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 All triple disturbances at the vertex Y(y1,y2...)  can be classified as a member of one of the elements T(xyx,xyz,zyx,zyz). The classification is determined by the equivalence relation  (ŷ.)

                                              Two disturbances (ayb) and (cyd) are related by (ŷ)  if (a,b) and (c,d) belong to same element from (X,Z)

Equivalence under ŷ is invariant to the complexity of Y disturbances. 

This section explores the nature and conservations of (Y) disturbances within each triple class.

Transformations which preserve class membership are first considered.

All triples are transformed under auto-convergence which has a Semigroup structure.

An easier reference frame considers the transformations of whole triple neighbourhoods. Under this reference frame there are only 4 permutations, designated( I,H,V,R)

The properties of these transformations have been established.

VV=HH=RR=I  and VHR=VRH=RHV=RVH=HRV=HVR=I

Further Identity preserving permutations are combinations of the above only.

and VH=R,  VR=H,  RH=V

Mapping Neighbourhood Group (N) with Triple Semigroup (T)

Triple/Transform	I	@		V	@		H	@		R	@
1	1I1=1	1@1=1		1V=2	1@2=2		1H=3	3@1=3		1R=4	3@2=4
2	2I2=2	2@2=2		2V=1	2@1=1		2H=4	4@2=4		2R=3	4@1=3
3	3I3=3	3@3=3		3V=4	3@4=4		3H=1	1@3=1		3R=2	1@4=2
4	4I4=4	4@4=4		4V=3	4@3=3		4H=2	2@4=2		4R=1	2@3=1

 eg    4@3@2@1= Computing from Left to Right =[4@3]@2@1=[3@2]@1=4@1=3  y*(zxx)

OR   [4V]VV=[3V]V=4V=3   or making use of  NG properties 4V[VV]=4VI=4V=3  y*(note 4=z,)same, opposite,opposite)

 

Mapping Triple Semigroup (T) elements with Neighbourhood Group (N) elements

@	1	2	3	4
1	1@1=1x1=1 maps to 1I=1(x)(z)	1@2=1x2=2 maps to  1V=2(x)(z)	1@3=1z3=1 maps to 3H=1(xz)	1@4=1z4=2 maps to 3R=2(xz)
2	2@1=2x1=1 maps to 2V=1(x)(z)	2@2=2x2=2 maps to   2I=2(x)(z)	2@3=2z3=1 maps to 4R=1(xz)	2@4=2z4=2 maps to 4H=2(xz)
3	3@1=3x1=3 maps to 1H=3(xz)	3@2=3x2=4 maps to  1R=4(xz)	3@3=3z3=3 maps to 3I=3(x)(z)	3@4=3z4=4 maps to 3V=4(x)(z)
4	4@1=4x1=3 maps to 2R=3(xz)	4@2=4x2=4 maps to  2H=4(xz)	4@3=4z3=3 maps to 4V=3(x)(z)	4@4=4z4=4 maps to 4I=4(x)(z)

 

Lemma1: Any sequence of triple (t) interactions can be written as a unique sequence of (n)  transformations.  Proof: Every (t) interaction has a 1:1 relationship with a neighbourhood transformation.

Mapping Correlation Sequence 4@(321) to 4(VVV)

4	V	(z)		4	@	3	(z)							1	@	4					1	V				4	@	2
	3	V	(x)			3	@	2	(x)							2	@	4				2	I					4	@	3
		4	V	(x)				4	@	1	(x)							2	@	1			2	V						3	@
			3							3										1				1

 

Algorithm For calculating (Y) disturbance Trail. (1) For (t) sequence  a@b@c@d...=a@(bcd..)  Ignore first number (a) and substitute letters for x,z according to rule: x=(1,2) z=(3,4) eg 4@3@2@1 leads to y*(zxx)

2. For (n) sequence a(ABCD...) Start with number (a) where x=(1,2) z=(3,4) then add symbols by either changing or keeping the previous symbol according to the rule: change=(H,R) same=(I,V)  The last symbol is ignored. eg 4(VVV) leads to 3y*(zxx) 

Category Preserving Transformations up to 3 Operations with Complex y* Content

N/Transform

I

II

VV

HH

RR

III

VHR

VRH

HVR

RVH

HRV

RHV

N1

(x)

(xx)

(xx)

(xz)

(xz)

(xxx)

(xxz)

(xxz)

(xzz)

(xzz)

(xzx)

(xzx)

N2

(x)

(xx)

(xx)

(xz)

(xz)

(xxx)

(xxz)

(xxz)

(xzz)

(xzz)

(xzx)

(xzx)

N3

(z)

(zz)

(zz)

(zx)

(xz)

(zzz)

(zzx)

(zzx)

(zxx)

(zxx)

(zxz)

(zxz)

N4

(z)

(zz)

(zz)

(zx)

(xz)

(zzz)

(zzx)

(zzx)

(zxx)

(zxx)

(zxz)

(zxz)

 

 Properties of the compositions are:

1.With neighbourhood ay*(a.....)a/b  the first bonded disturbance is (a)

2.The patterns for (1,2) and (3,4) are identical and x/z transposes the patterns.

3.(a) is more represented than (b) in all neighbourhoods.

 

This section considers triple changing single transformations

Single Transformations

Category	N/Transform	I	V		H	R
(x)Reflection	xy*(..)x=1	xy*(x)x=1 (1@1=1)	xy*(x)z=2 (1@2=2)		zy*(x)x=3 (3@1=3)	zy*(x)z=4 (3@2=4)
(x)Refraction	xy*(..)z=2	xy*(x)z=2 (2@2=2)	xy*(x)x=1 (2@1=1)		zy*(x)z=4 (4@2=4)	zy*(x)x=3 (4@1=3)
(z)Refraction	zy*(..)x=3	zy*(z)x=3 (3@3=3)	zy*(z)z=4 (3@4=4)		xy*(z)x=1 (1@3=1)	xy*(z)z=2 (1@4=2)
(z)Reflection	zy*(..)z=4	zy*(z)z=4 (4@4=4)	zy*(z)x=3 (4@3=3)		xy*(z)z=2 (2@4=2)	xy*(z)x=1 (2@3=1)

 

1.Only Identity (I) preserves the category class.

2.The (V) transformations convert (x/z)Refractions into (x/z)Reflections and (x/z)Reflections into (x/z)Refractions.

3. The (H) transformations convert (x/z)Refractions into (z/x)Reflections and (x/z)Reflections into (z/x)Refractions.

4. The (R) transformations convert (x/z) Refractions into (z/x) Reflections and (x/z) Reflections into (z/x) Reflections.

 

Neighbourhood changing 3-Transforms

		VHH	VRR	HVV	HRR
	N1
	N2
	N3
	N4

 

 

©Copyright 2007 Dr David McGrath. All rights reserved