I am looking for some references (especially a good recent book) that covers important topics involving partial orders such as: order polytopes, sorting/selection in partially ordered sets, upper and lower bounds on the amount of linear extensions in a width $w$ partially ordered set, and perhaps even algorithms for computing some of these things. Might anyone perhaps know a good book that covers some or all of these topics? For example, the Art of Computer Programming by Knuth covers a lot of the topics on sorting and selection, but only involving total orders - I would be looking for something similar involving partially ordered sets.
Thanks!
One of the best references I know on order theory is the third chapter of Richard Stanley's Enumerative Combinatorics, Volume I. A manuscript of this is available by the author here. The topics aren't computer science oriented though, and cover lattices, Möbius functions, and some other interesting topics. It requires a fair bit of mathematical maturity, but the exercises section at the end of the chapter basically summarizes a lot of interesting old and present research in the field.
For sorting on posets I think you're looking for the concept of topsort, which I'm pretty sure Knuth covers somewhere in TAOCP as the wikipedia article cites Volume 1, section 2.2.3 as a place for further recommended reading.