Geometry of flocks and $n$-ary groups

Sonia Dog

Algebra and discrete mathematics, Tome 28 (2019) no. 1, pp. 60-74.

Résumé

Semiabelinan flocks and $n$-ary groups are characterized by the properties of parallelograms and vectors of the affine geometry defined by these flocks and $n$-ary groups.

Détail

Keywords: $n$-ary group, flock, symmetry, affine geometry.

Bibliographie

[1] R. Baer, “Zur Einführung des Scharbegriffs”, J. Reine Angew. Math., 160 (1929), 199–207 | MR | Zbl

[2] D. Brǎnzei, “Structures affines et opérations ternaires”, An. Şti. Univ. Iaşi, Sect. I a Mat. (N.S.), 23 (1977), 33–38 | MR

[3] J. Certaine, “The ternary operation $(abc)=ab^{-1}c$ of a group”, Bull. Amer. Math. Soc., 49 (1943), 869–877 | DOI | MR | Zbl

[4] W. Dörnte, “Untersuchungen über einen verallgemeinerten Gruppenbegriff”, Math. Z., 29 (1928), 1–19 | DOI | MR | Zbl

[5] W. A. Dudek, “Remarks on $n$-groups”, Demonstratio Math., 13 (1980), 165–181 | MR | Zbl

[6] W. A. Dudek, “Varieties of polyadic groups”, Filomat (Niš), 9 (1995), 657–674 | MR | Zbl

[7] W. A. Dudek, “Ternary quasigroups connected with the affine geometry”, Algebras, Groups and Geometries, 16 (1999), 329–354 | MR | Zbl

[8] W. A. Dudek, “On some old and new problems in $n$-ary groups”, Quasigroups and Related Systems, 8 (2001), 15–36 | MR | Zbl

[9] W. A. Dudek, B. Gleichgewicht and K. Głazek, “A note on the axioms of $n$-groups”, Universal Algebra ((Esztergom, Hungary, 1977)), Colloquia Math. Soc. J. Bolyai, 29, 195–202 | MR | Zbl

[10] A. Knoebel, “Flocks, groups and heaps, joined with semilattices”, Quasigroups and Related Systems, 24 (2016), 43–66 | MR | Zbl

[11] Yu. I. Kulazhenko, “Geometry of parallelograms”, Vopr. Algeb. and Prik. Mat., Izdat. Belorus. Gos. Univ. Transp., Gomel, 1995, 47–64 (Russian)

[12] Yu. I. Kulazhenko, “Semi-commutativity criteria and self-coincidence of elements expressed by vector properties of $n$-ary groups”, Algebra Discrete Math., 9:2 (2010), 98–107 | MR

[13] Yu. I. Kulazhenko, “Geometry of semiabelian $n$-ary groups”, Quasigroups Related Systems, 19 (2011), 265–278 | MR | Zbl

[14] Yu. I. Kulazhenko, “Vectors and semiabelian criteria of $n$-ary groups”, Probl. Phys. Math. Tekh., 1:6 (2011), 65–68 (Russian) | Zbl

[15] Yu. I. Kulazhenko, “Sequences of vectors and criteria of semi-commutativity of $n$-ary groups”, Southeast Asian Bull. Math., 37 (2013), 745–752 | MR | Zbl

[16] Yu. I. Kulazhenko, “Semiabelianness and properties of vectors on $n$-ary groups”, Tr. Inst. Mat. Mekh., 20:1 (2014), 142–147 (Russian) | MR

[17] E. L. Post, “Polyadic groups”, Trans. Amer. Math. Soc., 48 (1940), 208–350 | DOI | MR

[18] H. Prüfer, “Theorie der Abelschen Gruppen”, Math. Z., 20 (1924), 166–187

[19] S. A. Rusakov, “A definition of $n$-ary group”, Doklady AN BSSR (Minsk), 23 (1972), 965–967 (Russian) | MR

[20] S. A. Rusakov, “Existence of $n$-ary $rs$-groups”, Voprosy Algebry, 6 (1992), 89–92 (Russian) | MR

[21] S. A. Rusakov, “Vectors of $n$-ary groups. Linear operations and their properties”, Vopr. Algeb. and Prik. Mat., Izdat. Belorus. Gos. Univ. Transp., Gomel, 1995, 10–30 (Russian)

[22] S. A. Rusakov, Some applications of $n$-ary groups, Belaruskaya navuka, Minsk, 1998 (Russian)

[23] W. Szmielew, Theory of $n$-ary equivalences and its application to geometry, Dissertationes Math., 191, 1980 | MR

[24] A. K. Sushkevich, The theory of generalized groups, Kharkov–Kiev, 1937 (Russian) | MR

[25] V. V. Vagner, “Theory of generalized heaps and generalized groups”, Mat. Sb., 32:74 (1953), 545–632 (Russian) | MR | Zbl