# 06-1350/Class Notes for Thursday November 16

## Today's Agenda

### Formulas are a Chore (Bore?)

• 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 TRΦB 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 (B1aB2aΦ1a;B1bB2b;B1cB2aΦ1b;B2cΦ1c) = (Φ2aB3a2aB3b2bB3c2cB3c).
• R3: $(1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star B = (1123)^\star B\cdot(1203)^\star B\cdot(1231)^\star B$ or (B1aB2aB3a;B1bB2b;B1cB2aB3b;B2cB3c) = (B4aB5aB6a;B4aB5bB6b;B4bB6c;B4cB5cB6a).
• R2: $(123)^\star B^\pm\cdot(132)^\star B^\mp=1_3$ or $(B^\pm_{1a}B^\mp_{2a}; B^\pm_{2b}B^\mp_{3c}; B^\pm_{1c}B^\mp_{2b}) = (1;1;1)$.
• R1: $(B^\pm_a; B^\pm_bB^\pm_c)=(1;T^{\pm 2})$.
• But for now, skip the writing of the following relations:
• Symmetry of Φ and of $B^{\pm}$.
• u, d and $\#$
• Idempotence for T, R, Φ and $B^{\pm}$.
• $B^{\pm}$ in terms of Φ and R and R in terms of T.

### Exponentiation is a Miracle

• Description of the problem.