A groupoid of the ternary ring of a projective plane

1991 ◽  
Vol 42 (1-2) ◽  
pp. 3-16 ◽  
Author(s):  
M. W. Al-Dhahir ◽  
W. Benz ◽  
K. Ghalieh
Keyword(s):  
Projective Plane ◽  
Ternary Ring

Projective planes of infinite but isolic order

1976 ◽  
Vol 41 (2) ◽  
pp. 391-404 ◽  
Author(s):  
J. C. E. Dekker
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Latin Squares ◽  
Combinatorial Theory ◽  
Ternary Ring ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Planar Ternary Ring ◽  
Partial Recursive Functions

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

Weakly Isotopic Planar Ternary Rings

1975 ◽  
Vol 27 (1) ◽  
pp. 32-36
Author(s):  
Frederick W. Stevenson
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Ternary Ring ◽  
Algebraic Relation ◽  
Planar Ternary Ring ◽  
Algebraic Properties

This paper introduces two relations both weaker than isotopism which may hold between planar ternary rings. We will concentrate on the geometric consequences rather than the algebraic properties of these relations. It is well-known that every projective plane can be coordinatized by a planar ternary ring and every planar ternary ring coordinatizes a projective plane. If two planar ternary rings are isomorphic then their associated projective planes are isomorphic; however, the converse is not true. In fact, an algebraic bond which necessarily holds between the coordinatizing planar ternary rings of isomorphic projective planes has not been found. Such a bond must, of course, be weaker than isomorphism; furthermore, it must be weaker than isotopism. Here we show that it is even weaker than the two new relations introduced.This is significant because the weaker of our relations is, in a sense, the weakest possible algebraic relation which can hold between planar ternary rings which coordinatize isomorphic projective planes.

Groupoids associated with the ternary ring of a projective plane

1998 ◽  
Vol 61 (1-2) ◽  
pp. 17-31 ◽  
Author(s):  
Walter Benz ◽  
Khuloud Ghalieh
Keyword(s):  
Projective Plane ◽  
Ternary Ring

The dual of addition in the ternary ring of a projective plane

1974 ◽  
Vol 25 (1) ◽  
pp. 536-539 ◽  
Author(s):  
M. W. Al-Dhahir ◽  
M. S. Abdul-Elah
Keyword(s):  
Projective Plane ◽  
Ternary Ring

scholarly journals A note on planes of characteristic three

1972 ◽  
Vol 7 (1) ◽  
pp. 105-111
Author(s):  
Michael J. Kallaher
Keyword(s):  
Projective Plane ◽  
Near Field ◽  
Ternary Ring ◽  
Finite Plane

A projective plane has characteristic three if in every ternary ring coordinatizing it all elements ≠ 0 of the additive loop have order 3. We show that if a finite plane of characteristic three is coordinatized by a near-field, then the plane is desarguesian.

Reidemeister Projective Planes

1968 ◽  
Vol 20 ◽  
pp. 1459-1464
Author(s):  
Michael J. Kallaher
Keyword(s):  
Projective Plane ◽  
Prime Power ◽  
Projective Planes ◽  
Ternary Ring ◽  
The Third

By a Reidemeister plane we mean a projective plane having the property that every ternary ring coordinatizing it has associative addition. Finite Reidemeister planes have been investigated by Gleason (2), Liineburg (6), and Kegel and Luneburg (4). In the first paper, Gleason proved that if the order of the plane is a prime power, then it is Desarguesian. Luneburg showed that this is still true if the order is not 60. In the third paper, this last restriction is removed. For infinite planes, the only result is the following theorem due to Pickert (7, p. 301).

Gilles Deleuze's Luminous Philosophy

2020 ◽  
Author(s):  
Hanjo Berressem
Keyword(s):  
Projective Plane ◽  
Visual Arts ◽  
Whole Body ◽  
Literary Studies ◽  
Chronological Order ◽  
Real Projective Plane ◽  
Cinema Studies ◽  
The Real

Providing a comprehensive reading of Deleuzian philosophy, Gilles Deleuze’s Luminous Philosophy argues that this philosophy’s most consistent conceptual spine and figure of thought is its inherent luminism. When Deleuze notes in Cinema 1 that ‘the plane of immanence is entirely made up of light’, he ties this philosophical luminism directly to the notion of the complementarity of the photon in its aspects of both particle and wave. Engaging, in chronological order, the whole body and range of Deleuze’s and Deleuze and Guattari’s writing, the book traces the ‘line of light’ that runs through Deleuze’s work, and it considers the implications of Deleuze’s luminism for the fields of literary studies, historical studies, the visual arts and cinema studies. It contours Deleuze’s luminism both against recent studies that promote a ‘dark Deleuze’ and against the prevalent view that Deleuzian philosophy is a philosophy of difference. Instead, it argues, it is a philosophy of the complementarity of difference and diversity, considered as two reciprocally determining fields that are, in Deleuze’s view, formally distinct but ontologically one. The book, which is the companion volume toFélix Guattari’s Schizoanalytic Ecology, argues that the ‘real projective plane’ is the ‘surface of thought’ of Deleuze’s philosophical luminism.

scholarly journals Group-labeled light dual multinets in the projective plane

2018 ◽  
Vol 341 (8) ◽  
pp. 2121-2130 ◽  
Author(s):  
Gábor Korchmáros ◽  
Gábor P. Nagy
Keyword(s):  
Projective Plane
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