Preorders and their special cases

\[ \boxed { \begin{array} {} & \text {preorders} & \\ & \text {reflexive:} \; x \leq x & \\ & \text {transitive:}\; x\leq y \;\&\; y\leq z \Rightarrow x\leq z & \\ \boxed { \begin{array} {} \text {partial order} \\ \text {skeletal:}\; x\leq y \;\&\; y\leq x \Rightarrow x=y \\ \text {example: power set } X\mathcal P \\ \end{array} } & \boxed { \begin{array} {} \text{intersection} \\ \text{discrete order: only $x=x$ allowed } \\ \end{array} } & \boxed { \begin{array} {} \text {equivalence relation} \\ \text {symmetric: } x\leq y \Rightarrow y\leq x \\ \text{example: congruence } m \equiv n \; (\text{mod } k) \\ \end{array} } \\ \end{array}  } \]

For a relation between partial orders and total orders, see

"Partial orders are the free conservative cocompletion of total orders"

Calin Tataru

http://www.tac.mta.ca/tac/volumes/44/1/44-01abs.html

Groups in linear algebra and elementary geometry

There are several groups that play an important role in linear algebra and elementary geometry.

The following table shows what they are and the relations between them.

It includes the simplest possible example, the one-dimensional real case.

In the table $\bf k$ is an arbitrary commutative field (like $\R$ or $\bf C$).

\[ \boxed {\begin{array} {l|cccccccc} & \text {rotations} & \subset & \text {isometries} & \subset & \text {affine transformations} \\ \hline \text{$n$-dimensional affine transformations} & \text {Isom}_n^+ ({\bf k}) =  \text {SO}_n ({\bf k}) \ltimes \text {Trans}_n ({\bf k}) & \subset & \text {Isom}_n ({\bf k}) = \text {O}_n ({\bf k})  \ltimes \text {Trans}_n ({\bf k}) & \subset & \text {Aff}_n ({\bf k}) = \text {GL}_n ({\bf k})  \ltimes \text {Trans}_n ({\bf k}) \\ & \cup &&  \cup && \cup \\ \text{$n$-dimensional linear transformations} & \text {SO}_n ({\bf k}) & \subset & \text {O}_n ({\bf k})  & \subset & \text {GL}_n ({\bf k}) \\ \\ \text{one-dimensional affine transformations} & \text {Isom}_1^+ (\R) = \{x \mapsto x+b \mid b \in \R \} & \subset & \text {Isom}_1 (\R) = \{x \mapsto \pm x+b \mid b \in \R \} & \subset & \text {Aff}_1 (\R) = \{x \mapsto a x+b \mid a, b \in \R, a\not=0 \} \\ & \cup && \cup && \cup \\ \text{one-dimensional linear transformations} & \text {SO}_1(\R) = \text {O}_1^+(\R) \cong \{1\in\R\} & \subset & \text {O}_1 (\R) \cong \{1,-1\in\R\} & \subset & \text {GL}_1 (\R) \cong \{a\in\R: a\not= 0\}  \\ \hline  & \text{translations} \\ \end{array} } \]

The $\ltimes$ symbol denotes semi-direct product.

The reason for the notation $a$ as opposed to $A$ 

is because $A$ is customary notation for matrices, while $a$ is more appropriate for real numbers 

(which of course can be viewed as $1\times 1$ real matrices).

Reference: Andrew Baker, Matrix Groups