User:William Ellison/sandbox 3

PROJECTIVE INCIDENCE STRUCTURES

Introduction

The object of this article is to describe the axiomatic foundations of the theory of projective incidence structures.

The main, and historically first, application is in the foundations of projective and affine geometry. However, the theory of such structures has applications in communication theory and in cryptography.

The aim is to develope the classification of projective incidence structures up to the point where one is lead to the study of specific algebraic structures such as : GL_n (R), where specific R is a division ring or specific ternary rings.

The first classifying facto is the dimension of the structure.

• Structures of dimension greater than 2
• Structures of dimension 2

For structures of dimension 2 the classifications is by the Desargues axiom

• Desarguesian planes
• Non-desarguesian planes

A projective incidence structure can be associated with an algebraic coordinate structure and the "isotopic" equivalence classes of these algebraic structures classify all the projective incidence structures.

Geometry :

• Affine geometry
• Affine space
• Space (mathematics)
• Affine geometry
• Affine group
• Affine transformation
• Affine variety
• Affine hull
• Affine transformation
• Heap (mathematics)
• Equipollence (geometry)
• Interval measurement, an affine observable in statistics
• Exotic affine space
• Complex affine space
• Complex projective space
• Real projective space
• Projection (mathematics)
• Projective geometry
• Projective line
• Projective plane
• Projective space
• Incidence
• Cross-ratio
• Möbius transformation
• Projective transformation
• Homogeneous coordinates
• Duality (projective geometry)
• Fundamental theorem of projective geometry
• Projective configuration
• Complete quadrangle
• Desargues' theorem
• Pasch's axiom
• Pappus's hexagon theorem
• Pascal's theorem
• Projective line over a ring
• Joseph Wedderburn
• Grassmann–Cayley algebra
• Planar ternary ring

Combinatorics

• Cryptography
• Error correction code
• Kirkman's schoolgirl problem
• Steiner system
• Block design
• Modular lattice
• De Bruijn–Erdős theorem (incidence geometry)
• Sylvester–Gallai theorem
• Sylvester–Gallai configuration
• Happy ending problem
• Difference set

What we will describe

• If N > 2, then any projective incidence structure of dimension N is isomorphic to GLN(R), where R is a division ring. It is isomorphic to GLN(F), where F is a field (finite or infinite) if and only if the Pappus theorem is true.
• If N = 2, then a projective incidence structure is isomorphic to GL2(R), R a division ring, if and only Desargues' theorem holds.
• If N = 2 and Desargues' theorem does not hold, then there is a rich family of Non-Desarguesian planes.

Vocabulary

The unique line k containg the distinct points A and B is called the line joining A and B. It will be noted as AB.

The unique point P contained in two distinct lines l and k will be called the intersection of l and k.

The axioms

A projective incidence structure is a set of objects called "points" , denoted by ${Points}$ and a set of distinguished sub-sets of ${Points}$ called "lines", denoted by ${Lines}$ which satisfy a certain number of simple axioms. We denote points by upper-case latin letters, lines by lower-case latin letters, and later, planes (to be defined later) by upper-case greek letters.

The pair ${\mathcal {P}}=(Points,Lines)$ satisfying the following axioms is called a projective incidence structure.

A1. If $A,B\in {Points},A\neq B$, then there is at least one $\ell \in {Lines}$ such that $A\in \ell ,B\in \ell$.

A2. If $A,B\in {Points},A\neq B$, then there is at most one $\ell \in {Lines}$ such that $A\in \ell ,B\in \ell$.

A3. There are at least three points on any line.

A4. There is at least one point $A$ and at least one line $l$ such that $A\notin l$

A5. If $A,B,C\in {Points}$ are not all on the same line and $D,E(D\neq E)$ are such that $D$ is on the line $AB$ and $E$ on the line $AC$, then the lines $BC$ and $DE$ have a point in common.

Axiom 5 can be thought of as saying : If ABC is a triangle and a line intersects two sides of the triangle, then it intersects the third side.'

In the mathematical litterature there are many variations for the set of axioms to be used to define a projective incidence structure. However, they are all equivalent and given two sets of axioms it is a simple exercice to deduce one set from the other.

The following theorems, some of which are used as axioms in other expositions, are simple consequences of the above axioms.

• Theorem 1 - Two distinct points are on one, and only one, line.
• Theorem 2 - There are at least two distinct lines.
• Theorem 3 - If $C,D$ are distinct points on the line $AB$, then $A,B$ are distinct points on the line $CD$.
• Theorem 4 - Two distinct lines cannot have more than one common point.
• Theorem 5 - There exists four points $A,B,C,D$ no three of which are collinear.

The following theorem is slightly less evident.

• Theorem 6 - All ${\mathcal {l}}\in Lines$ have the same cardinality.

Corollary - If the set $Points$ is finite, then all lines in ${\mathcal {P}}$ have the same number of points.

Proof

If ${\mathcal {l_{1}}},{\mathcal {l_{2}}}\in Lines$, we show that there exists a bijective map $\phi :{\mathcal {l_{1}}}\mapsto {\mathcal {l_{2}}}$ from the set of points on ${\mathcal {l_{1}}}$ to the set of ponts on ${\mathcal {l_{2}}}$. We can suppose that ${\mathcal {l_{1}}}\neq {\mathcal {l_{2}}}$.

Case 1 : The lines ${\mathcal {l_{1}}},{\mathcal {l_{2}}}$ intersect at a point $S$.

Let $P_{1}\in {\mathcal {l_{1}}},P_{1}\neq S;P_{2}\in {\mathcal {l_{2}}},P_{1}\neq S$. By axiom A3 there is a point $Q$ on the line $P_{1}P_{2}:Q\notin {\mathcal {l_{1}}},Q\notin {\mathcal {l_{2}}}$.

