Saturday, January 25, 2020

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)  ; kR-{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 kD such that xi=kyi. Similarly, we say (l1,l2,l3) and (m1,m2,m3)  in T are right proportional if there exist kD such that  li=kmi for i=1,2,3
Now define
            PD = {[x1,x2,x3]: (x1,x2,x3)T}
            LD = {<l1,l2,l3>: (l1,l2,l3)T}
            ID = {({[x1,x2,x3], <l1,l2,l3>): x1l1+x2l2+x3l3=0}
Then π D =(PD,LD,ID) is incidence structure, called plane over D.

Theorem

Prove that π D =(PD,LD,ID)  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)   PD = {[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)   LD = {<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)   ID = {({[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
            PF = {[x1,x2,x3]: (x1,x2,x3)T}
            LF = {<l1,l2,l3>: (l1,l2,l3)T}
            IF = {({[x1,x2,x3], <l1,l2,l3>): x1l1+x2l2+x3l3=0}
Then π F =(PF,LF,IF) is incidence structure, called plane over F.
Here, π F =(PF,LF,IF) is a projective plane over F.

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