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1.
If E is a vector space over a field K, then any regular symmetric bilinear form on E induces a polarity
on the lattice of all subspaces of E. In the particular case where E is 3-dimensional, the set of all subspaces M of E such that both M and
are not N-subspaces (which, in most cases, is equivalent to saying that M is nonisotropic), ordered by inclusion and endowed with the restriction of the above polarity, is an orthomodular lattice T(E, ). We show that if K is a proper subfield of K, with K F2, and E a 3-dimensional K -subspace of E such that the restriction of to E × E is, up to multiplicative constant, a bilinear form on the K -space E , then T(E , ) is isomorphic to an irreducible 3-homogeneous proper subalgebra of T(E, ). Our main result is a structure theorem stating that, when K is not of characteristic 3, the converse is true, i.e., any irreducible 3-homogeneous proper subalgebra of T(E, ) is of this form. As a corollary, we construct infinitely many finite orthomodular lattices which are minimal in the sense that all their proper subalgebras are modular. In fact, this last result was our initial aim in this paper.Received June 4, 2003; accepted in final form May 18, 2004. 相似文献
2.
3.
Franz Kalhoff 《Journal of Geometry》1987,28(1):7-40
The question, whether the Archimedean ordering of only one of the ternary rings of a projective plane implies that is Archimedean, i.e. that every ternary ring of is Archimedean, is answered in the negative by the construction of local-Archimedean orderings of free planes. There exists even Archimedean affine planes with non-Archimedean associated projective planes. 相似文献
4.
Harald Löwe 《Geometriae Dedicata》1995,58(1):45-51
We construct symmetric planes associated with an arbitrary locally compact connected nearfield
. If
is a proper nearfield, i.e.
{;;}, then the tangent translation plane of this symmetric plane is not classical. All previously known examples of symmetric planes have classical tangent translation planes.Herrn Professor Dr. H. Salzmann zum 65. Geburtstag gewidmet 相似文献
5.
Günter F. Steinke 《Geometriae Dedicata》1993,48(3):337-348
This paper concerns 4-dimensional (topological locally compact connected) Minkowski planes that admit a 7-dimensional automorphism group . It is shown that such a plane is either classical or has a distinguished point that is fixed by the connected component of and that the derived affine plane at this point is a 4-dimensional translation plane with a 7-dimensional collineation group.This research was supported by a Feodor Lynen Fellowship. 相似文献
6.
Klaus Metsch 《Geometriae Dedicata》1995,54(2):171-197
A partial projective plane of ordern consists of lines andn
2 +n + 1 points such that every line hasn+1 points and distinct lines meet in a unique point. Suppose that two essentially different partial projective planes and of ordern, n a perfect square, that are defined on the same set of points cover the same pairs of points. For sufficiently largen we show that this implies that and have at leastn(n+1) lines. This bound is sharp and there exist essentially two different types of examples meeting the bound.As an application, we can show that derived planes provide an example for a pair of projective planes of square order with as much structure as possible in common, that is, as many lines as possible in common. Furthermore, we present a new method (twisted derivations) to obtain planes from one another by replacing the same number of lines as in a derivation. 相似文献
7.
Hermann Hähl 《Monatshefte für Mathematik》1984,97(1):23-45
This paper is one of the final steps in a classification program to determine all eight-dimensional, locally compact translation planes having large collineation groups. Here, we describe all such planes whose collineation group contains a semidirect product ·N, whereN is an at least 3-dimensional normal subgroup consisting of shears with fixed axis, and is isomorphic to SO3 (). 相似文献
8.
Joachim Otte 《Geometriae Dedicata》1995,58(2):203-212
A projective plane is called smooth if both the point space and the line space are smooth manifolds such that the geometric operations are smooth. We prove that every smooth projective translation plane is isomorphic to one of the classical planes over , , or
.Dedicated to Professor Dr. H. Salzmann on the occasion of his 65th birthday 相似文献
9.
Barbara Priwitzer 《Geometriae Dedicata》1995,58(3):245-258
LetP=(P, L) be a compact projective plane with 0P< and let be a compact connected subgroup of Aut(P). If dim dimE – dimP, whereE is the elliptic motion group of the corresponding classical plane, then E or is isomorphic to a point stabilizerE
0 inE, cf. [31]. Here we consider the case E
0. It is shown that the action of on the point spaceP is equivalent to the classical action ofE
0. For dimP {8, 16} the planeP is uniquely determined by a 2-dimensional subplane with SO2 Aut().Für H. Reiner Salzmann zum 65. Geburtstag 相似文献
10.
11.