If $X\in {\mathcal {l_{1}}},X\neq S$, then the line $XQ$ intersects ${\mathcal {l_{2}}}$ in a unique point $\phi _{Q}(X)$. That is, the map $\phi _{Q}$ defined by : $\forall X\in {\mathcal {l_{1}}},X\neq S:\phi _{Q}(X)\mapsto XQ\cap {\mathcal {l_{2}}}$ and $\phi _{Q}(S)=S$ is an injection from ${\mathcal {l_{1}}}$ into ${\mathcal {l_{2}}}$. Similarly we can construct an injection $\psi :{\mathcal {l_{2}}}\mapsto {\mathcal {l_{1}}}$. By the Schröder-Bernstein theorem there exists a bijection between the sets ${\mathcal {l_{1}}},{\mathcal {l_{2}}}$.

Case 2 : The lines ${\mathcal {l_{1}}},{\mathcal {l_{2}}}$ do not intersect. Let $A\in {\mathcal {l_{1}}},B\in {\mathcal {l_{2}}}$ and ${\mathcal {h}}$ the line $AB$. By case 1 the bijections $\phi _{1}({\mathcal {l_{1}}})\mapsto {\mathcal {h}},\phi _{2}({\mathcal {h}})\mapsto {\mathcal {l_{2}}}$ exist, and so the composite map $\phi _{2}\circ \phi _{1}$ is a bijection between ${\mathcal {l_{1}}},{\mathcal {l_{2}}}$.

Dimension of an incidence structure

Definition : If $P,Q,R$ are three points not all on the same line and ${\mathcal {l}}$ is the line joining $Q$ and $R$, the class of all points on the lines joining $P$ to the points on the line ${\mathcal {l}}$l is called the plane determined by $P$ and ${\mathcal {l}}$. Small greek letters will be used to denote planes.

Theorem - Any two lines on the same plane $\pi$ have a common point.

Theorem - The plane $\alpha$ determined by a line ${\mathcal {l}}$ and a point $P$ is identical with the plane $\beta$ determined by a line ${\mathcal {m}}$ and a point $Q$, provided ${\mathcal {m}}$ and $Q$ are on $\alpha$.

Corollary - There is one and only one plane determined by three non-collinear points, or by a line and one point not on the line or by two intersecting lines.

Theorem - Two distinct points planes which have two distinct points $A,B$ in common contain all the points on the line $AB$ and have no other points in common.

Corollary - Two distinct planes cannot have more than one common line.

Points have dimension 0

Lines have dimension 1

Planes have dimension 2

If $S(n-1)$ is an incidence structure of dimension $n-1$ and $P\in {Points},P\notin S(n-1)$. The set of all points on the lines joining $P$ to the points of $S(n-1)$ is an n-dimensional structure $S(n)$

The analogous theorems to the above are straightforward.

Definition : If all the points of $Points$ are in $S(N)$, the the incidence structure has dimension $N$.

Thus the dimension of an incidence structure is either a positive integer or infinity.

Infinite dimensional incidence structures exist.

The collineation group and some sub-groups

Depuis Félix Klein and his Erlanger programme, it is always a fruitful occupation in mathematics when studying a mathematical structure to examine in detail those properties which are conserved by symmetries, i.e. subgroups of the automorphism group.

In the case of projective incidence structures the exercise yields some beautiful mathematics and is still an active source of research.

An intuitive motivation for some of the formel definitions

Imagine two planes $\Pi _{1},\Pi _{2}$ in Euclidean three-space, which intersect in a line ${\mathcal {l}}$ and two points $P_{1},P_{2}$ not on either plane.

We project the point $Q\in \Pi _{1}$ onto the point $R\in \Pi _{2}$ by drawing the line $P_{1}Q$ and $R$ is the point of intersection with the plane $\Pi _{2}$. This procedure gives an isomorphisme between the points of $\Pi _{1}$ and $\Pi _{2}$ and lines in $\Pi _{1}$ are mapped into lines on $\Pi _{2}$. Call this isomorphism $\phi _{P_{1}}$.

We can define a similar isomorphism, $\phi _{P_{2}}$, from $\Pi _{2}\mapsto \Pi _{1}$.

The combined map $\phi _{P_{1}}\circ \phi _{P_{2}}$ is an automorphism of $\Pi _{1}$. This automorphism has several interesting features :

1. Every point on the line ${\mathcal {l}}$ is mapped onto itself. The line is fixed by the automorphism.
2. The point of intersection, $C$ of the line $P_{1}P_{2}$ with the plane $\Pi _{1}$ is also a fixed point.
3. Any line in the plane $\Pi _{1}$ which passes through $C$ is mapped into itself (the points are not fixed).

Automorphisms with these properties, which arise from a simple and intuitive geometrical construction, will play a very important role in the study of projective incidence structures.

Definitions

• A collineation is an automorphism $\alpha$ of ${Points}$ which also maps ${Lines}$ onto itself in the sense that if ${\mathcal {l}}\in {Lines}$, then the image of the set $\{P\in {\mathcal {l}}\}$, that is $m=\{\alpha (P)\forall P\in {\mathcal {l}}\}$ is also $\in {Lines}$.

The set of collineations form a group under composition, ${\mathcal {Coll({\mathcal {P}})}}$.

Note : The identity is a collineation and there there exist incidence structures for which it is the ONLY collineation.

Lemma - If $\alpha$ is a collineation of ${\mathcal {P}}$, then for two distinct points $X,Y\in {\mathcal {P}}$ the image of the line $XY,\alpha (XY)$ is the line $\alpha (X)\alpha (Y)$.

