We discuss representations of the projective line over a ring R with 1 in a projective space over some (not necessarily commutative) field K. Such a representation is based upon a (K, R)-bimodule U . The points of the projective line over R are represented by certain subspaces of the projective space P(K, U × U ) that are isomorphic to one of their complements. In particular, distant points go over to complementary subspaces, but in certain cases, also non-distant points may have complementary images. Mathematics Subject Classification (1991): 51C05, 51A45, 51B05.

. This additional degree of freedom makes it possible to view any perp-set as a set-theoretic union of the corresponding points of the associated projective line.

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group G, we first construct vector spaces over GF(p), p a prime, by factorising G over appropriate normal subgroups. Then, by expressing GF(p) in terms of the commutator subgroup of G, we construct alternating bilinear forms, which reflect whether or not two elements of G commute. Restricting to p = 2, we search for ''refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of G is ≤ 2. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ''condensation'' of several distinct elements of G. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism

As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single d-dimensional qudit is made, with d being any positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring Z d . Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of Z 2 d . We find, remarkably, that a perp-set is not a set-theoretic union of the corresponding points of the associated projective line unless d is a product of distinct primes. The operators are also seen to be structured into disjoint 'layers' according to the degree of their representing vectors. A brief comparison with some multiple-qudit cases is made.

The set G of all m-dimensional subspaces of a 2m-dimensional vector space V is endowed with two relations, complementarity and adjacency. We consider bijections from G onto G ′ , where G ′ arises from a 2m ′ -dimensional vector space V ′ . If such a bijection ϕ and its inverse leave one of the relations from above invariant, then also the other. In case m ≥ 2 this yields that ϕ is induced by a semilinear bijection from V or from the dual space of V onto V ′ .As far as possible, we include also the infinite-dimensional case into our considerations.2000 Mathematics Subject Classification: 51A10, 51A45, 05C60.