\part{Orders}

\chapter{Partial orders}


\begin{definition}
A \emph{partial order} $\leq$ is a homogeneous relation on a set $X$ that obeys the following properties.
\begin{itemize}
	\item Reflexive
	\item Antisymmetric
	\item Transitive
\end{itemize}
\end{definition}


\begin{definition}
A \emph{strict partial order} $<$ is a homogeneous relation on a set $X$ that obeys the following properties.
\begin{itemize}
	\item Irreflexive
	\item Antisymmetric
	\item Transitive
\end{itemize}
The ordered pair $(X,\leq)$ is called a \emph{partially ordered set} or \emph{poset}.
\end{definition}


\begin{definition}
A \emph{total order} $\leq$ is a homogeneous relation on a set $X$ that obeys the following properties.
\begin{itemize}
	\item Reflexive
	\item Antisymmetric
	\item Transitive
	\item Strongly connected
\end{itemize}
The ordered pair $(X,\leq)$ is called a \emph{totally ordered set} or \emph{toset}.
\end{definition}


\begin{definition}
A \emph{strict total order} $<$ is a homogeneous relation on a set $X$ that obeys the following properties.
\begin{itemize}
	\item Irreflexive
	\item Antisymmetric
	\item Transitive
	\item Connected
\end{itemize}
\end{definition}





- hasse diagram

- product order
- dense order
- quasiorder

- minimal element
- maximal element

- order convex set
A \emph{order convex set} is a set $S$ such that if $a,b \in S$ satisfy $a \leq x \leq b$, then $x \in S$

- 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
