Baer Subplanes in Finite Projective and Affine Planes

1972 ◽  
Vol 24 (1) ◽  
pp. 90-97 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Baer Subplane ◽  
Affine Planes ◽  
Baer Subplanes ◽  
Finite Planes

Let π be a projective or an affine plane ; a configuration C of π is a subset of points and a subset of lines in π such that a point P of C is incident with a line I of C if and only if P is incident with I in π. A configuration of a projective plane π which is a projective plane itself is called a projective subplane of π, and a configuration of an affine plane π’ which is an affine plane with the improper line of π‘ is an affine subplane of π‘.Let π be a finite projective (respectively, an affine) plane of order n and π0 a projective (respectively, an affine) subplane of π of order n0 different from π; then n0 ≦ . If n0 = , then π0 is called a Baer subplane of π. Thus, Baer subplanes are the “biggest” possible proper subplanes of finite planes.

scholarly journals Embedding Cycles in Finite Planes

10.37236/3377 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Felix Lazebnik ◽  
Keith E. Mellinger ◽  
Oscar Vega
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Projective Planes ◽  
Finite Projective Planes ◽  
Finite Affine ◽  
Finite Planes

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$,  a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$,  a $k$-cycle can be embedded in any projective plane of order $q$.

Characterizations of the Generalized Hughes Planes

1976 ◽  
Vol 28 (2) ◽  
pp. 376-402 ◽  
Author(s):  
Heinz Lüneburg
Keyword(s):  
Projective Plane ◽  
Baer Subplane ◽  
Desarguesian Planes ◽  
Finite Planes

Let be a projective plane and a subplane of . If l is a line of , we let denote the group of all elations in that have as axis and leave Q invariant. In [12, p. 921], Ostrom asked for a description of all finite planes that have a Baer subplane with the property that for all lines l of . Here denotes the order of G. Both the desarguesian planes of square order and the generalized Hughes planes have this property (Hughes [10], Ostrom [14], Dembowski [6]). One of the aims of this paper is to show that these are the only planes having such a Baer subplane.

Divisible Semiplanes, Arcs, and Relative Difference Sets

1987 ◽  
Vol 39 (4) ◽  
pp. 1001-1024 ◽  
Author(s):  
Dieter Jungnickel
Keyword(s):  
Projective Plane ◽  
Difference Sets ◽  
Difference Set ◽  
Relative Difference ◽  
List Type ◽  
Relative Difference Set ◽  
Large Collection ◽  
Baer Subplane ◽  
Relative Difference Sets ◽  
Definition Of

In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following:(i) S is the empty set.(ii) S consists of a line L with all its points and a point p with all the lines through it.(iii) S is a Baer subplane of Π.We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs).

On a conjecture of Hughes

1967 ◽  
Vol 63 (3) ◽  
pp. 647-652 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Permutation Group ◽  
Translation Plane ◽  
Collineation Group ◽  
Affine Plane ◽  
Finite Projective Plane ◽  
Transitive Permutation Group ◽  
Condition C

D. R. Hughes stated the following conjecture: If π is a finite projective plane satisfying the condition: (C)π contains a collineation group δ inducing a doubly transitive permutation group δ* on the points of a line g, fixed under δ, then the corresponding affine plane πg is a translation plane.

scholarly journals BRUCK NETS AND PARTIAL SHERK PLANES

2017 ◽  
Vol 104 (1) ◽  
pp. 1-12
Author(s):  
JOHN BAMBERG ◽  
JOANNA B. FAWCETT ◽  
JESSE LANSDOWN
Keyword(s):  
Affine Plane ◽  
Finite Geometries ◽  
Affine Planes ◽  
Incidence Structures ◽  
Finite Affine ◽  
Odd Order ◽  
Finite Incidence ◽  
Desarguesian Affine Plane

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

scholarly journals Blocking and Double Blocking Sets in Finite Planes

10.37236/5717 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jan De Beule ◽  
Tamás Héger ◽  
Tamás Szőnyi ◽  
Geertrui Van de Voorde
Keyword(s):  
Projective Planes ◽  
Blocking Set ◽  
Blocking Sets ◽  
Desarguesian Plane ◽  
Baer Subplanes ◽  
Desarguesian Projective Planes ◽  
Finite Planes ◽  
Value Sets ◽  
Minimal Blocking ◽  
Desarguesian Affine Plane

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.

