Problem Set - Ordered sets

## Problem Set - Ordered sets

 Show that if a greatest lower bound exists, then it is unique. Show that if $S$ is a lattice then the intersection of two intervals is an interval. A poset $S$ is left filtered if every subset $E$ of $S$ has an upper bound. A poset $S$ is right filtered if every subset $E$ of $S$ has an lower bound. Let $S$ be a poset and let $E$ be a subset of $S$. A minimal element of $E$ is an element $x\in E$ such that if $y\in E$ then $x\le y$. A poset $S$ is well ordered if every subset $E$ of $S$ has a minimal element. Show that every well ordered set is totally ordered. Show that there exist totally ordered sets that are not well ordered.