Loops, reflection structures and graphs with parallelism

被引：0

作者：

Karzel H. ^{[1
]}

Pianta S. ^{[2
]}

Zizioli E. ^{[3
]}

机构：

[1] Zentrum Mathematik, Technische Universität München, München

[2] Dip. di Matematica e Fisica, Università Cattolica, Via Trieste, 17, Brescia

[3] Dip. di Matematica - Facoltà di Ingegneria, Università di Brescia, Via Valotti, 9, Brescia

来源：

Results in Mathematics | 2002年 / 42卷 / 1-2期

关键词：

Graphs with parallelism; Right loops;

D O I：

10.1007/BF03323555

中图分类号：

学科分类号：

摘要：

The correspondence between right loops (P, +) with the property “(*) ∀a, b ∈ P: a − (a − b) − b” and reflection structures described in [4] is extended to the class of graphs with parallelism (P, ε, ∥). In this connection K-loops correspond with trapezium graphs, i.e. complete graphs with parallelism satisfying two axioms (T1), (T2) (cf. §3). Moreover (P, ε, ∥ +) is a structure loop (i.e. for each a ∈ P the map a +: P → P; x → a + x is an automorphism of the graph with parallelism (P, ε, ∥)) if and only if (P, +) is a K-loop or equivalently if (P, ε, ∥) is a trapezium graph. © Birkhäuser Verlag, Basel, 2002.

引用

页码：74 / 80

页数：6

共 7 条

[1] Ellers E., Karzel H., Kennzeichnung elliptischer Gruppenräume, Abh. Math. Sem. Univ. Hamburg, 26, pp. 55-77, (1963)
[2] Gabrieli E., Karzel H., Webs related to K-loops and Reflection Structures, Abh. Math. Sem. Univ. Hamburg, 69, pp. 89-102, (1999)
[3] Karzel H., Bericht über projective Inzidenzgruppen, J. Ber. DMV, 67, pp. 58-92, (1964)
[4] Karzel H., Recent Developments on Absolute Geometries and Algebraization by KLoops, Discr. Math, 208-209, pp. 387-409, (1999)
[5] Kreuzer A., Wefelscheid H., On K-loops of Finite Order, Result. Math, 25, pp. 79-102, (1994)
[6] Wahling H., Projective Inzidenzgruppoide und Fastalgebren, J. of Geom, 9, pp. 109-126, (1977)
[7] Zizioli E., Connections Between Loops of Exponent 2, Reflection Structures and Complete Graphs with Parallelism, Result. Math, 38, pp. 187-194, (2000)

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