• If $\alpha \in {\mathcal {Coll({\mathcal {P}})}}$ and for $P\in Points,\alpha (P)=P$ then $P$ is a fixed point of $\alpha$.

A collineation maps $Lines$ onto $Lines$ and there are two possibilities for the notion of fixed line :

• If $\alpha \in {\mathcal {Coll({\mathcal {P}})}}$ and ${\mathcal {l}}\in {Lines}:\alpha ({\mathcal {l}})={\mathcal {l}}$ as sets, then we say $\alpha$ preserves the line ${\mathcal {l}}$.

• If $\alpha \in {\mathcal {Coll({\mathcal {P}})}}$ and ${\mathcal {l}}\in {Lines}:\forall P\in {\mathcal {l}},\alpha (P)=P$, then we say that $\alpha$ fixes the line ${\mathcal {l}}$.

If $\Pi _{n}$ is a hyperplane of dimension $n$, then we can define the collineations $\alpha$ which preserve or which fix $\Pi _{n}$ in the obvious way.

Simple consequences of the definitions

Theorem If $\alpha \in Coll({\mathcal {P}})$, then :(i) $\alpha$ has at least one fixed point. (ii) $\alpha$ preserves at least one line.

Proof - Since any ${\mathcal {P}}$ contains at least one plane, it suffices to prove the case for planes.

(i) Suppose that there is a line ${\mathcal {l}}$ that is not preserved by $\alpha$, then $\alpha ({\mathcal {l}})\neq {\mathcal {l}}\Rightarrow P={\mathcal {l}}\cap \alpha ({\mathcal {l}})$, hence $P$ is a fixed point of $\alpha$.

Now suppose that all lines are preserved by $\alpha$. Then $\exists {\mathcal {l_{1}}},{\mathcal {l_{2}}},{\mathcal {l_{1}}}\neq {\mathcal {l_{2}}},\alpha ({\mathcal {l_{1}}})={\mathcal {l_{1}}},\alpha ({\mathcal {l_{2}}})={\mathcal {l_{2}}}$ which implies that $P={\mathcal {l_{1}}}\cap {\mathcal {l_{2}}}$ is a fixed point.

(ii) Suppose that no line is preserved by $\alpha$. Let ${\mathcal {l_{1}}},{\mathcal {l_{2}}}$ be distinct lines, hence $\alpha ({\mathcal {l_{1}}})\neq {\mathcal {l_{1}}},\alpha ({\mathcal {l_{2}}})\neq {\mathcal {l_{2}}};P={\mathcal {l_{1}}}\cap \alpha ({\mathcal {l_{1}}});Q={\mathcal {l_{2}}}\cap \alpha ({\mathcal {l_{2}}})$. The points $P,Q$ are fixed by $\alpha$.

If $P\neq Q$, the the line $PQ$ is preserved by $\alpha$. This is contrary to the hypotheses, hence $P=Q$ and so all the lines in the plane must pass through $P$.

Let $A\in {\mathcal {l_{1}}},B\in \alpha ({\mathcal {l_{1}}})$ and different from $P$.

The line $AB$ must pass through $P$. The lines $AB,{\mathcal {l_{1}}}$ have $A$ in common. If they are distinct they cannot have $P$ in common, so they are not distinct, and this implies that $\alpha ({\mathcal {l_{1}}})={\mathcal {l_{1}}}$, a contradiction, since we are assuming that no lines are preserved.

Hence $\alpha$ preserves at least one line.

Theorem The set of $\alpha \in {Coll({\mathcal {P}})}$ which preserve a line ${\mathcal {l}}$ form a sub-group ${Coll}_{pres}({\mathcal {l}})\subseteq {Coll({\mathcal {P}})}$.

Theorem The set of $\alpha \in {Coll}_{pres}({\mathcal {l}})$ which fix a line ${\mathcal {l}}$ form a sub-group ${Coll}_{fix}({\mathcal {l}})\subseteq {Coll}_{pres}({\mathcal {l}})$.

The groups ${Coll}_{fix}({\mathcal {l}})$ are important ; they will give rise to affine geometry, the carcterisation of n-dimensional projective structures. We state and prove some simple theorems which will enable us to better understant the collineations which fix a given line.

Clarify the details of  :

• Given any two lines of P there is a collineation that maps L1 to L2
• Given any line in P, then there is a collineation, different from the identity, which preserves the line.

This shows that the collineation group is not trivial and is 'large'.

Theorem - If ${\mathcal {l}},{\mathcal {m}}\in {Lines},{\mathcal {l}}\neq {\mathcal {m}}$, then ${Col}_{fix}({\mathcal {l}})\cap {Coll}_{fix}({\mathcal {m}})=\iota$, the identity collineation. i.e. A collineation different from the identity cannot fix more than one line.

Proof

Case 1 : The two lines have a point in common.

Proof.

Let $Q$ be the intersection of the lines ${\mathcal {l_{1}}},{\mathcal {l_{2}}}$ fixed by $\alpha$.

Let $P$ be a point of the plane not on these lines.

Let $R,S$ be two points on ${\mathcal {l_{1}}}$ distinct from $Q$.

Suppose that the line $PR$ intersects ${\mathcal {l_{2}}}$ in $T$ and that the line $PS$ intersects ${\mathcal {l_{2}}}$ in $U$.

The points $R,S,T,U$, are fixed by $\alpha$, so the lines $RT$ and $SU$ are preserved by $\alpha$. Hence their intersection $P$ is fixed by $\alpha$. But $P$ is any point in the plane not on ${\mathcal {l_{1}}}$ or on ${\mathcal {l_{2}}}$. Thus all the points of the plane are fixed by $\alpha$. Hence, $\alpha$ is the identity collineation.