Cécile Huybrechts 《Geometriae Dedicata》1996,59(2):163-179
Let be a [L.Af*]-geometry, that is a rank 3 geometry with linear spaces as plane residues, with dual affine planes as point residues and with generalized digons as line residues. Assume that (LL) holds in . In the particular case where the plane residues are finite circles, the structure of such geometries has been strongly restricted by A. P. Sprague. Moreover, C. Lefèvre and L. Van Nypelseer have given a complete classification of such geometries under the assumption that the plane residues are affine planes. We generalize these two results for [L.Af*]-geometries.Aspirant du Fonds National Belge de la Recherche Scientifique. 相似文献
12.
Hauke Klein 《Geometriae Dedicata》1995,58(1):53-62
We consider 4-dimensional compact projective planes with a solvable 6-dimensional collineation group and with orbit type (2, 1), i.e. fixes a flagv W, acts transitively onL
\{W} and fixes no point in the setW\{v}. We We prove a series of lemmas concerning the action of invariant subgroups of . These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of has dimension at least 4.Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday 相似文献
13.
Hermann Hähl 《Geometriae Dedicata》1995,58(2):213-226
As a contribution to a classification of all sixteen-dimensional translation planes whose collineation group
has dimension at least 38, this paper deals with the case that
contains a subgroup (locally) isomorphic to SU2Spin5. Under various further assumptions, it is shown that such a plane satisfying dim
38 is necessarily isomorphic to the classical plane over the octonions.The complete classification will reveal that these further assumptions may in fact be omitted, except for the case that
even contains a subgroup isomorphic to Spin7. The latter planes have been explicitly determined in previous papers.
Meinem verehrten Lehrer Helmut Salzmann zum 65. Geburtstag 相似文献
14.
Let be a projective space. By H() we denote the graph whose vertices are the non-incident point-hyperplane pairs of , two vertices (p,H) and (q,I) being adjacent if and only if p I and q H. In this paper we give a characterization of the graph H() (as well as of some related graphs) by its local structure. We apply this result by two characterizations of groups G with PSL
n
(
)GPGL
n
(
), by properties of centralizers of some (generalized) reflections. Here
is the (skew) field of coordinates of . 相似文献
15.
Let X be a Banach space with a countable unconditional basis (e.g., X=2), X open. We show that is pseudoconvex if and only if for each affine complex line L in X the sheaf cohomology group H
1
(,I) vanishes, where I is the ideal sheaf of all holomorphic functions on that vanish on L. We also give an example that the condition H
q
(,)=0 for all q1 unlike in finite dimensions does not imply the pseudoconvexity of . Lastly, we prove an interpolation result.
Mathematics Subject Classification (2002): 32T05, 46G20. 相似文献
16.
It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(x + (1 – )y) g(x) + (1 – )g(y) andg(x + (1 – )y) f(x) + (1 – )f(y) for allx, y I and [0, 1], iff there exists an affine functionh: I such thatf h g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions. 相似文献
17.
V. A. Andrienko 《Analysis Mathematica》1996,22(4):243-266
( ) . .
Dedicated to Professor K. Tandori on his seventieth birthday
This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation. 相似文献
Dedicated to Professor K. Tandori on his seventieth birthday
This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation. 相似文献
18.
Barbara Priwitzer 《Geometriae Dedicata》1994,52(1):33-40
We prove the following theorem: LetP be an 8-dimensional compact topological projective plane. If the connected component of its automorphism group has dimension at least 12, then is a Lie group. 相似文献
19.
Richard Bödi 《Geometriae Dedicata》1994,53(2):201-216
LetT be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed fieldF
of is connected, then is either isomorphic to one of the compact Lie groupsG
2 or SU3, or the (covering) dimension of is at most 7. 相似文献
20.
Rolf Riesinger 《Journal of Geometry》1997,60(1-2):127-145
Throughout this paper, the underlying projective space is 3-dimensional and Pappian. A spreadL admits aregulization , if is a collection of reguli contained inL and if each element ofL, except at most two lines, is contained either in exactly one regulus of or in all reguli of . Replacement of each regulus of by its complementary regulus (exceptional lines remain unchanged) produces thecomplementry congruence
L
c
of
L
with respect to . IfL
c
is an elliptic linear congruence of lines, then we call anelliptic regulization. Applying a method due to Thas and Walker we construct topological spreads of PG(3,) which admit one elliptic and no further regulization. For each of these spreads we determine the group of automorphic collineations. Among others we obtain also spreads which are the complete intersection of a general linear complex of lines and of a cubic complex of lines.In conclusion, I would like to thank H. H{upavlicek} (Vienna) for valuable suggestions in the preparation of this article. 相似文献