The Hausdorff Maximal Principle states:

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> Every partially ordered set $\left(X,\leqslant\right)$ has a linearly ordered subset $\left(E,\leqslant\right)$ such that no subset of $X$ that properly includes $E$ is linearly ordered by $\leqslant$.     *[Folland]*
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I know that this principle is equivalent to the axiom of choice, which means that it is undecidable. If we assume its negation (or the negation of axiom of choice, or of the well ordering principle, or anything equivalent), can we actually construct a counterexample?

My question is: is it possible *[or even easy? :)]* to write down an example of a partially ordered set such that no linearly ordered subset is maximal?

I know that to construct such an example, we need to use the negation of the axiom of choice.