On stable geometries

Within the concept of projective lattice geometry we are considering the class of stable geometries which have also been introduced in [14]. The investigation of their basic properties will result in fundamental structure theorems which especially give a lattice-geometric characterization of free left modules of rank 6 over proper right Bezout rings of stable rank 2. This yields a proper generalization of previous results of ours.     

Global-Affine Morphisms of Projective Lattice Geometries

The fundamental theorem of projective geometry gives an algebraic representation of isomorphisms between projective geometries of dimension at least 3 over vector spaces and has been generalized in different ways. This note briefly presents some further generalizations which will be proved in the author’s thesis. We introduce the notion of global-affine morphisms between projective lattice geometries. Our investigations result in a general partial representation of global-affine morphisms which yields a complete representation of global-affine homomorphisms between large classes of module-induced projective geometries by semilinear mappings between the underlying modules.     

On Meet-Complements in Cohn Geometries

Within the frame of projective lattice geometry, the present paper investigates classes of meet-complements in Cohn geometries and especially in Ore and Bezout geometries. The algebraic background of these geometries is given by torsion free modules over domains — in particular Ore and Bezout domains. ¹     

Morphisms of projective geometries and semilinear maps

Morphisms between projective geometries are introduced; they are partially defined maps satisfying natural geometric conditions. It is shown that in the arguesian case the morphisms are exactly those maps which in terms of homogeneous coordinates are described by semilinear maps. If one restricts the considerations to automorphisms (collineations) one recovers the so-called fundamental theorem of projective geometry, cf. Theorem 2.26 in [2].Supported by a grant from the Fonds National Suisse de la Recherche Scientifique.     

A structural characterization of non-Arguesian lattices

This is the first of a planned series of papers on the structure of non-Arguesian modular lattices. Apart from the (subspace lattices of) non-Arguesian projective planes, the best known examples of such lattices are obtained via the Hall-Dilworth construction by badly gluing together two projective planes of the same order. Our principal result shows that every non-Arguesian modular lattice L retains some of the flavor of these examples: There exist in the ideal lattice of L 20 intervals, not necessarily distinct, that form non-degenerate projective plains, and 10 points and 10 lines in these planes that constitute in a natural sense a classical non-Arguesian configuration.Research supported by NSERC Operating Grant A8190.Research supported by NSF Grant DMS-8300107.     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

Projective geometry over rings with stable range condition

A generalization of the fundamental theorem of projective geometry is established for non-injective mappings between projective geometries defined over rings which satisfy a stable range condition. A geometrical characterization of the stable range condition is also given.     

Induced residuated maps

The purpose of this paper is to introduce a class of mappings from a lattice L, whose elements are residuated maps, into itself. The main results of this paper identify certain injective residuated mappings of L and order automorphisms of a sublattice of L with mappings from this class.     

On point-irreducible projective lattice geometries

Within the conceptual frame of projective lattice geometry (as introduced in [5]) we are considering the class of all point-irreducible geometries. In the algebraic context these geometries are closely connected with unitary modules over local rings. Besides several synthetic investigations we obtain a lattice-geometric characterization of free left modules over right chain rings which allows a purely lattice-theoretic version in the Artinian case.This paper results from a joint work of the authors at the Hungarian Academy of Sciences (Budapest) in the fall of 1991, supported by the DAAD.     

Characterizing Combinatorial Geometries by Numerical Invariants

We show that the projective geometry PG(r − 1,q ) for r & 3 is the only rank- r(combinatorial) geometry with (q^r − 1) / (q − 1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice.     

Combinatorial representation and convex dimension of convex geometries

We develop a representation theory for convex geometries and meet distributive lattices in the spirit of Birkhoff's theorem characterizing distributive lattices. The results imply that every convex geometry on a set X has a canonical representation as a poset labelled by elements of X. These results are related to recent work of Korte and Lovász on antimatroids. We also compute the convex dimension of a convex geometry.Supported in part by NSF grant no. DMS-8501948.     

The design of linear algebra and geometry

Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.     

Join-semidistributive lattices and convex geometries

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice S_P(A) of algebraic subsets of a suitable algebraic lattice A. This latter construction, S_P(A), is a key example of a convex geometry that plays an analogous role in hierarchy of join-semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries.     

On Barbilian spaces in projective lattice geometries

We introduce the notion of a Barbilian space of a projective lattice geometry in order to investigate the relationship between lattice-geometric properties and the properties of point-hyperplane structures associated with. We obtain a characterization of those projective lattice geometries, the Barbilian space of which is a Veldkamp space.     

A unified approach to projective lattice geometries

The interest in pursuing projective geometry on modules has led to several lattice theoretic generalizations of the classical synthetic concept of projective geometry on vector spaces.Introduced in this paper is an approach that is capable of unifying various attempts within a new conceptual frame. This approach reflects algebraic properties from a lattice-geometric point of view. Together with new results we are presenting results from previous publications which have been improved in the frame of this work.     

Lattice theory and metric geometry

K. Menger and G. Birkhoff recognized 70 years ago that lattice theory provides a framework for the development of incidence geometry (affine and projective geometry). We show in this article that lattice theory also provides a framework for the development of metric geometry (including the euclidean and classical non-euclidean geometries which were first discovered by A. Cayley and F. Klein). To this end we introduce and study the concept of a Cayley–Klein lattice. A detailed investigation of the groups of automorphisms and an algebraic characterization of Cayley–Klein lattices are included. The authors would like to thank an unknown referee for his helpful suggestions.     

Dimension versus size

We investigate the behavior of f(d), the least size of a lattice of order dimension d. In particular we show that the lattice of a projective plane of order n has dimension at least n/ln(n), so that f(d)=O(d) ² log² d. We conjecture f(d)=(d ² ), and prove something close to this for height-3 lattices, but in general we do not even know whether f(d)/d.Supported in part by NSF grant MCS 83-01867, AFORS grant number 0271 and a Sloan Research Fellowship.     

Automorphisms of Mn,partially ordered by the star order

Let Mn be the space of all n × n matrices with coefficients in [image omitted] or [image omitted], where n ≥ 3. The star order on Mn is defined by [image omitted] iff [image omitted], where A* is the Hermitian adjoint (i.e., the conjugate transpose) of A. We characterize surjective mappings Φ on Mn such that [image omitted] iff [image omitted]. The tools we use are the Fundamental theorem of projective geometry, Wigner's theorem, and the Penrose decomposition, which we need to describe the main result as well.     

Uniform binary geometries

In a geometric lattice every interval can be mapped isomorphically into an upper interval (containing 1) by a strong map. A natural question thus arises as to what extent certain assumptions on the upper interval structure determine the whole lattice. We consider conditions of the following sort: that above a certain levelm any two upper intervals of the same length be isomorphic. This property, called uniformity, is studied for binary geometries. The geometries satisfying the strongest uniformity condition (m = 1) are determined (except for one open case). As is to be expected the corresponding problem for lower intervals is easier and is solved completely.     

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.