06-1350/Class Notes for Thursday November 16

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In Preparation

The information below is preliminary and cannot be trusted! (v)

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 $TR\Phi B$ relations in various algebraic notations.
  • R4: $(1230)^{\star }B^{\pm }\cdot (1213)^{\star }B^{\pm }\cdot (1023)^{\star }\Phi =(1123)^{\star }\Phi \cdot (1233)^{\star }B^{\pm }$.
  • $B^{\pm }$ in terms of $\Phi$ and $R$ and $R$ in terms of $T$.
  • R3, R2, R1
  • Symmetry of $\Phi$ and of $B^{\pm }$.
  • $u$, $d$ and $\#$
  • Idempotence for $T$, $R$, $\Phi$ and $B^{\pm }$.