Compactness in Topological Hjelmslev Planes

1984 ◽  
Vol 27 (4) ◽  
pp. 423-429 ◽  
Author(s):  
J. W. Lorimer
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Projective Planes ◽  
Coordinate Ring ◽  
Locally Compact ◽  
Compact Projective Plane ◽  
Negative Results ◽  
Connected Projective Plane ◽  
Topological Affine

AbstractIn the theory of ordinary topological affine and projective planes it is known that (1) An affine plane is never compact (2) a locally compact ordered projective plane is compact and archimedean (3) a locally compact connected projective plane is compact and (4) a locally compact projective plane over a coordinate ring with bi-associative multiplication is compact. In this paper we re-examine these results within the theory of topological Hjelmslev Planes and observe that while (1) remains valid (2), (3) and (4) are false. At first glance these negative results seem to suggest we are working in too general a setting. However a closer examination reveals that the absence of compactness in our setting is a natural and expected feature which in no way precludes the possibility of obtaining significant results.

scholarly journals On multiple blocking sets in Galois planes

2007 ◽  
Vol 7 (1) ◽  
pp. 39-53 ◽  
Author(s):  
A Blokhuis ◽  
L Lovász ◽  
L Storme ◽  
T Szőnyi
Keyword(s):  
Pairwise Disjoint ◽  
London Math ◽  
Algebraic Curves ◽  
Blocking Sets ◽  
Baer Subplane ◽  
Baer Subplanes ◽  
Galois Planes

AbstractThis article continues the study of multiple blocking sets in PG(2,q). In [A. Blokhuis, L. Storme, T. Szőnyi, Lacunary polynomials, multiple blocking sets and Baer subplanes.J. London Math. Soc. (2)60(1999), 321–332. MR1724814 (2000j:05025) Zbl 0940.51007], using lacunary polynomials, it was proven thatt-fold blocking sets of PG(2,q),qsquare,t<q¼/2, of size smaller thant(q+ 1) +cqq⅔, withcq= 2−⅓whenqis a power of 2 or 3 andcq= 1 otherwise, contain the union oftpairwise disjoint Baer subplanes whent≥ 2, or a line or a Baer subplane whent= 1. We now combine the method of lacunary polynomials with the use of algebraic curves to improve the known characterization results on multiple blocking sets and to prove at(modp) result on smallt-fold blocking sets of PG(2,q=pn),pprime,n≥ 1.

scholarly journals The modal logic of affine planes is not finitely axiomatisable

2008 ◽  
Vol 73 (3) ◽  
pp. 940-952
Author(s):  
Ian Hodkinson ◽  
Altaf Hussain
Keyword(s):  
Modal Logic ◽  
Affine Plane ◽  
Kripke Frame ◽  
Affine Planes ◽  
Modal Language ◽  
Line Point ◽  
Point Line

AbstractWe consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable.

Quasigraphs

1977 ◽  
Vol 29 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Norman D. Lane ◽  
Peter Scherk ◽  
Jean M. Turgeon
Keyword(s):  
Differential Geometry ◽  
Projective Plane ◽  
Affine Plane ◽  
Conic Sections

In the study of direct differential geometry, families of oriented arcs and curves have been employed extensively to define the differentiability of an arc at a point in various kinds of planes; cf. [2]. In [6], P. Scherk used lines in the projective plane; in [3] and [4], N. D. Lane and P. Scherk used circles in the conformai plane; conic-sections in the projective plane were employed in [5] and [7] by N. D. Lane and K. D. Singh; in [1], M. Gupta and N. D. Lane used the graphs of polynomials of degree at most n in the affine plane.

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