# Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

^{1}

^{2}

^{*}

## Abstract

**:**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Veldkamp, F.D. Geometry over Rings, Handbook of Incidence Geometry; Buekenhout, F., Ed.; Elsevier: Amsterdam, The Netherlands, 1995; pp. 1033–1084. [Google Scholar]
- Payne, S.E.; Thas, J.A. Finite Generalized Quadrangles; Pitman: Boston, MA, USA, 1984. [Google Scholar]
- Saniga, M.; Havlicek, H.; Planat, M.; Pracna, P. Twin “Fano-Snowflakes” over the Smallest Ring of Ternions. Symmetry Integr. Geom.
**2008**, 4, 50. [Google Scholar] [CrossRef] - Havlicek, H.; Saniga, M. Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions. J. Geom.
**2009**, 92, 79–90. [Google Scholar] [CrossRef] - Saniga, M.; Planat, M. Multiple Qubits as Symplectic Polar Spaces of Order Two. Adv. Stud. Theor. Phys.
**2007**, 1, 1–4. [Google Scholar] - Lévay, P.; Holweck, F.; Saniga, M. Magic Three-Qubit Veldkamp Line: A Finite Geometric Underpinning for Form Theories of Gravity and Black Hole Entropy. Phys. Rev. D
**2017**, 96, 026018. [Google Scholar] [CrossRef] - Borsten, L.; Duff, M.J.; Lévay, P. The Black-Hole/Qubit Correspondence: An Up-To-Date Review. Class. Quant. Gravity
**2012**, 29, 224008. [Google Scholar] [CrossRef] - Holweck, F.; Saniga, M. Contextuality with a Small Number of Observables. Int. J. Quant. Inf.
**2017**, 15, 1750026. [Google Scholar] [CrossRef]

**Figure 1.**A pictorial representation of the doily, with its points labeled by duads (

**left**) and non-unimodular vectors (

**right**); both a duad $\{a,b\}$ and a vector $(a,b)$ are abbreviated to $ab$.

**Figure 2.**A graphical illustration of “Jacobson traces” of individual non-unimodular free cyclic submodules in the core doily; consecutively from top left to bottom right, they are shown the traces of $R(6,8)$, $R(8,14)$, $R(8,6)$, $R(8,3)$, $R(8,11)$, $R(3,8)$, $R(8,5)$, $R(8,13)$, and $R(5,8)$. In each subfigure, the corresponding concurrent lines are shown in boldface, the point of concurrence being encircled.

**Figure 3.**Visualisation of the full geometric structure formed by the nine nonunimodular free cyclic submodules, with the doily lying in its center. For each submodule there are shown all of its vectors except for the trivial one. For six submodules pairs of identical symbols are employed to identify the corresponding points of concurrence; the remaining three cases are obtained by swapping the figure with respect to the vertical axis passing through the center. The distinguished GQ(2,1) of the doily is shown in bold.

$\mathit{\alpha}$ | $\mathit{R}(3,8)$ | $\mathit{R}(5,8)$ | $\mathit{R}(6,8)$ | $\mathit{R}(8,11)$ | $\mathit{R}(8,13)$ | $\mathit{R}(8,14)$ | $\mathit{R}(8,6)$ | $\mathit{R}(8,5)$ | $\mathit{R}(8,3)$ |
---|---|---|---|---|---|---|---|---|---|

0 | $(0,0)$ | $(0,0)$ | $(0,0)$ | $(0,0)$ | $(0,0)$ | $(0,0)$ | $(0,0)$ | $(0,0)$ | $(0,0)$ |

1 | $(3,8)$ | $(5,8)$ | $(6,8)$ | $(8,11)$ | $(8,13)$ | $(8,14)$ | $(8,6)$ | $(8,5)$ | $(8,3)$ |

2 | $(3,11)$ | $(5,11)$ | $(6,11)$ | $(11,8)$ | $(11,14)$ | $(11,13)$ | $(11,6)$ | $(11,5)$ | $(11,3)$ |

3 | $(0,3)$ | $(0,3)$ | $(0,3)$ | $(3,3)$ | $(3,3)$ | $(3,3)$ | $(3,0)$ | $(3,0)$ | $(3,0)$ |

4 | $(3,13)$ | $(5,13)$ | $(6,13)$ | $(13,14)$ | $(13,8)$ | $(13,11)$ | $(13,6)$ | $(13,5)$ | $(13,3)$ |

5 | $(0,5)$ | $(0,5)$ | $(0,5)$ | $(5,5)$ | $(5,5)$ | $(5,5)$ | $(5,0)$ | $(5,0)$ | $(5,0)$ |

6 | $(0,6)$ | $(0,6)$ | $(0,6)$ | $(6,6)$ | $(6,6)$ | $(6,6)$ | $(6,0)$ | $(6,0)$ | $(6,0)$ |

7 | $(3,14)$ | $(5,14)$ | $(6,14)$ | $(14,13)$ | $(14,11)$ | $(14,8)$ | $(14,6)$ | $(14,5)$ | $(14,3)$ |

8 | $(0,8)$ | $(0,8)$ | $(0,8)$ | $(8,8)$ | $(8,8)$ | $(8,8)$ | $(8,0)$ | $(8,0)$ | $(8,0)$ |

9 | $(3,0)$ | $(5,0)$ | $(6,0)$ | $(0,3)$ | $(0,5)$ | $(0,6)$ | $(0,6)$ | $(0,5)$ | $(0,3)$ |

10 | $(3,3)$ | $(5,3)$ | $(6,3)$ | $(3,0)$ | $(3,6)$ | $(3,5)$ | $(3,6)$ | $(3,5)$ | $(3,3)$ |

11 | $(0,11)$ | $(0,11)$ | $(0,11)$ | $(11,11)$ | $(11,11)$ | $(11,11)$ | $(11,0)$ | $(11,0)$ | $(11,0)$ |

12 | $(3,5)$ | $(5,5)$ | $(6,5)$ | $(5,6)$ | $(5,0)$ | $(5,3)$ | $(5,6)$ | $(5,5)$ | $(5,3)$ |

13 | $(0,13)$ | $(0,13)$ | $(0,13)$ | $(13,13)$ | $(13,13)$ | $(13,13)$ | $(13,0)$ | $(13,0)$ | $(13,0)$ |

14 | $(0,14)$ | $(0,14)$ | $(0,14)$ | $(14,14)$ | $(14,14)$ | $(14,14)$ | $(14,0)$ | $(14,0)$ | $(14,0)$ |

15 | $(3,6)$ | $(5,6)$ | $(6,6)$ | $(6,5)$ | $(6,3)$ | $(6,0)$ | $(6,6)$ | $(6,5)$ | $(6,3)$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Saniga, M.; Bartnicka, E. Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules. *Axioms* **2019**, *8*, 28.
https://doi.org/10.3390/axioms8010028

**AMA Style**

Saniga M, Bartnicka E. Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules. *Axioms*. 2019; 8(1):28.
https://doi.org/10.3390/axioms8010028

**Chicago/Turabian Style**

Saniga, Metod, and Edyta Bartnicka. 2019. "Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules" *Axioms* 8, no. 1: 28.
https://doi.org/10.3390/axioms8010028