# Order theory
---
orders
- partial order
\begin{definition}
A \emph{partially ordered set} or \emph{poset} is an ordered pair $(X, \leq)$.
\begin{itemize}
\item $X$ is a set
\item $\leq$ is a relation on $X$
\end{itemize}
$\leq$ is referred to as the poset's \emph{partial order}, and it obeys the following properties.
\begin{itemize}
\item Reflexive
\item Antisymmetric
\item Transitive
\end{itemize}
\[(X,\leq) \text{ is a poset } \iff \forall a,b,c \in X [ a \leq a \land [ a \leq b \land b \leq a \implies a = b] \land [ a \leq b \land b \leq c \implies a \leq c ]]\]
\end{definition}

- total order
- product order
- strict order
- dense order

- Least upper bound property
- infimum
- supremum
\begin{definition}
\[ M \in \mathbb{R} \text{ is an upper bound of } U \iff \forall u\in U [M \geq u ] \]
\[ U \text{ is bounded above } \iff \exists M [  M \text{ is an upper bound of } U ]  \]

\[ M \in \mathbb{R} \text{ is a lower bound of } U \iff \forall u\in U [M \leq u ] \]
\[ U \text{ is bounded below } \iff \exists M [  M \text{ is a lower bound of } U ]  \]
\end{definition}

\begin{definition}
Let $U$ be a subset of \mathbb{R} bounded above. The \emph{supremum} of a set $U$ is the smallest upper bound of $U$.
	\[ \sup U = \min \{ M : M \text{ is a lower bound of } U \} \]
\end{definition}

\begin{definition}
Let $U$ be a subset of \mathbb{R} bounded below. The \emph{infimum} of a set $U$ is the largest lower bound of $U$.
	\[ \inf U = \max\{ M : M \text{ is an upper bound of } U \} \]
\end{definition}

\begin{definition}[Least upper bound property for the total order of real numbers]
	\[ U \text{ is bounded above } \implies  \exists k \in \mathbb{R} [ k = \sup U ] \]
\end{definition}

- dual order


lattice theory
- boolean algebra
- heyting algebra

Topology and orders
- banach lattice
- frechet lattice