A Classification of the Projective Lines over Small Rings II. Non-Commutative Case

A list of different types of a projective line over non-commutative rings with unity of order up to thirty-one inclusive is given. Eight different types of such a line are found. With a single exception, the basic characteristics of the lines are identical to those of their commutative counterparts. The exceptional projective line is that defined over the non-commutative ring of order sixteen that features ten zero-divisors and it most pronouncedly differs from its commutative sibling in the number of shared points by the neighbourhoods of three pairwise distant points (three versus zero), that of "Jacobson" points (zero versus five) and in the maximum number of mutually distant points (five versus three).Comment: 2 pages, 1 tabl

Cayley-Dickson Algebras and Finite Geometry

Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \leq a \leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$ if one regards these imaginary units as points and distinguished triads of them $\{e_a, e_b, e_c\}$, $1 \leq a < b <c \leq 2^N -1$ and $e_ae_b = \pm e_c$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b = c$ or $a+b \neq c$. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG$(N-1,2)$, the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a binomial $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration ${\cal C}_N$; in particular, ${\cal C}_3$ (octonions) is isomorphic to the Pasch $(6_2,4_3)$-configuration, ${\cal C}_4$ (sedenions) is the famous Desargues $(10_3)$-configuration, ${\cal C}_5$ (32-nions) coincides with the Cayley-Salmon $(15_4,20_3)$-configuration found in the well-known Pascal mystic hexagram and ${\cal C}_6$ (64-nions) is identical with a particular $(21_5,35_3)$-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. We also draw attention to a remarkable nesting pattern formed by these configurations, where ${\cal C}_{N-1}$ occurs as a geometric hyperplane of ${\cal C}_N$. Finally, a brief examination of the structure of generic ${\cal C}_N$ leads to a conjecture that ${\cal C}_N$ is isomorphic to a combinatorial Grassmannian of type $G_2(N+1)$.Comment: 26 pages, 20 figures; V2 - the basis made explicit, a footnote and a couple of references adde

Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions

Given a finite associative ring with unity, $R$, any free (left) cyclic submodule (FCS) generated by a $uni$modular ($n+1$)-tuple of elements of $R$ represents a point of the $n$-dimensional projective space over $R$. Suppose that $R$ also features FCSs generated by ($n+1$)-tuples that are $not$ unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for $n=2$ when $R$ is the (unique) non-commutative ring of order eight. The corresponding geometry is dubbed a "Fano-Snowflake" due to its diagrammatic appearance and the fact that it contains the Fano plane in its center. There exist, in fact, two such configurations -- each being tied to either of the two maximal ideals of the ring -- which have the Fano plane in common and can, therefore, be viewed as twins. Potential relevance of these noteworthy configurations to quantum information theory and stringy black holes is also outlined.Comment: 6 pages, 1 table, 1 figure; v2 -- standard representation of the ring of ternions given, 1 figure and 3 references added; v3 -- published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

A Classification of the Projective Lines over Small Rings

A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of points on the line, the number of points represented by coordinates with at least one entry being a unit, the cardinality of the neighbourhood of a generic point of the line as well as those of the intersections between the neighbourhoods of two and three mutually distant points, the number of `Jacobson' points per a neighbourhood, the maximum number of pairwise distant points and, finally, a list of representative/base rings. The classification is presented in form of a table in order to see readily not only the fine traits of the hierarchy, but also the changes in the structure of the lines as one goes from one type to the other. We hope this study will serve as an impetus to a search for possible applications of these remarkable geometries in physics, chemistry, biology and other natural sciences as well.Comment: 7 pages, 1 figure; Version 2: classification extended up to order 20, references updated; Version 3: classification extended up to order 31, two more references added; Version 4: references updated, minor correctio

A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

The Fano-Snowflake, a specific $non$-unimodular projective lattice configuration associated with the smallest ring of ternions $R_{\diamondsuit}$ (arXiv:0803.4436 and 0806.3153), admits an interesting partitioning with respect to the Jacobson radical of $R_{\diamondsuit}$. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left $R_{\diamondsuit}$-module $R_{\diamondsuit}^{3}$ are shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding "ternion-induced" factorization of the lines of the Fano plane sitting in the middle of the Fano-Snowflake (6 -- 7 -- 3) is found to $differ fundamentally$ from the natural one, i. e., from that with respect to the Jacobson radical of the Galois field of two elements (3 -- 3 -- 1).Comment: 7 pages, 3 figure

'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon

Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13 pages] have given a number of distinct sets of three-qubit observables, each furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that two of these sets/configurations, namely the $18_{2} - 12_{3}$ and $2_{4}14_{2} - 4_{3}6_{4}$ ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types ${\cal V}_{22}(37; 0, 12, 15, 10)$ and ${\cal V}_{4}(49; 0, 0, 21, 28)$ in the classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773-797]. Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained

Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties

Let $S_{(N)} \equiv PG(1,\,2) \times PG(1,\,2) \times \cdots \times PG(1,\,2)$ be a Segre variety that is $N$-fold direct product of projective lines of size three. Given two geometric hyperplanes $H'$ and $H''$ of $S_{(N)}$, let us call the triple $\{H', H'', \overline{H' \Delta H''}\}$ the Veldkamp line of $S_{(N)}$. We shall demonstrate, for the sequence $2 \leq N \leq 4$, that the properties of geometric hyperplanes of $S_{(N)}$ are fully encoded in the properties of Veldkamp {\it lines} of $S_{(N-1)}$. Using this property, a complete classification of all types of geometric hyperplanes of $S_{(4)}$ is provided. Employing the fact that, for $2 \leq N \leq 4$, the (ordinary part of) Veldkamp space of $S_{(N)}$ is $PG(2^N-1,2)$, we shall further describe which types of geometric hyperplanes of $S_{(N)}$ lie on a certain hyperbolic quadric $\mathcal{Q}_0^+(2^N-1,2) \subset PG(2^N-1,2)$ that contains the $S_{(N)}$ and is invariant under its stabilizer group; in the $N=4$ case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type $LG(4,8)$, to the set of 2295 maximal subspaces of the symplectic polar space $\mathcal{W}(7,2)$.Comment: 16 pages, 8 figures and 7 table