Projective plane over Ring/Division Ring/Field

Proportionality class

Proportionality class is a set of triples (x1,x2,x3) of reals in which ordinates preserves proportionality. Proportionality class of (x1,x2,x3) is denoted by [𝑥1,𝑥2,𝑥3 ]

Example:

The proportionality class [1,2,3] is a set given as below.

                [1,2,3]={(x,y,z) ∶ (1,2,3)=k(x,y,z)  ; k∈R-{0}}

In particular,

                [1,2,3]={(1,2,3),(2,4,6),(-1,-2,-3),(1/2,1,3/2)…}

                [2,4,6]={(2,4,6),(1,2,3),(-1,-2,-3),(1/2,1,3/2)…}

Thus,     [1,2,3]=[2,4,6], However, (1,2,3)≠ (2,4,6)

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Projective plane over Ring/Division Ring     

Let D be a division ring and T = D × D× D/(0,0,0) be a set of all non-zero triples of D.

Also we say (x₁,x₂,x₃) and (y₁,y₂,y₃) in T are left proportional if there exist k∈D such that x_i=ky_i. Similarly, we say (l₁,l₂,l₃) and (m₁,m₂,m₃)  in T are right proportional if there exist k∈D such that  l_i=km_i for i=1,2,3

Now define

            P_D = {[x₁,x₂,x₃]: (x₁,x₂,x₃)∈T}

            L_D = {<l₁,l₂,l₃>: (l₁,l₂,l₃)∈T}

            I_D = {({[x₁,x₂,x₃], <l₁,l₂,l₃>): x₁l₁+x₂l₂+x₃l₃=0}

Then π _D =(P_D,L_D,I_D) is incidence structure, called plane over D.

Theorem

Prove that π _D =(P_D,L_D,I_D)  is projective plane.

Since, π _R is projective plane satisfying the statements

(1)   Each point is proportionality class [x₁,x₂,x₃] of triples of real numbers not all zero

(2)   Each line is proportionality class <l₁,l₂,l₃> of triples of real numbers not all zero

(3)   Point <l₁,l₂,l₃> is on line <l₁,l₂,l₃> if and only if x₁l₁+x₂l₂+x₃l₃=0

Now, we need to show that π _D ~ π _R, for this

      (1)   P_D = {[x₁,x₂,x₃]: (x₁,x₂,x₃)∈T}

Each point [x₁,x₂,x₃] is left proportionality class of triples (x₁,x₂,x₃) not all zero

            Implies statement 1 of π _R

      (2)   L_D = {<l₁,l₂,l₃>: (l₁,l₂,l₃)∈T}

Each line <l₁,l₂,l₃>  is right proportionality class of triples (l₁,l₂,l₃) not all zero

Implies statement 2 ofπ _R

       (3)   I_D = {({[x₁,x₂,x₃], <l₁,l₂,l₃>): x₁l₁+x₂l₂+x₃l₃=0}

The point [x₁, x₂, x₃] is on the line <l₁,l₂,l₃> if and only if

       x₁l₁+x₂l₂+x₃l₃=0

Implies statement 3 of π _R

Hence the theorem.

Projective plane over Field

Let F be a division ring and T = F × F× F/(0,0,0) be a set of all non-zero triples of F.

Then

            P_F = {[x₁,x₂,x₃]: (x₁,x₂,x₃)∈T}

            L_F = {<l₁,l₂,l₃>: (l₁,l₂,l₃)∈T}

            I_F = {({[x₁,x₂,x₃], <l₁,l₂,l₃>): x₁l₁+x₂l₂+x₃l₃=0}

Then π _F =(P_F,L_F,I_F) is incidence structure, called plane over F.

Here, π _F =(P_F,L_F,I_F) is a projective plane over F.