\chapter{Monotone functions}


As with many areas of modern mathematics, it is useful to understabd wgat s


\begin{definition}[Monotone function]
Let $(P,\leq_P),(Q,\leq_Q)$ be posets, a \emph{monotone function} $ f : P \to Q$ preserves order.
\[f : P \to Q \text{ is a monotone function } \iff \forall p \in P \forall q \in Q [ p \leq_P q \implies f(p) \leq_Q f(q) ] \]
\end{definition}


\begin{definition}[Order isomorphism]
An \emph{order isomorphism} is a bijective monotone function whose inverse is also a monotone function
\end{definition}


\begin{definition}[Order automorphism]
An \emph{order automorphism} is an order isomorphism onto the same poset.
\end{definition}



Wosets are isomorphic to $(\mathbb{N} , \leq)$



\
\begin{proposition}[Not sure about this one]
Let $(W,\leq)$ be a woset and $f :W \to W$ a monotone function, then $f(x) \geq x$ for all $x \in W$.
\end{proposition}

\begin{proposition}
The only order automorphism on a woset is the identity function.
\end{proposition}

- hasse diagram

- product order
- dense order
- quasiorder


- 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 $x \in S$

\section{Least upper bounds}
\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 $X$ 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 $X$ 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]
\[ (X, \leq) \text{ obeys least upper bound property } \iff  \forall U \subseteq X [  U \text{ is bounded above } \implies  \exists k \in \mathbb{X} [ k = \sup U ] ] \]
\end{definition}