\begin{definitionOf}{chain}
    Given a \Poset $\langle \LL, \lte \rangle$, a {\em chain} is a (possibly empty) set $C \subseteq \LL$ that is totally ordered, that is, 
    for all $x, y \in C$ either $x \lte y$ or $y \lte x$.
    
\end{definitionOf}