Case 2 : The two lines have no point in common.

Proof.

Theorem - If $\alpha \in Coll_{fix}({\mathcal {l}})$, then there is at most one point $C\notin {\mathcal {l}}:\alpha (C)=C$.

Proof

Lemma 2 : A collineation of a plane $\Pi$ which fixes a line ${\mathcal {l}}$ and two points $P_{1},P_{2}$ not on ${\mathcal {l}}$ is the identity collineation

Proof Let $P$ be a point in the plane not on the line ${\mathcal {l}}$ and not on the line $P_{1}P_{2}$.

Let $PP_{1}$ intersect ${\mathcal {l}}$ at $Q_{1}$ and $PP_{2}$ at $Q_{2}$.

The lines $P_{1}Q_{1}$ and $P_{2}Q_{2}$ are distinct and $P$ is their unique point of intersection.

The points $P_{1},Q_{1}$ are fixed by the collineation, hence the line $P_{1}Q_{1}$ is preserved by the collineation, likewise the line $P_{2}Q_{2}$ is preserved by the collineation.

This implies that the intersection of these two lines is fixed by the collineation. But $P$ is any point not on ${\mathcal {l}}$ and not on the line $P_{1}P_{2}$. This implies that the line $P_{1}Q_{1}$ is fixed by the collineation, and by lemma 1, the collineation must be the identity.

The next theorem proves that if a collineation has a fixed line, then it must have a unique fixed point with the property that all lines through the fixed point are preserved by the collineation.

Theorem - If $\alpha \in Coll_{Fix}({\mathcal {l}})$, then $\exists C_{\mathcal {l}}\in Points:\alpha (C_{\mathcal {l}})=C_{\mathcal {l}}$ and any line ${\mathcal {m}}$ through $C_{\mathcal {l}}$ is preserved by $\alpha$. The point $C_{\mathcal {l}}$ is unique.

N.B. $C_{\mathcal {l}}$ may or may not be on the line ${\mathcal {l}}$.

Proof

By lemma 2 there is at most one point not on the fixed line which has the stated properties.

We now show that there cannot exist a point on the line L with these properties. Suppose that such a point, C1 exists. If Q is a point not on L and different from C, then the line C2.Q is preserved by alpha. The line CQ is preserved by alpha. The intersection of these two lines is then a fixed point of alpha, nameley Q, and Q is different from C. Thus alpha has twou fixed points not on L, hence alpha is the identity by lemma 2.

Similarly there cannot be two distinct fixed points C1, C2 on L which preseve all lines throug C1 and C2.

Thus, the point C, if it exists is unique. We now show that such a point does, in fact, exist.

Suppose that alpha has a fixed point C, C not on L. Any line m through C intersects L at a point Q. The points P and Q are fixed by alpha, hence the line m is preserved by alpha. Thus, any line through C is preserved by alpha.

Now suppose that alpha does not fix any point not on the line L.

Let P be any point, then $\alpha (P)\neq P$ and $\alpha (P)\notin {\mathcal {l}}$.

The line ${\mathcal {m}}=P.P(\alpha )$ intersects ${\mathcal {l}}$ in a point $C\neq P$.

The line m = P.C. and so alpha (m) = alpha (P).alpha (C) = PC. Thus m is preserved by alpha.

Let Q be a point not on L or m, then Q is on a preserved line n. The intersection of m and n must be a fixed point and by hypothesis there are no fixed points not on L. Hence the point of intersection must be the point C. Thus every line preserved by alpha must pass through C and every line through C is preserved.

The 'dual' theorem is also true.

Theorem - If $\alpha \in Coll$ and there exists a $C\in Points:\alpha (C)=C$ and all lines through $C$ are preserved by $\alpha$, then there exists a line ${\mathcal {l_{C}}}$ that is fixed by $\alpha$. The line ${\mathcal {l_{C}}}$ is unique.

Lemma Let $\alpha$ be a collineation of ${\mathcal {P}}$ and $\Pi$ a hyperplane such that each point is fixed by $\alpha$. Then :

(i) There exists a point $C$ such that each line through $C$ is preserved by $\alpha$.

(ii) If $C\notin {\mathcal {H}}$, then $C$ is unique.

Proof - If $C\notin {\mathcal {H}},\alpha (C)=C$, the $C$ is a centre : for each line through $C$ is preserved (if ${\mathcal {l}}$ is on C, then ${\mathcal {l}}\cap {\mathcal {H}}=P,\alpha (P)=P$. The line $CP={\mathcal {l}}$ is preserved.

Suppose now that no point outside ${\mathcal {H}}$ is fixed.

Let $P\notin {\mathcal {H}}\Rightarrow \alpha (P)\neq P$. Let $C=P\alpha (P)\cap {\mathcal {H}}$, then the lines : $P\alpha (P)=C\alpha (P)=CP$ and $\alpha (P\alpha (P))=\alpha (PC)=\alpha (P)\alpha (C)=\alpha (P)C=\alpha (P)P$ i.e. the line $P\alpha (P)$ is preserved.

We now show that any line through $C$ is preserved.

Let ${\mathcal {g}}\in Lines,C\in {\mathcal {g}},{\mathcal {g}}\notin {\mathcal {H}},Q\notin {\mathcal {H}},Q\notin P\alpha (P)$. The line $Q\alpha (Q)$ passes through $C=P\alpha (P)\cap {\mathcal {H}}$. For let $S=PQ\cap {\mathcal {H}}$, then $S=\alpha (S)\in \alpha (PQ)=\alpha (P)\alpha (Q)$

Hence the points $S,P,Q,\alpha (P),\alpha (Q)$ are contained in a common plane $\Pi$.

