## 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 [š„1,š„2,š„3
]
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)
Video 1

Video 2

Video 3

Projective plane over Ring/Division Ring

Video 2

Video 3

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 (

*x*_{1},x_{2},x_{3}) and (*y*_{1},y_{2},y_{3}) in*T*are left proportional if there exist k∈D such that x_{i}=ky_{i}. Similarly, we say (l_{1},l_{2},l_{3}) and (*m*_{1},m_{2},m_{3}) in*T*are right proportional if there exist k∈D such that l_{i}=*k*m_{i}for i=1,2,3
Now
define

P

_{D}= {[*x*_{1},x_{2},x_{3}]: (*x*_{1},x_{2},x_{3})∈T}
L

_{D}= {<*l*_{1},l_{2},l_{3}>: (*l*_{1},l_{2},l_{3})∈T}
I

_{D}= {({[*x*_{1},x_{2},x_{3}], <*l*_{1},l_{2},l_{3}>): x_{1}l_{1}+x_{2}l_{2}+x_{3}l_{3}=0}
Then Ļ

##

_{D}=(P_{D,}L_{D,}I_{D}) is incidence structure, called plane over*D.*##
*Theorem*

##
Prove that Ļ _{D}
=(P_{D,}L_{D,}I_{D})
is projective plane.

Since, Ļ

_{R}is projective plane satisfying the statements
(1)
Each point is proportionality class [

*x*_{1},x_{2},x_{3}] of triples of real numbers not all zero
(2)
Each line is proportionality class <

*l*_{1},*l*_{2},*l*_{3}> of triples of real numbers not all zero
(3)
Point <

*l*_{1},*l*_{2},*l*_{3}> is on line <*l*_{1},*l*_{2},*l*_{3}> if and only if x_{1}*l*_{1}+x_{2}*l*_{2}+x_{3}*l*_{3}=0
Now, we need to show that
Ļ

_{D }~_{ }Ļ_{R}, for this*(1)*P

_{D}= {[

*x*

_{1},x

_{2},x

_{3}]: (

*x*

_{1},

*x*

_{2},

*x*

_{3})∈T}

Each
point [

*x*_{1},x_{2},x_{3}] is left proportionality class of triples (*x*_{1},x_{2},x_{3}) not all zero
Implies statement 1 of Ļ

_{R}
(2)
L

_{D}= {<*l*_{1},*l*_{2},*l*_{3}>: (*l*_{1},*l*_{2},*l*_{3})∈T}
Each
line <

*l*_{1},*l*_{2},*l*_{3}> is right proportionality class of triples (*l*_{1},*l*_{2},*l*_{3}) not all zero
Implies
statement 2 ofĻ

_{R}*(3)*I

_{D}= {({[

*x*

_{1},x

_{2},x

_{3}], <

*l*

_{1},l

_{2},l

_{3}>): x

_{1}l

_{1}+x

_{2}l

_{2}+x

_{3}l

_{3}=0}

The point [

*x*_{1}, x_{2}, x_{3}] is on the line <*l*_{1},*l*_{2},*l*_{3}> if and only if
x

_{1}l_{1}+x_{2}l_{2}+x_{3}l_{3}=0
Implies
statement 3 of Ļ

_{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}= {[*x*_{1},x_{2},x_{3}]: (*x*_{1},x_{2},x_{3})∈T}
L

_{F}= {<*l*_{1},l_{2},l_{3}>: (*l*_{1},l_{2},l_{3})∈T}
I

_{F}= {({[*x*_{1},x_{2},x_{3}], <*l*_{1},l_{2},l_{3}>): x_{1}l_{1}+x_{2}l_{2}+x_{3}l_{3}=0}
Then Ļ

_{F}=(P_{F,}L_{F,}I_{F}) is incidence structure, called plane over*F.*
Here, Ļ

_{F}=(P_{F,}L_{F,}I_{F}) is a projective plane over*F*.
## No comments:

## Post a Comment