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Web loops and webs with reflections

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Abstract

We introduce a web loop (P,\(\mathfrak{L}\), +) which is precisely a loop (P,+) provided with the structure\(\mathfrak{L}\) of a web such that both structures are compatible in the sense that for eacha ∈P the map a+∶P→P;x↦a+x is an automorphism of (P,\(\mathfrak{L}\)). Exploiting the results of [6, 7, 9, Z] for the case of a web, we study web loops, webs with reflection structures and webs with point reflection structures.

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References

  1. Barlotti, A., andStrambach, K.: The geometry of binary systems. Adv. Math.49 (1983) 1–105.

    Google Scholar 

  2. Barlotti, A., andStrambach, K.: The geometry of double loops. Adv. Math.72 (1988) 1–58.

    Google Scholar 

  3. Blaschke, W., andBol, G.: Geometrie der Gewebe. Springer-Verlag, Berlin 1938

    Google Scholar 

  4. Bol, G.: Gewebe und Gruppen. Math Ann.114 (1937) 414–431.

    Google Scholar 

  5. Bruck, R.H.: A survey of binary systems. Springer-Verlag, Berlin-Heidelberg-New York 1966.

    Google Scholar 

  6. Gabrieli, E., andKarzel, H.: Reflection geometries over loops. Resultate Math.32(1997) 61–65.

    Google Scholar 

  7. Gabrieli, E., andKarzel, H.: Point-reflection geometries, geometric K-loops and unitary geometries. Resultate Math.32 (1997) 66–72.

    Google Scholar 

  8. Karzel, H.: Kinematic spaces. Symp. Mathematica, Ist. Naz. Alta Mathematica11 (1973) 413–439.

    Google Scholar 

  9. Karzel, H.: Recent developments on absolute geometries and algebraization by K- loops. Submitted to Discrete Math. (1996)

  10. Karzel, H., andKonrad, A.: Reflection groups and K-loops. J. Geometry52 (1995) 120–129.

    Google Scholar 

  11. Karzel, H., andKist, G.P.: Kinematic algebras and their geometries. Rings and geometry, NATO ASI Series, Ser. C160 (1985) 437–509.

    Google Scholar 

  12. Karzel, H.,and Kroll, H.-J.: Geschichte der Geometrie seit Hilbert. Wiss. Buchgesellschaft, Darmstadt 1988

  13. Kist, G.P.: Theorie der verallgemeinerten kinematischen Räume. Habilitationsschrift Techn. Univ. München 1980

  14. Kreuzer, A.: Inner mappings of Bol loops. To appear in Math. Proc. Cambridge Philos. Soc.121 (1997)

  15. Kreuzer, A., andWefelscheid, H.: OnK-loops of finite order. Resultate Math.25 (1994) 79–102.

    Google Scholar 

  16. Pflugfelder, H.O.: Quasigroups and Loops: Introduction. Heldermann-Verlag, Berlin 1990

    Google Scholar 

  17. Reidemeister, K.: Grundlagen der Geometrie. Springer-Verlag, Berlin-Heidelberg-New york 1968

    Google Scholar 

  18. Zizioli, E.: Fibered incidence loops and kinematic loops. J. Geometry30 (1987) 144–156.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Chonnam National University, 500-757, Kwangju, Korea

    Bokhee Im & Hayi-Joo Ko

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  1. Bokhee Im
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  2. Hayi-Joo Ko
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Dedicated to Professor Dr. Dr. h. c. Helmut Karzel on the occasion of his seventieth birthday

This work was supported by CHONNAM NATIONNAL UNIVERSITY RESEARCH FUND, 1996

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Im, B., Ko, HJ. Web loops and webs with reflections. J Geom 61, 62–73 (1998). https://doi.org/10.1007/BF01237496

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  • DOI: https://doi.org/10.1007/BF01237496

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