Jantosciak, James Transposition hypergroups: Noncommutative join spaces. (English) Zbl 0874.20051 J. Algebra 187, No. 1, 97-119 (1997). A transposition hyperqroup is a hypergroup \(H\) for which: \((b\setminus a)\cap(c/d)\neq\emptyset\) implies \(ad\cap bc\neq\emptyset\), for every \(a\), \(b\), \(c\), \(d\) in \(H\), where \(b\setminus a=\{x\in H\mid a\in bx\}\) and \(c/d=\{x\in H\mid c\in xd\}\).Introducing this notion, the author treats from a unique point of view several algebraic hyperstructures (join spaces, weak cogroups, double coset spaces, polygroups, canonical hypergroups, groups) instead to study them separately (as usually), all of them being transposition hypergroups. The (closed and reflexive) subhypergroups and the quotient spaces of a transposition hypergroup are studied. The three isomorphism theorems and the theorem of Jordan-Hölder of group theory are derived in the context of transposition hypergroups. The paper is very clear and very well written. Reviewer: M.Guţan (Aubière) Cited in 1 ReviewCited in 15 Documents MSC: 20N20 Hypergroups Keywords:join spaces; weak cogroups; double coset spaces; polygroups; canonical hypergroups; transposition hypergroups; quotient spaces; isomorphism theorems; Jordan-Hölder theorem PDFBibTeX XMLCite \textit{J. Jantosciak}, J. Algebra 187, No. 1, 97--119 (1997; Zbl 0874.20051) Full Text: DOI Link References: [1] Barlotti, A.; Strambach, K., Multigroups and the foundations of geometry, Rend. Circ. Mat. Palermo (2), 40, 5-68 (1991) · Zbl 0731.51009 [2] Comer, S. D., Combinatorial aspects of relations, Algebra Universalis, 18, 77-94 (1984) · Zbl 0549.20059 [3] Comer, S. D., Polygroups derived from cogroups, J. Algebra, 89, 397-405 (1984) · Zbl 0543.20059 [4] Corsini, P., Prolegomena of Hypergroup Theory. Prolegomena of Hypergroup Theory, Rivista di matematica pura ed applicata (1993), Aviani Editore · Zbl 0785.20032 [5] Dresher, M.; Ore, O., Theory of multigroups, Amer. J. Math., 60, 705-733 (1938) · JFM 64.0056.01 [6] Eaton, J. E., Theory of cogroups, Duke Math. J., 6, 101-107 (1940) · Zbl 0027.00804 [7] Haddad, L.; Sureau, Y., Les cogroupes et la construction de Utumi, Pacific J. Math., 145, 17-58 (1990) · Zbl 0663.20070 [8] Harrison, D. K., Double coset and orbit spaces, Pacific J. Math., 80, 451-491 (1979) · Zbl 0415.20022 [9] Jantosciak, J., Classical geometries as hypergroups, Convegno su: Ipergruppi, altre strutture multivoche e loro applicazioni, Udine, Italy (1985), p. 93-104 [10] Jantosciak, J., Homomorphisms, equivalences and reductions in hypergroups, Riv. Mat. Pura Appl., 9, 23-47 (1991) · Zbl 0739.20037 [11] Krasner, M., La loi de Jordan-Hölder dans les hypergroupes et les suites génératices des corps de nombres \(p\), Duke J. Math., 6, 120-140 (1940) · JFM 66.1207.02 [12] Massouros, C. G., Quasicanonical hypergroups, (Vougiouklis, T., Algebraic Hyperstructures and Applications (1991), World Scientific: World Scientific Singapore), 129-136 · Zbl 0791.20088 [13] Mittas, J., Hypergroups canoniques, Math. Balk. Beograd, 2, 165-179 (1972) · Zbl 0261.20064 [14] Prenowitz, W.; Jantosciak, J., Geometries and join spaces, J. Reine Angew. Math., 257, 100-128 (1972) · Zbl 0264.50002 [15] Prenowitz, W.; Jantosciak, J., Join Geometries, UTM (1979), Springer-Verlag: Springer-Verlag New York · Zbl 0421.52001 [16] Roth, R. L., Character and conjugacy class hypergroups of a finite group, Ann. Mat. Pura Appl., 105, 295-311 (1975) · Zbl 0324.20011 [17] Sureau, Y., Hypergroupes de type \(C\), Rend. Circ. Mat. Palermo (2), 40, 421-437 (1991) · Zbl 0771.20025 [18] Utumi, Y., On hypergroups of group right cosets, Osaka Math. J., 1, 73-80 (1949) · Zbl 0036.29302 [19] Wall, H. S., Hypergroups, Amer. J. Math., 59, 77-98 (1937) · Zbl 0016.10302 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.