Sid/Sahiti: March 22nd 2020
===

### Semidirect products

- $(\alpha \equiv \{ a, b, \dots\}, +, 0)$
- $(\omega \equiv \{ X, Y, \dots\}, \times, 1)$
- $\cdot ~: ~\omega \rightarrow Automorphisms(\alpha)$

---
- [How to twist pointers without breaking them](https://www.cse.iitk.ac.in/users/ppk/research/publication/Conference/2016-09-22-How-to-twist-pointers.pdf)

- rotations: $\mathbb Z 5$
- reflection: $\mathbb Z 2$

- $D_5 = \mathbb Z5 \rtimes \mathbb Z2$
---

\begin{align*}
\begin{bmatrix}
1 & 0 \\
a & X
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
b & Y
\end{bmatrix}
= \begin{bmatrix}
1 & 0 \\
a + X \cdot b & XY
\end{bmatrix}
\end{align*}

- $(Y \mapsto b) \xrightarrow{act} (X \mapsto a)$

- $XY \mapsto a + X \cdot b$

### A walkway of lanterns

- Imagine $\mathbb Z$ as a long walkway. you start at 0. You are but a poor lamp lighter.
- Where are the lamps? At each $i \in \mathbb Z$, you have a lamp that is either on, or off. So you have $\mathbb Z2$.

- $L \equiv \mathbb Z \rightarrow \mathbb Z2$ is our space of lanterns. You can act on this space by either moving using $\mathbb Z$, or toggling a lamp using $\mathbb Z2$. $\mathbb Z2^{\mathbb Z} \rtimes \mathbb Z$

- $g = (lights:\langle-1, 0, 1\rangle, loc:10)$
- $move_3: (lights: \langle \rangle, loc: 3)$
- $move_3 \cdot g = (lights:\langle-1, 0, 1\rangle, loc:13)$
- $togglex = (lights:\langle 0, 2 \rangle, loc: 0)$
- $togglex \cdot g = (lights: \langle -1, 0, 1, 13, 15 \rangle, loc:13)$
- $toggley = (lights: \langle -13, -12 \rangle, loc:0)$
- $toggley\cdot g= (lights:\langle -1 \rangle, loc:13)$

### Wreath products

### This is automata theoretic? :O
- cascade finite automata.

### Krohn-rhodes, AKA how to model Freudian psychoanalysis using Lagrangians over semigroups.

> [name=sahiti] tremendously titillating titling, truly tremor inducing