Given a partial ordered set (poset) $S$, is there a known procedure or algorithm to find the set of chains (i.e. subsets of $S$ where every two elements are comparable)? Note: I am asking here instead of math.SE because i'm looking for an algorithm for the problem.  
Extracting the set of chains from a partial order
algorithms;partial order
Yes there is, have a look at :Chen, Yangjun, and Yibin Chen. On the Decomposition of Posets. Computer Science & Service System (CSSS), 2012 International Conference on. IEEE, 2012.and also this: Daskalakis, C., Karp, R. M., Mossel, E., Riesenfeld, S. J., & Verbin, E. (2011). Sorting and selection in posets. SIAM Journal on Computing, 40(3), 597-622.