Therefore the lines $Q\alpha (Q),P\alpha (P)$ intersect at $X\in \Pi$.

Since the lines $Q\alpha (Q),P\alpha (P)$ are preserved by $\alpha ,X$ satisfies $\alpha (X)=\alpha (P\alpha (P))\cap \alpha (Q\alpha (Q))=P\alpha (P)\cap Q\alpha (Q)=X$.

Thedrefore $X\in {\mathcal {H}}$, thus it must coincide with $P\alpha (P)\cap {\mathcal {H}}=C$.

Hence all the lines of the form $Q\alpha (Q)$ pass through the point $C$. Thus each line trough $C$ is preserved.

Lemma Let $\alpha$ be a collineation of ${\mathcal {P}}$ and $C$ a point such that each line through $C$ is preserved by $\alpha$. Then :

(i) There exists a hyperplane ${\mathcal {H}}$ which is fixed by $\alpha$

(ii) The hyperplane is unique.

Proof

Suppose that the line ${\mathcal {l}}$ does not pass by $C$ but is preserved by $\alpha$. If $P\in {\mathcal {l}}$, the the line $CP$ is preserved by definition. Hence the point of intersection of ${\mathcal {l}}$ and $CP$ is a fixed point. But $P$ is any point of ${\mathcal {l}}$ , hence ${\mathcal {l}}$ is a fixed line.

Suppose now the \mathcal{l_{1}} , \mathcal{l_{2}} are two lines which are NOT preserved by \alpha , we will construct a line tha IS preserved by \alpha .

Let $P={\mathcal {l_{1}}}\cap \alpha ({\mathcal {l_{1}}},Q={\mathcal {l_{2}}}\cap \alpha ({\mathcal {l_{2}}})$. The points $P,Q$ are fixed by $\alpha$, hence $PQ$ is preserved by $\alpha$.

Since there at least three lines in ${\mathcal {P}}$ there is at least one preserved line that does not pass by $C$ and so is fixed.

We now have to build up a fixed hyperplane.

The above theorems were proved on the assumption that a colleation with a fixed line exists. We investigate the conditions of their existance in the next section.

Central collineations

Definition : An $\alpha \in Coll$ with the above properties is called a Central collineation ; $C$ is the centre and ${\mathcal {l}}$ the axis of the collineation.

Definition : A collineation $\alpha$ of ${\mathcal {P}}$ is a central collineation if there exists a hyperplane ${\mathcal {H}}$ (the axis if $\alpha$ and a oint $C$, the centre of $\alpha$ such that :

- For every point $X\in {\mathcal {H}}\Rightarrow \alpha (X)=X$ i.e. is a fixed point.

- For every line${\mathcal {l}}$ through $C$ $\alpha ({\mathcal {l}})={\mathcal {l}}$ i.e. the line is preserved.

Lemma - Let ${\mathcal {H}}$ be a hyperplane and $C$ a point of ${\mathcal {P}}$. The set of central collineations with axis ${\mathcal {H}}$ and centre $C$ form a group, ${\mathcal {Coll(H,C)}}$ with respect to composition.

N.B. The group is not empty since it contains the identity collineation.

Lemma - Let $\alpha \in {\mathcal {Coll(H,C)}}$ and suppose that $P\neq C,P\notin H,\alpha (P)=P'$. Then $\alpha$ is uniquely determined. In particular the image of any $X,X\notin {\mathcal {H}},X\notin PP'(=PC)$ satisfies : $\alpha (X)=CX\cap FP'$, where $F=PX\cap X$

Proof The image $X'$ of $X$ is subject to the following restrictions :

-The line $CX$ is mapped onto itself, so $\alpha (X)$ is on the line $CX$.

- Consider the point $F=PX\cap {\mathcal {H}}$ is on the axis of \alpha , so is fixed. X is on the line FP.

Since $X$ is not on $PP',F$ is not on $CX$ , thus $X'$ is on the intersection of two distinct lines and so uniquely determined. It follows that $X'$ is on the image of the line $FP$. The image of $\alpha (FP)=F\alpha (P)$.

Thus $X$ is the intersection of the lines $CX,FP'$ , hence uniquely determined.

We now show that any $Y$ on the line $CP$ is also uniquely determined. Replace the pair of points $(P,P')$ by any other pair of points $(R,R')$ with $R\neq R',R\in CP$, the repeat the above construction to determine $\alpha (Y)$ (need to quote axioms to say there exists a point $R$ not fixed and not on $CP$, then use the construction to determine $\alpha (R)=R')$.

Corollary (Uniqueness of central collineations) - If $\alpha \in {\mathcal {Coll(H,C)}},\alpha \neq \iota$ :

(a) If $P\neq C,P\notin {\mathcal {H}}$, the P is not fixed by $\alpha$;

(b) $\alpha$ is uniquely determined by one pair of points $P,\alpha (P),P\neq C$.

Proof (a) Supose that Q is fixed by \alpha , then we will show that any P \in \mathcal{P} is fixed i.e. \alpha is the identity collineation.

Let X \notin CP , then \alpha (X) = CX \cap FP' = CX \cap FP = X, since X is on FP, so every point not on CP is fixed. We pick a pont X_{0} not on CP and repeat the argument to show that all points of CP are fixed. hence \alpha = \iota.

(b) follows directly from the lemma.

Note : We will be considering the group of central collineations with centre on ${\mathcal {H}}$ (the elations or translations). To show that two elations with the same centre are the same, it suffices to show that for just one point $X\notin {\mathcal {H}}$, that $\alpha (X)=\beta (X)$.

QUESTION : Is it always true that : A collination is the product of a fine number of central collineations.

