I am currently beginning to read Folland after finishing chapters 1 - 6 in baby Rudin. I am a little confused about the section on partially ordered sets. Firstly, I think I know the answer to this, but if $x \leq y$ is false, then that doesn't necessarily imply $y < x$ i.e. $y < x$ isn't necessarily the negation of $x \leq y$. Clearly, this will imply in the Real numbers, which I am used to.
Also, I am a bit confused about the Hausdorff maximal principle, which is "Every partially ordered set has a maximal linearly ordered subset".
When it's saying maximal linearly ordered subset does that mean that it has an element that is maximal to the entire set or specifically for the subset?