I have to prove this if and only if statement and I cant get anything written down.. Let le(X,≼) denote the number of linear extensions of a partially ordered set (X,≼). Prove le(X,≼)=1 if and only if ≼ is a linear ordering.
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One direction is easy. If $(X,\leq)$ is a linear order then it only has one linear extension (show that if $(X,\leq')$ is a linear extension then it adds nothing new).
In the other direction, if $(X,\leq)$ is not a linear order then there are $a,b\in X$ such that $a\nleq b$ and $b\nleq a$. Show that you can extend $\leq$ by deciding $a\leq'b$ or you can extend it in the other direction. This must give you at least two different extensions for $(X,\leq)$.