On the rank of incidence matrices in projective Hjelmslev spaces

Let be a finite chain ring with , , and let . Let be an integer sequence satisfying . We consider the incidence matrix of all shape versus all shape subspaces of with . We prove that the rank of over is equal to the number of shape subspaces. This is a partial analog of Kantor's result about the rank of the incidence matrix of all dimensional versus all dimensional subspaces of . We construct an example for shapes and for which the rank of is not maximal

The Projective Line Over the Finite Quotient Ring GF(2)[xx]/<x3−x>< x^{3} - x> and Quantum Entanglement I. Theoretical Background

The paper deals with the projective line over the finite factor ring $R\_{\clubsuit} \equiv$ GF(2)[$x$]/$$. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As $R\_{\clubsuit}$ is not a local ring, the neighbour (or parallel) relation is not an equivalence relation so that the sets of neighbour points to two distant points overlap. There are nine neighbour points to any point of the line, forming three disjoint families under the reduction modulo either of two maximal ideals of the ring. Two of the families contain four points each and they swap their roles when switching from one ideal to the other; the points of the one family merge with (the image of) the point in question, while the points of the other family go in pairs into the remaining two points of the associated ordinary projective line of order two. The single point of the remaining family is sent to the reference point under both the mappings and its existence stems from a non-trivial character of the Jacobson radical, ${\cal J}\_{\clubsuit}$, of the ring. The factor ring $\widetilde{R}\_{\clubsuit} \equiv R\_{\clubsuit}/ {\cal J}\_{\clubsuit}$ is isomorphic to GF(2) $\otimes$ GF(2). The projective line over $\widetilde{R}\_{\clubsuit}$ features nine points, each of them being surrounded by four neighbour and the same number of distant points, and any two distant points share two neighbours. These remarkable ring geometries are surmised to be of relevance for modelling entangled qubit states, to be discussed in detail in Part II of the paper.Comment: 8 pages, 2 figure

A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2

Using Singer polygons, we construct locally finite affine buildings of types ~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This construction produces very explicit descriptions of these buildings as well as very short presentations of the lattices. All but one of the ~C_2-buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices in buildings of type ~C_2. Integral and rational group homology for the lattices is also calculated.Comment: 42 pages, small corrections and cleanup. Results are unchanged