Home
Surgery location
Articles
Contact us
Disclaimer
Dear patient
Presentations
Doctor Information
Insurer Information
Testimonials
Administration
Movement Experiments
Movement Demo
 
 

The vertex semigroup structure is a 4semigroup (one of 126 possibilities) 

NxN =4^4^2 =4^16. Under B2(Y)=2^4=16 (Rectangular Band)

LR Ideals	1=xyx L	2=xyz L	3=zyx L	4=zyz L
1=xyx R	1	2	1	2
2=xyz R	1	2	1	2
3=zyx R	3	4	3	4
4=zyz R	3	4	3	4

1.Idempotent aa=a

2.Associative (ab)c=a(bc)

3.Rectangular abc=ac.  aba=a   (aa =a) (point1)

Greens Relation L/R Class

	1	2	3	4
L	1133	2244	1133	2244
R	1212	1212	3434	3434
J	xxxx	yyyy	yyyy	xxxx

L,R Classes L={ (13), (24) } and R={ (12), (34) }

(1L3, 2L4) and (1R2, 3R4)   

J Classes  J={ (14) , (23) } 

SaS=SbS

H  Classes

aLb=aRb  eg 1L2=1R2   ie not possible no H class, except trivial aLa=aRa

D Class D={ (14), (23) }

(1R2,2L4) and/or (4R3 ,3L1) implies (1,4) belong same D class.

(3R4,4L2) and/or  (2R1,1L3) implies (2,3) belong same D class.

 

(16=2^4) Transformation (Semigroup/Monoid under Function Composition)            with 5 subgroups/1 subsemigroup (+further subsemigroups ) 

R Classes	11	22
33	L Class (f) 1133 ff=f(2)	D Class (g) 2233 ff=f(2)	(a) Z2 1233* ff=f (1) (2133) fff=f (3)
44	D Class (h) 1144 ff=f(2)	L Class (e) 2244 ff=f (2)	(b) Z2 1244* ff=f (1) (2144) fff=f(3)
	(c) Z2 1134* ff=f(1) (1143) fff=f(3)	(d) Z2 2234* ff=f(1) (2243)fff=f(3)	(i) Klein 4-Group (2134) fff=f(2)      (2143) fff=f(4) 1234* ff=f(0)   (1243) fff=f(2)

 (brackets)=number of category changes and generative set.  ff..=cycle length

Generative Set

Set  fff=f	2134	2143	1243	1143	2243	2133	2144
Inverses	2134	2143	1243	-	-	-	-
Single Permutation Set  ff=f			1234	1134	2234	1244	1233

A generative set can be identified. Each Z2 group has one generative permutation (1143,2243,2133,2144).

The K4 group has 3 (2134,2143,1243) (all produce the identity). All others can be derived. The generative set are all 3power idempotent. 

Four Combinations of (a,b,c,d)=(ac,ad,bc,bd) (=16x4=64)

(b,d) Matrix (not closed)

bd		1244	(2144)	2234	(2243)
1244		1244	(2144)	2244	2233
(2144)		(2144)	1244	2244	2233
2234		2244	1144	2234	(2243)
(2243)		2444	1144	(2243)	2234

 Non Closed structure with 2 -Z2 subgroups (1244,2144) (2234, 2243)

(b,c) Matrix (not closed)

bc		1244	(2144)	1134	(1143)
1244		1244	(2144)	1144	1133
(2144)		(2144)	1244	1144	1133
1134		1144	2244	1134	(1143)
(1143)		1144	2244	(1143)	1134

 

Klein 4-Group (i=1234,aa=i,ab=ba)

ii		#1234	#2143	1243	2134
#1234		#1234	#2143	1243	2134
#2143		#2143	#1234	2134	1243
1243		1243	2134	#1234	#2143
2134		2134	1243	#2143	#1234

4-Group Z2xZ2 Klein 4-Group with three subgroups (#1234,#2143) (1234,1243) (1234,2134) 

 

(K4,b) Matrix (not closed)

ib		1234	(2143)	(1243)	(2134)	1244	(2144)
						L	L
1234		1234	2143	1243	2134	1244	(2144)
2143		2143	1234	2134	1243	(2144)	1244
1243		1243	2134	1234	2143	1244	(2144)
2134		2134	1243	2143	1243	(2144)	1244
1244	R	1244	(2133)	1233	(2144)	1244	(2144)
(2144)	R	(2144)	1233	(2133)	1244	(2144)	1244

 

(K4,e) Matrix (not closed)

ie		1234	(2143)	(1243)	(2134)	2244
						L
1234		1234	2143	1243	2134	2244
(2143)		2143	1234	2134	1243	2244
(1243)		1243	2134	1234	2143	2244
(2134)		2134	1243	2143	1243	2244
2244	R	2244	1133	2233	1144	2244

 

(e,f,g,h) Idempotent 4-Semigroup and (b) Z2 group

		1144	2233	1133	2244		1244	(2144)
							L	L
1144		1144	2233	1133	2244		1144	2244
2233		1144	2233	1133	2244		2244	1144
1133		1144	2233	1133	2244		1144	2244
2244		1144	2233	1133	2244		2244	1144
1244	R	1144	2233	1133	2244		1244	(2144)
(2144)	R	1144	2233	1133	2244		(2144)	1244

Idempotent 4-Semigroup with (2) 2-subsemigroups (1133,2244) and (1144,2233)

(c,d)  4-Semigroup

cd		2234	2243	1134	1143
2234		2234	2243	1134	1143
2243		2243	2234	1143	1134
1134		2234	2243	1134	1143
1143		2243	2234	1143	1134

4-Semigroup with 2 Z2 groups (1134,1143) and (2234,2243)

 

Closed under Binary Function composition.  (fg) belongs set of 16 functions.  

Transformation Semigroup is a pair (N,S)  N=(1,3,2,4) and S=(16 functions, closed under composition)

Not the full transformation semigroup, S4xS4=256 functions (n^n) 

 

Only (0,1) functions supported by B2(Y). ie (1234), (1134), (1233),(2234),(1244).

 

 

 

 

©Copyright 2007 Dr David McGrath. All rights reserved