scholarly journals Semiaffine Spaces

10.37236/107 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Hendrik Van Maldeghem
Keyword(s):  
Projective Plane ◽  
Desarguesian Projective Plane ◽  
Planar Space ◽  
Infinite Case

In this paper we improve on a result of Beutelspacher, De Vito & Lo Re, who characterized in 1995 finite semiaffine spaces by means of transversals and a condition on weak parallelism. Basically, we show that one can delete that condition completely. Moreover, we extend the result to the infinite case, showing that every plane of a planar space with at least two planes and such that all planes are semiaffine, comes from a (Desarguesian) projective plane by deleting either a line and all of its points, a line and all but one of its points, a point, or nothing.

scholarly journals Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

10.37236/2582 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tamás Héger ◽  
Marcella Takáts
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
The Other ◽  
Blocking Set ◽  
Metric Dimension ◽  
Desarguesian Projective Plane ◽  
Resolving Set ◽  
Incidence Graph ◽  
Vertex Set

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane

2013 ◽  
Vol 104 (1) ◽  
pp. 11-43 ◽  
Author(s):  
Daniele Bartoli ◽  
Alexander A. Davydov ◽  
Giorgio Faina ◽  
Stefano Marcugini ◽  
Fernanda Pambianco
Keyword(s):  
Projective Plane ◽  
Upper Bounds ◽  
Desarguesian Projective Plane ◽  
Complete Arc ◽  
Finite Desarguesian Projective Plane

scholarly journals Finite groups as automorphism groups of orthocomplemented projective planes

1978 ◽  
Vol 25 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Richard J. Greechie
Keyword(s):  
Finite Group ◽  
Projective Plane ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Absolute Point

AbstractA construction is given for a non-desarguesian projective plane P and an absolute-point free polarity on P such that the group of collineations of P which commute with the polarity is isomorphic to an arbitrary preassigned finite group.

A Class of Non-Desarguesian Projective Planes

1957 ◽  
Vol 9 ◽  
pp. 378-388 ◽  
Author(s):  
D. R. Hughes
Keyword(s):  
Positive Integer ◽  
Projective Plane ◽  
Collineation Group ◽  
Difference Sets ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Desarguesian Plane ◽  
Desarguesian Projective Planes

In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian projective plane of order 9; we shall show that this plane is self-dual and can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets. Furthermore, the technique used in (7) will be generalized and we will construct a new non-Desarguesian plane of order p2n for every positive integer n and every odd prime p.

Extension of collineations defined on certain sets of a desarguesian projective plane

1971 ◽  
Vol 6 (1) ◽  
pp. 59-65 ◽  
Author(s):  
B. Orbán
Keyword(s):  
Projective Plane ◽  
Desarguesian Projective Plane

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

2015 ◽  
Vol 107 (1) ◽  
pp. 89-117 ◽  
Author(s):  
Daniele Bartoli ◽  
Alexander A. Davydov ◽  
Giorgio Faina ◽  
Alexey A. Kreshchuk ◽  
Stefano Marcugini ◽  
...  
Keyword(s):  
Projective Plane ◽  
Upper Bounds ◽  
Computer Search ◽  
Desarguesian Projective Plane ◽  
Complete Arc ◽  
Finite Desarguesian Projective Plane

Unitals in the Desarguesian projective plane of order 16

2014 ◽  
Vol 144 ◽  
pp. 110-122 ◽  
Author(s):  
John Bamberg ◽  
Anton Betten ◽  
Cheryl E. Praeger ◽  
Alfred Wassermann
Keyword(s):  
Projective Plane ◽  
Desarguesian Projective Plane

On Baer Involutions of Finite Projective Planes

1970 ◽  
Vol 22 (4) ◽  
pp. 878-880 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Finite Projective Planes ◽  
Special Cases ◽  
Collineation Groups ◽  
Baer Involution ◽  
Finite Desarguesian Projective Plane

1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.

Extension of collineations defined on certain sets of a Desarguesian projective plane

1970 ◽  
Vol 5 (1) ◽  
pp. 127-127
Author(s):  
B. Orbán
Keyword(s):  
Projective Plane ◽  
Desarguesian Projective Plane
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