Theorem - There is at most one central collineation $\alpha$ with given centre $C_{0}$, axis ${\mathcal {l}}$ and pair of points $P_{0},\alpha (P_{0})$.

Proof

Lemma - A central collineation $\alpha$ of a plane is completely determined by its axis ${\mathcal {l}}$, centre $C_{0}$ and a pair of points $P_{0},\alpha (P_{0})$.

Proof

Let $Q$ be any point in the plane. We will show how to determine $\alpha (Q)$.

The line $P_{0}Q$ intersects ${\mathcal {l}}$ at $R$, a fixed point. The line $\alpha (P_{0})R$ intersects the line $C_{0}Q$ at $X$.

We claim that $\alpha (Q)=X$, because $\alpha (Q)$ must be on the line $C_{0}Q$, since $\alpha$ preserves all lines through $C_{0}$ ; $\alpha (Q)$ must also lie on the line $R\alpha (P_{0})$ and $X$ is the unique intersection of these lines.

Question : What if L, C, P, Q are not in the plane C, L ?

Theorem If N > 2 there exists a central collineation with any given centre, axis and (P, image(P) ).

Theorem Desargues theorem is true if and only if all the possible central collineations exist.

Cor For N > 2 Desargues theorem is true. For N = 2 we have to introduce as an axion that all possible central collineations exist.

N.B. Check out the little an big Desargues theorems to get the logical structure clear.

If ${\mathcal {P_{2}}}$ is non-Desarguenian, the there exist at least one line which is not fixed by any collineation of ${\mathcal {P_{2}}}$.

Definition :

Central collineations with centre on the axis are called elations.

A central collineation with centre NOT on the axis is called a homology/

N.B. In many papers Central collineations are called perspectivities.

Theorem - The homologies with centre C and axis L form an abelian group.

If $\sigma _{1},\sigma _{3}$ are homologies with cetres $C_{1},C_{2}$ and axes ${\mathcal {l_{1}}},{\mathcal {l_{2}}}$, then $\sigma _{1}*\sigma _{2}$ is a collineation with centre ${\mathcal {l_{1}}}\cap {\mathcal {l_{2}}}$ and preserves the line $C_{1}C_{2}$. The line is not necessarily fixed.

Theorem - The elations with axis ${\mathcal {l}}$ and centre $C_{i}$ form a group $El({\mathcal {l}},C_{i})$

Theorem - The elations with axis ${\mathcal {l}}$ form a group $El({\mathcal {l}})$, the $El({\mathcal {l}},C_{i})$ are subgroups.

Theorem - If $El({\mathcal {l}})$ has at least two non-trivial subgroups $El({\mathcal {l}},C_{1})$ and $El({\mathcal {l}},C_{2})$, then $El({\mathcal {l}})$ is an abelian group. The orders of the elements of $El({\mathcal {l}})$ is either infinity or all are of finite order $p$, a fixed prime number.

There exists examples where the group $El({\mathcal {l}})$ does not have two such sub-groups.

Question : What are the examples ?

Definition : The group generated by all the elations of ${\mathcal {P}}$ is often called the little projective group.

Question : Is this for different line or for one fixed line ?

Desarguesian and non-desarguesian projective incidence structures

The first major classification of projective incidence structures is a binary distinction :

- A desarguesian structure.

- A non-desarguesian structure.

A desarguesian projective structure satisfies an additional axiom, which we give in two forms (Our first theorem will be to show that the two forms of the axiom are equivalent.)

- Axiom 5a : - Axiom 5b :

Theorem - The two forms of axiom 5 are equivalent.

There exist projective incidence structures which satisfy axiom 5 and structures which do not satisfy axiom 5, so the concept of desarguesian structres is a useful one.

Example 1 : (A desarguesian plane)

Example 2 : (Moulton plane)

Example 3 : (Finite not-desarguesian plane)

Theorem A projective incidence structure of dimension N > 2 is desarguesian.

We will see in the section : Algebraic structures associated with projective incidence structures that a desarguesian projection structure of dimension n is isomorphic GLn(R), where R is a division ring.

For two dimensionnel projective incidence structures a great deal of research has centered on finding criteria (usually in termes of the collineation group) which ensure that the plane is desarguesian.

THE BARSOTTI CLASSIFICATION

Artin's axioms 4a, 4b 4b P Get the inter-relations clear !

Axiom 4a Given any two points p, q there is a translation that sends p to q.

Axiom 4b If tau1, tau2 are dfferent non-identity translations with the same traces, then there is a homomorphism that sends tau1 to tau2

Axiom 4b P - For a given P, then given Q, R such that P, Q, R are distict and PQR are collinear, then there is a dilatation which has P as fixed point and sends Q to R

Are thes axioms equivalent ?

Is axiom 4b P equivalent to Desargues theorem ,

Give the Moulton plane example of a non-Desarguesian plane (Refractive index)

QUESTIONS TO CLARIFY

1. If N >2 are all collineations central , all collineations of a Desarguesian plane ?
2. Does there exist at least one line that is fixed by by some alpha ?
3. If alpha has a fixed point does it have a preserved line ?
4. If alpha has a centre does it have a fixed line ?

Definitions

1. A collineation of order 2 is an involution

1. A affine plane is a projective plane with a distinguished lin ${\mathcal {l}}_{\infty }$

1. An elation of an affine plane with axis ${\mathcal {l}}_{\infty }$ is translation.

1. The group generated by all the translations of a plane is the Translation group.
2. It the translation group of an affine plane is transitive, then the plane is a 'Translation plane.

