyperrings. The general hyperfield. Fourth Internat. Congress Algebraic Hyperstructures and Applications, Xanthi 1990. World Scientific (1991), 209-217.
[30] Vougiouklis T.: \(H_ v\)-groups defined on the same set. Combinatorics 92, preprint. · Zbl 0855.20057
[31] Vougiouklis T.: \(H_ v\)-vector spaces. Algebraic Hyperstructures and Applications, Proc. Fifth Internat. Congress on AHA lasi, Roumania 1993, Hadronic Press, Inc. Palm Harbor (1994), 181-190. · Zbl 0843.16043
[32] Vougiouklis T.: Hyperstructures and their Representations. Hadronic Press Monographs in Mathematics, Palm Harbor Florida 1994. · Zbl 0828.20076
[33] Vougiouklis T.: A new class of hyperstructures. Journ. Comb. Inf. System Sciences, V. 20, (1995), 229-235. · Zbl 0874.20052
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.