Exercise 6

A linear extension of a poset $\bfP=(X,P)$ is a total order $L$ on $X$ such that if $x\leq y$ in $P$, then $x\leq y$ in $L$. Give linear extension of the three posets shown in this figure. If you feel very ambitious, try to count the number of linear extensions of the poset on the left side of the figure. Don't list them. Just provide an integer as your answer.