An example of posets from analysis. If a measure space \$$(X,\mu)\$$ is \$$\sigma\$$-finite, then its \$$L^p\$$ spaces (the collection of measurable functions \$$f\colon X\to\Bbb C\$$ such that \$$\int_X|f|^p\,\rm{d}\mu<\infty\$$) can be preordered by set inclusion. In general this is not a total order. However, consider a finite measure space \$$(X,\mu)\$$. As a consequence of Hölder's inequality, we can actually put a total order on the \$$L^p\$$ spaces with inclusion: $$L^q(X,\mu)\subseteq L^p(X,\mu)\qquad\text{if 1\le p\le q\le \infty.}$$

I would be interested if anybody knew of any interesting follow-up points to this post, since while it's true that we can do this, it feels somewhat contrived.