06-1350/Class Notes for Thursday November 16

#	Week of...	Links
1	Sep 11	About, Tue, Thu
2	Sep 18	Tue, Kurlin(P), Thu
3	Sep 25	Tue, Photo, Thu
4	Oct 2	HW1, Tue, Thu
5	Oct 9	Tue(P), Thu(P)
6	Oct 16	HW2, Tue(P), Thu(P)
7	Oct 23	Tue(P), Thu
8	Oct 30	HW3, Tue, Thu
9	Nov 6	Tue ( $\nu$ ), Thu
10	Nov 13	Tue, Thu
11	Nov 20	HW4(P), Thu
12	Nov 27	Thu
13	Dec 4	Syzygies in Asymptote, Final
	Jan 8	Grades
Note. (P) means "contains a problem that Dror cares about".
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On to 07-1352

Today's Agenda

Sweeping clean a tree and ${\mathcal {A}}(\Gamma )\equiv {\mathcal {A}}(\uparrow _{b_{1}(\Gamma )})$ .

${\mathcal {A}}(\uparrow _{n})$ is a VS-algebra (see more at VS, TS and TG Algebras).

In the coordinates above, write the $TR\Phi B$ $TR\Phi B$ relations in various algebraic notations.
- R4: $(1230)^{\star }B\cdot (1213)^{\star }B\cdot (1023)^{\star }\Phi =(1123)^{\star }\Phi \cdot (1233)^{\star }B$ or $(B_{1a}B_{2a}\Phi _{1a};B_{1b}B_{2b};B_{1c}B_{2a}\Phi _{1b};B_{2c}\Phi _{1c})=(\Phi _{2a}B_{3a};\Phi _{2a}B_{3b};\Phi _{2b}B_{3c};\Phi _{2c}B_{3c})$ .
- R3: $(1230)^{\star }B\cdot (1213)^{\star }B\cdot (1023)^{\star }B=(1123)^{\star }B\cdot (1203)^{\star }B\cdot (1231)^{\star }B$ or $(B_{1a}B_{2a}B_{3a};B_{1b}B_{2b};B_{1c}B_{2a}B_{3b};B_{2c}B_{3c})=(B_{4a}B_{5a}B_{6a};B_{4a}B_{5b}B_{6b};B_{4b}B_{6c};B_{4c}B_{5c}B_{6a})$ .
- R2: $(123)^{\star }B^{\pm }\cdot (132)^{\star }B^{\mp }=1_{3}$ or $(B_{1a}^{\pm }B_{2a}^{\mp };B_{2b}^{\pm }B_{3c}^{\mp };B_{1c}^{\pm }B_{2b}^{\mp })=(1;1;1)$ .
- R1: $(B_{a}^{\pm };B_{b}^{\pm }B_{c}^{\pm })=(1;T^{\pm 2})$ .
But for now, skip the writing of the following relations:
- Symmetry of $\Phi$ and of $B^{\pm }$ .
- $u$ , $d$ and $\#$
- Idempotence for $T$ , $R$ , $\Phi$ and $B^{\pm }$ .
- $B^{\pm }$ in terms of $\Phi$ and $R$ and $R$ in terms of $T$ .