Dr David McGrath

Spine Physician

MB BS (Hons) FAFOM, RACP, FAFMM
Master of Pain Medicine

Vertex Semigroup
	N1 (xyx)	N2 (xyz)	N3 (zyx)	N4 (zyz)
N1 (xyx)	1____	2___	1____	2____
N2 (xyz)	1____	2___	1____	2____
N3 (zyx)	3____	4____	3____	4____
N4 (zyz)	3____	4____	3____	4____

Canonical Representation
	N1	N4	N2	N3
Recursion	1z3=1	4x2=4	2z4=2	3x1=3
Recursion/Participation	1x1=1	4z4=4	2x2=2	3z3=3
Creation/Participation	2x1=1	3z4=4	1x2=2	4z3=3
Creation	2z3=1	3x2=4	1z4=2	4x1=3

Pattern A (C-D 1/2 and 3/4)
	Creation	Destruction	Recursion	Participation	Total
N1	2x1=1	1x2=2	1z3=1	3x1=3	4
N1*	2z3=1	1z4=2	1x1=1	4x1=3	4
N2	1x2=2	2x1=1	2z4=2	4x2=4	4
N2*	1z4=2	2z3=1	2x2=2	3x2=4	4

N4	3z4=4	4z3=3	4x2=4	2z4=2	4
N4*	3x2=4	4x1=3	4z4=4	1z4=2	4
N3	4z3=3	3z4=4	3x1=3	1z3=1	4
N3*	4x1=3	3x2=4	3z3=3	2z3=1	4
Total	8	8	8	8	32

(1,[2,3]), (4,[2,3]) (3,[1,4]),(2,[1,4]) =Transform Triples

Pattern B (R-P 1/3 and 2/4)
	Recursion	Participation	Creation	Destruction	Total
N1	1z3=1	3x1=3	2x1=1	1x2=2	4
N3	3x1=3	1z3=1	4z3=3	3z4=4	4
					4
N2	2z4=2	4x2=4	1x2=2	2x1=1	4
N4	4x2=4	2z4=2	3z4=4	4z3=3	4

N1*	1x1=1	4x1=3	2z3=1	1z4=2	4
N4*	4z4=4	1z4=2	3x2=4	4x1=3	4

N2*	2x2=2	3x2=4	1z4=2	2z3=1	4
N3*	3z3=3	2z3=1	4x1=3	3x2=4	4
Total	8	8	8	8	32

N1 Neighbourhood Object=O1
	Creation 2x1=1		Creation 2z3=1
Participation 3x1=3				Recursion 1z3=1
		N1Neighbourhood
Participation 4x1=3				Recursion 1x1=1
	Destruction 1x2=2		Destruction 1z4=2

N4 Neighbourhood Object=O4
	Creation 3z4=4		Creation 3x2=4
Participation 2z4=2				Recursion 4x2=4
		N4Neighbourhood
Participation 1z4=2				Recursion 4z4=4
	Destruction 4z3=3		Destruction 4x1=3

N3 Neighbourhood Object=O3
	Creation 4z3=3		Creation 4x1=3
Participation 1z3=1				Recursion 3x1=3
		N3Neighbourhood
Participation 2z3=1				Recursion 3z3=3
	Destruction 3z4=4		Destruction 3x2=4

N2 Neighbourhood Object=O2
	Creation 1x2=2		Creation 1z4=2
Participation 4x2=4				Recursion 2z4=2
		N2Neighbourhood
Participation 3x2=4				Recursion 2x2=2
	Destruction 2x1=1		Destruction 2z3=1

There are four (4) even permutations on the Object set. Permutations are created by exchanging triples T(1234) and (xz) bonding.

Identity=P0 (1234,xz)=(1)(2)(3)(4) and (x)(z)

Vertical Reflection=P1(1234,xz)=(2143)=(12)(34) and (x)(z)

Rotation=P2(1234,xz)=(4321)=(14)(23) and (xz)

Horizontal Reflection=P3(1234,xz)=(3412)=(13)(24) and (xz)

The permutations form a group under composition and

PxPx=Po, PxPy=PyPx, PxPo=Px (inverse,abelian,identity)

Reflection+Reflection=Rotation

Reflection1+Rotation=Reflection2

Reflection2+Rotation=Reflection1

The permutation matrix is as follows.

Permutation Matrix
f1f2	(1234)	(12)(34)	(14)(23)	(13)(24)
(1234)	(1234)	(12)(34)	(14)(23))	(13)(24)
(12)(34)	(12)(34)	(1234)	(13)(24)	(14)(23)
(14)(23)	(14)(23)	(13)(24)	(1234)	(12)(34)
(13)(24)	(13)(24)	(14)(23)	(12)(34)	(1234)

The group can be recognised as the Klein4 group which is the automorphism group of

Klein Four-Group (Cayley Table)
	Po	P1	P2	P3
P0	Po	P1	P2	P3
P1	P1	Po	P3	P2
P2	P2	P3	Po	P1
P3	P3	P2	P1	Po

The permutation group on the neighbourhood objects:

1.Equivalent to a Klein 4-Group. The smallest non-cyclic group. Order 4.

2.Isomorphic to Dih2. (Dihedral group of order 2) (groups of polygon symmetry)

3.Subgroup of A4 {(1234),(2143),(4321),(3412): (123),(213),(124),(421),(234),(243),(134),(143) }=4!/2 Also subgroup of S4.

4.Normal Subgroup because operation is invariant under conjugation. gag¯¹(inverse)=a eg(2143)(3)(2143)=(3)

5. Symmetry group of Rectangle. With elements identity (1234),vertical reflection (13)(24),horizontal reflection (12)(34),180 rotation (14)(23).

6.All neighbourhood objects are equivalent under K4group action, in the sense that there is a K4 element , such that for or all x,y gx=y. Thus K4 is transitive and all neighbourhoods belong to the one ORBIT.

7.K4=Z2xZ2 with 3 Z2 (cyclic) subgroups (Po,P1)(Po,P2)(Po,P3)