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: $(1230)^\star B^\pm\cdot(1213)^\star B^\pm\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B^\pm$ or $(B^\pm_{1a}B^\pm_{2a}\Phi_{1a}; B^\pm_{1b}B^\pm_{2b}; B^\pm_{1c}B^\pm_{2b}\Phi_{1b}; B^\pm_{2c}\Phi_{1c}) = (\Phi_{2a}B^\pm_{3a}; \Phi_{2a}B^\pm_{3b}; \Phi_{2b}B^\pm_{3c}; \Phi_{2c}B^\pm_{3c})$.
  • $B^{\pm}$ 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 $B^{\pm}$.