1. A projective plane is $(C-{\mathcal {l}})$ transitive with fixed point $C$ and fixed line ${\mathcal {l}}$ if for any pair of points $Q_{1},Q_{2}$ with $Q_{1},Q_{2}\ni {\mathcal {l}};Q_{1},Q_{2}\neq C;CQ_{1}neCQ_{2}$, there exists a $(C-{\mathcal {l}})$ perspectivity $\alpha :\alpha (Q_{1})=Q_{2}$.

Ternary algebraic structures

If ${\mathcal {G}}$ is a sub-group of ${\mathcal {Coll}}$ and ${Hom(}{\mathcal {G}},{\mathcal {G}}{)}$ the semi-group of homomorphisms of ${\mathcal {G}}\rightarrow {\mathcal {G}}$ we define two binary operations "$\boxtimes$" and $\boxplus$" as follows :

If $\alpha ,\beta \in {Hom(}{\mathcal {G}},{\mathcal {G}}{)}$ then $\alpha \boxtimes \beta =\gamma \in {Hom(}{\mathcal {G}},{\mathcal {G}})$ defined by $\gamma (g)=\alpha (\beta (g))\forall g\in {\mathcal {G}}$

If $\alpha ,\beta \in {Aut(}{\mathcal {G}},{\mathcal {G}}{)}$ then $\alpha \boxplus \beta =\delta \in {Hom(}{\mathcal {G}},{\mathcal {G}})$ defined by $\gamma (g)=\alpha (g)*\beta (g)\forall g\in {\mathcal {G}}$, where "$*$" is the group operation in ${\mathcal {G}}$

The binary operations $\boxtimes ,\boxplus$ are a priori not assumed to be communtative, associative or distributative.

However, there always exists

• A two sided 'multiplicative identity' denoted by $\iota \in {Aut(}{\mathcal {G}},{\mathcal {G}}{)}$such that $\alpha \boxtimes \iota =\iota \boxtimes \alpha =\alpha$ and defined $\forall g\in {\mathcal {G}}$ by $\iota (g)={g}$.

If $\alpha \in {Aut(}{\mathcal {G}},{\mathcal {G}}{)}$ then $\exists \beta ,\gamma \in {Aut(}{\mathcal {G}},{\mathcal {G}}{)}$ such that $\alpha \boxtimes \beta =\iota$ and $\gamma \boxtimes \alpha =\iota$ i.e. every element has a right inverse and a left inverse.

Thus, "$\boxtimes$" defines a loop.

• An 'additive zero', denoted by $o\in {Aut(}{\mathcal {G}},{\mathcal {G}}{)}$ and defined $\forall g\in {\mathcal {G}}$ by $o(g)={I}$, the identity element of ${\mathcal {G}}$, such that $\alpha \boxplus o=o\boxplus \alpha =\alpha$ $\forall \alpha \in {Aut(}{\mathcal {G}},{\mathcal {G}}{)}$

Il est à noter que $\iota \neq o$.

An 'additive inverse' : the homomorphism $inv\in {Aut(}{\mathcal {G}},{\mathcal {G}}{)}$ and defined by $inv(g)\rightarrow g^{-1}\forall g\in {\mathcal {G}}$ is such that $\alpha \boxplus inv(\alpha )=o$.

It is suggestive to denote $\boxplus inv$ by $\boxminus$ and write : $\alpha \boxminus \alpha =o$.

$\boxplus$ is associative : $\alpha \boxplus (\beta \boxplus \gamma )=(\alpha \boxplus \beta )\boxplus \gamma$

However, $\boxplus$ is commutative if and only if the group ${\mathcal {G}}$ is abelian.

By restricting attention to special groups ${\mathcal {G}}$ this ternary algebraic structure acquires more structure and can become a division ring or even a field. It will be used later to introduce coordinates into projective incidence structures and to characterise them.

Bibliography

Books

• Artin, Emil (1957). Geometric Algebra. New York: Interscience Publishers Inc. pp. 1–103.
• Beutelspacher, Albrecht; Rosenbaum, Ute (1998). Projective Geometry: From Foundations to Applications (PDF). Cambridge: Cambridge University Press. pp. 1–103. ISBN 0 521 48364 6.

• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275

• Hall Jr., Marshall (1959). The Theory of Groups. New York: The Macmillan Company. pp. 346–420.

• Hilbert, David (1950) [1902]. The Foundations of Geometry [Grundlagen der Geometrie] (PDF). Translated by Townsend, E.J. (2nd ed.). La Salle, IL: Open Court Publishing.

• Veblen, Oswald; Young, John Wesley (1910). Projective Geometry, volume 1. New York: Ginn & Co.

• Veblen, Oswald; Young, John Wesley (1946). Projective Geometry, volume 2. New York: Blaisdel.

• Whitehead, A.N. (1907). The axioms of projective geometry. Cambridge: Cambridge University Press. pp. 1–76.

Papers

• Baer, R. (1942). "Homogeneity of projective planes". American Journal of Mathematics. 64: 137–152.

• Baer, R. (1947). "Projectivities of finite projective planes". Amer. J. of Maths. 69: 653–684.

• Hall Jr., M. (1943). "Projective planes". Trans. Amer. Math. Soc. 54: 229–277.

• Hall Jr., M. (1949). "Projective planes (corrections)" (PDF). Trans. Amer. Math. Soc. 65: 473–474.
• Weibel, C. (2007). "Survey of Non-Desarguesian Planes". Notices of the AMS. 54 (10): 1294–1303.

• Veblen, O; Wedderburn, J.H.M (1907). "Non-desarguesian and non-pascalian geometries". Trans. Amer. Math. Soc. 8 (10): 379–388.

