On projective Hjelmslev planes of level n

In this paper, we establish a new (but equivalent) definition of projective Hjelmslev planes of level n. This shows that the nth floor of a triangle building is a projective Hjelmslev plane of level n (a result already announced in [9], but left unproved). This will allow us to characterize Artmann-sequences by means of their inverse limits and to construct new ones. We also deduce a new existence theorem for level n projective Hjelmslev planes. All results hold in the finite as well as in the infinite case.

Dual Numbers and Topological Hjelmslev Planes

In 1929 J. Hjelmslev introduced a geometry over the dual numbers ℝ+tℝ with t2 = Q. The dual numbers form a Hjelmslev ring, that is a local ring whose (unique) maximal ideal is equal to the set of 2 sided zero divisors and whose ideals are totally ordered by inclusion. This paper first shows that if we endow the dual numbers with the product topology of ℝ2, then we obtain the only locally compact connected hausdorfT topological Hjelmslev ring of topological dimension two. From this fact we establish that Hjelmslev's original geometry, suitably topologized, is the only locally compact connected hausdorfr topological desarguesian projective Hjelmslev plane to topological dimension four.

Locally Compact Hjelmslev Planes and Rings

Affine and projective Hjelmslev planes are generalizations of ordinary affine and projective planes where two points (lines) may be joined by (may intersect in) more than one line (point). The elements involved in multiple joinings or intersections are neighbours, and the neighbour relations on points respectively lines are equivalence relations whose quotient spaces define an ordinary affine or projective plane called the canonical image of the Hjelmslev plane. An affine or projective Hjelmslev plane is a topological plane (briefly a TH-plane and specifically a TAH-plane respectively a TPH-plane) if its point and line sets are topological spaces so that the joining of non-neighbouring points, the intersection of non-neighbouring lines and (in the affine case) parallelism are continuous maps, and the neighbour relations are closed.In this paper we continue our investigation of such planes initiated by the author in [38] and [39].

Embedding Right Chain Rings in Chain Rings

The following problem was the starting point for this investigation: Can every desarguesian affine Hjelmslev plane be embedded into a desarguesian projective Hjelmslev plane [8]? An affine Hjelmslev plane is called desarguesian if it can be coordinatized by a right chain ring R with a maximal ideal J(R) consisting of two-sided zero divisors. A projective Hjemslev plane is called desarguesian if the coordinate ring is in addition a left chain ring, i.e. a chain ring. This leads to the algebraic version of the above problem, namely the embedding of right chain rings into suitable chain rings. We can prove the following result.

Ordered Desarguesian Affine Hjelmslev Planes

A Desarguesian affine Hjelmslev plane (D.A.H. plane) may be coordinatized by an affine Hjelmslev ring (A.H. ring), which is a local ring whose radical is equal to the set of two-sided zero divisors and whose principal right ideals are totally ordered (cf. [3]). In his paper on ordered geometries [4], P. Scherk discussed the equivalence of an ordering of a Desarguesian affine plane with an ordering of its coordinatizing division ring. We shall define an ordered D.A.H. plane and follow Scherk's methods to extend his results to D.A.H. planes and their A.H. rings i.e., we shall show that a D.A.H. plane is ordered if and only if its A.H. ring is ordered. We shall also give an example of an ordered A.H. ring. Finally, we shall discuss some infinitesimal aspects of the radical of an ordered A.H. ring.

Hjelmslev Planes Derived from Modular Lattices

In several papers, W. Klingenberg has elaborated the connections between Hjelmslev planes and a class of rings, called H-rings (4; 5; 6), which are rings of coordinates for the corresponding Hjelmslev planes. Certain homomorphic images of valuation rings are examples of H-rings. In these examples, the lattice of (right) ideals of the ring, say R,is a chain, and the coordinatization of the corresponding Hjelmslev plane yields a natural embedding of the plane in the lattice L(R3) of (right) submodules of the module R3. Now, L(R3) is a modular lattice with a homogeneous basis of order 3 given by the submodules a1 = (1, 0, 0)R, a2 = (0, 1, 0)R, a2 = (0, 0, 1)R, and the sublattices L(N, ai) of elements less than or equal to ai are chains. Forgetting about the ring, we find ourselves in the situation of a problem suggested by Skornyakov (7, Problem 23, p. 166), namely, to study modular lattices with a homogeneous basis of chains. Baer (2) and Inaba (3) investigated lattices of this kind with Desarguesian properties and assuming that the chains L(N, ai) were finite. Representations of the lattices by means of certain rings can be found in both articles.