IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v37y2008i2p337-345.html
   My bibliography  Save this article

On the fine structure of the projective line over GF(2)⊗GF(2)⊗GF(2)

Author

Listed:
  • Saniga, Metod
  • Planat, Michel

Abstract

The paper gives a succinct appraisal of the properties of the projective line defined over the direct product ring R▵≡GF(2)⊗GF(2)⊗GF(2). The ring is remarkable in that except for unity, all the remaining seven elements are zero-divisors, the non-trivial ones forming two distinct sets of three; elementary (‘slim’) and composite (‘fat’). Due to this fact, the line in question is endowed with a very intricate structure. It contains twenty-seven points, every point has eighteen neighbour points, the neighbourhoods of two distant points share twelve points and those of three pairwise distant points have six points in common. Algebraically, the points of the line can be partitioned into three groups: (a) the group comprising three distinguished points of the ordinary projective line of order two (the ‘nucleus’), (b) the group composed of twelve points whose coordinates feature both the unit(y) and a zero-divisor (the ‘inner shell’) and (c) the group of twelve points whose coordinates have both the entries zero-divisors (the ‘outer shell’). The points of the last two groups can further be split into two subgroups of six points each; while in the former case there is a perfect symmetry between the two subsets, in the latter case the subgroups have a different footing, reflecting the existence of the two kinds of a zero-divisor. The structure of the two shells, the way how they are interconnected and their link with the nucleus are all fully revealed and illustrated in terms of the neighbour/distant relation. Possible applications of this finite ring geometry are also mentioned.

Suggested Citation

  • Saniga, Metod & Planat, Michel, 2008. "On the fine structure of the projective line over GF(2)⊗GF(2)⊗GF(2)," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 337-345.
  • Handle: RePEc:eee:chsofr:v:37:y:2008:i:2:p:337-345
    DOI: 10.1016/j.chaos.2006.09.056
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077906009052
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2006.09.056?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Saniga, Metod & Planat, Michel, 2008. "Projective planes over “Galois” double numbers and a geometrical principle of complementarity," Chaos, Solitons & Fractals, Elsevier, vol. 36(2), pages 374-381.
    2. El Naschie, M. Saladin, 2006. "Advanced prerequisite for E-infinity theory," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 636-641.
    3. El Naschie, M.S., 2006. "E-infinity theory—Some recent results and new interpretations," Chaos, Solitons & Fractals, Elsevier, vol. 29(4), pages 845-853.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Liang, Y.S. & Su, W.Y., 2007. "The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus," Chaos, Solitons & Fractals, Elsevier, vol. 34(3), pages 682-692.
    2. He, Ji-Huan & Wan, Yu-Qin & Xu, Lan, 2007. "Nano-effects, quantum-like properties in electrospun nanofibers," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 26-37.
    3. Yuksel, S. & Gursel Caylak, E. & Acikgoz, A., 2009. "On fuzzy α-I-continuous and fuzzy α-I-open functions," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1691-1696.
    4. Saniga, Metod & Planat, Michel, 2008. "Projective planes over “Galois” double numbers and a geometrical principle of complementarity," Chaos, Solitons & Fractals, Elsevier, vol. 36(2), pages 374-381.
    5. Nasef, Arafa A. & Hatir, E., 2009. "On fuzzy pre-I-open sets and a decomposition of fuzzy I-continuity," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1185-1189.
    6. Saniga, Metod & Planat, Michel & Kibler, Maurice R. & Pracna, Petr, 2007. "A classification of the projective lines over small rings," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1095-1102.
    7. El Naschie, M.S., 2008. "Exact non-perturbative derivation of gravity’s G¯4 fine structure constant, the mass of the Higgs and elementary black holes," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 346-359.
    8. Ahmed, Sherif & Emam, Waled, 2009. "Dark matter candidates in U(1)B-L models," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 430-434.
    9. Yao, K. & Liang, Y.S. & Zhang, F., 2009. "On the connection between the order of the fractional derivative and the Hausdorff dimension of a fractal function," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2538-2545.
    10. Agop, M. & Radu, Cristina & Bontas, T., 2008. "El Naschie’s ε(∞) space–time and scale relativity theory in the topological dimension D=3," Chaos, Solitons & Fractals, Elsevier, vol. 38(5), pages 1243-1253.
    11. El Naschie, M.S., 2006. "Is Einstein’s general field equation more fundamental than quantum field theory and particle physics?," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 525-531.
    12. El Naschie, M.S., 2008. "Fuzzy knot theory interpretation of Yang–Mills instantons and Witten’s 5-Brane model," Chaos, Solitons & Fractals, Elsevier, vol. 38(5), pages 1349-1354.
    13. Ješić, Siniša N., 2009. "Convex structure, normal structure and a fixed point theorem in intuitionistic fuzzy metric spaces," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 292-301.
    14. Chen, Qingjiang & Cao, Huaixin & Shi, Zhi, 2009. "Construction and decomposition of biorthogonal vector-valued wavelets with compact support," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2765-2778.
    15. Zahran, A.M. & Abbas, S.E. & Abd El-baki, S.A. & Saber, Y.M., 2009. "Decomposition of fuzzy continuity and fuzzy ideal continuity via fuzzy idealization," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3064-3077.
    16. Yao, K. & Liang, Y.S. & Fang, J.X., 2008. "The fractal dimensions of graphs of the Weyl-Marchaud fractional derivative of the Weierstrass-type function," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 106-115.
    17. Agop, M. & Murgulet, C., 2007. "El Naschie’s ε(∞) space–time and scale relativity theory in the topological dimension D=4," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1231-1240.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:37:y:2008:i:2:p:337-345. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.