Mathematiciens who contributed

• Emil Artin
• Reinhold Baer
• Leonard Eugene Dickson
• Marshall Hall
• David Hilbert
• Félix Klein
• Donald Knuth
• Oswald Veblen
• Joseph Wedderburn
• Alfred North Whitehead
• John Wesley Young

Notes for WJE

• Baer, R. (1944). "The fundamental theorems of elementary geometry" (PDF). Trans. Amer. Math. Soc. 56: 94–129.

• Baer, R. (1946). "Projectivities with fixed points on every line of the plane" (PDF). Bull. Amer. Math. Soc. 52: 273–286.

Let ${\mathcal {P}}$ be a projective plane and $\sigma$ an involutory collineation, then either $\sigma$ is a perspectivity or $\sigma$ leaves fixed pointwise a proper subplane ${\mathcal {D}}\subset {\mathcal {P}}$.

• Wagner, A. (1958). "On Projective Planes Transitive on Quadrangles". J. Lond. Math. Soc. 33: 25–33.

Fundamental result : In a desaguesian plane there is a collineation that maps any quadrangle to any other quadrangle. The same is true for alternative planes. He examines the converse problem : what configurations existe in projective structures whose collineation group is transitive on quadrangles ? He proves that if the plane is finite, then it is desarguesian and conjectures that if it is infnite the plane is alternative.

Finite planes

• Ostrom, T.G. (1956). "Double transivity in finite projective planes". Canad. J. Maths. 8 (2): 563–567.

Let ${\mathcal {P}}$ be a projective plane, $\sigma _{1},\sigma _{2}$ two involutory homologies with centres $C_{1},C_{2}$ and axes ${\mathcal {l}}_{1},{\mathcal {l}}_{2}$. If $C_{1}\in {\mathcal {l}}_{2},C_{2}\in {\mathcal {l}}_{1}$ then $\sigma _{1}\sigma _{2}$ is an involutory homology with centre ${\mathcal {l}}_{1}\cap {\mathcal {l}}_{2}$ and axis $C_{1}C_{2}$.

• Ostrom, T.G.; Wagner, A (1959). "On projective and affine planes with transitive collineation groups". Math. Zeit. 71: 188–199.

THEOREM A. Let P be a finite projective plane admitting a collineation group doubly transitive on the points of P. Then, P is desarguesian.

THEOREM B. Let P be a finite affine plane admitting a collineation group doubly transitive on the a]fine, then P is a translation plane.

• Ostrom, T.G. (1970). "A Class of Translation Planes Admitting Elations which are not Translations". Arch. Math. 21: 214-.

• Piper, F.C. (1963). "Elations of finite projective planes". Math. Zeitschr. 82: 247–258.

• Siebenmann, L. (1965). "A characterization of free projective planes". Pacific J. of Maths. 15 (1): 293–298.

• Coulter, R.S. (2019). "On coordinatising planes of prime power order using finite fields". J. of th Australien Math. Soc. 106 (2): 184–199. arXiv:1609.01337v1.

Algebraic structure

• Martin, G.E. (1967). "Projective Planes and Isotopic Ternary Rings". American Math. Monthly. 74 (10): 1185–1195. JSTOR 2315659.

• Argunov, B.I.; Emel'chenkov, E.P. (1982). "Incidence structures and ternary rings". Uspekhi Mat. Nauk (in Russian). 37 (2): 3–37.

• Aschbacher, Michael (2006). "Projective planes, loops, and groups". J. of Algebra. 300: 396–432.

• Aschbacher, Michael (2008). "Isotopy and geotopy for ternary rings of projective planes". J. of Algebra. 319: 868–892.

• Ivanov, Nicolai V. (2016). "Affine planes, ternary rings, and examples of non-Desarguesian planes". ArXiv. pp. 1–25.

• Bertram, Walter; Kinyon, Michael (2012). "Torsors and ternary Moufang loops arising in projective geometry". HAL-Archives ouvertes.
• Landquist, Eric J. (2000). On Nonassociative Division Rings and Projective Planes (PDF) (MSc). Virginia State University.

• Kramer, Linus (1994). "The Topology of smooth projective planes". Arch. Math. 63: 85–91.

Infinite dimension structures

• Frink, Orrin (1946). "Complemented modular lattices and projective spaces of infinite dimmension" (PDF). Trans. Amer. Math. Soc: 452–467.

• Cameron, P.J. (1994). "Infinite linear spaces". Discrete Mathematics. 129: 29–41.

When does a planar ternary ring uniquely coordinitise a projective plane?

https://mathoverflow.net/questions/106888/when-does-a-planar-ternary-ring-uniquely-coordinitise-a-projective-plane/160978

projective plane over algebraic structure

https://math.stackexchange.com/questions/734288/projective-plane-over-algebraic-structure

• A. Wagner , Perspectivities and the little projective group, Algebraic and Topological Foundations of Geometryt, Proc. of a Coll., Utrecht, august 1959, 1962, pages 199-208.

The little projective group = Elation group, often a simple group.

• T.G. Room, P.B. Kirkpatrick, "Miniquaternion geometry" , Cambridge Univ. Press (1971)

• W.M. Kantor, "Primitive permutation groups of odd degree, and an application to finite projective planes" J. Algebra , 106 (1987) pp. 15–45
• G. Pickert, "Projective Ebenen" , Springer (1975)
• D.R. Hughes, F.C. Piper, "Projective planes" , Springer (1973)
• H. Lüneburg, "Translation planes" , Springer (1979)
• K.G.C. von Staudt, "Beiträge zur Geometrie der Lage" , I , Korn , Nürnberg (1865)
• G. Fano, "Sui postulati fondamentali della geometria proiettiva" Giornale di Mat. , 30 (1892) pp. 106–132
• I. Singer, "A theorem in finite projective geometry and some applications to number theory" Trans. Amer. Math. Soc. , 43 (1938) pp. 377–385