Editing Partially ordered set (section)

=== Sums of partially ordered sets ===
{{anchor|sum}}
Another way to combine two (disjoint) posets is the '''ordinal sum'''<ref>
{{citation
 | last1 = Neggers | first1 = J.
 | last2 = Kim | first2 = Hee Sik
 | contribution = 4.2 Product Order and Lexicographic Order
 | isbn = 9789810235895
 | pages = 62–63
 | publisher = World Scientific
 | title = Basic Posets
 | year = 1998
}}</ref> (or '''linear sum'''),{{sfnp|Davey|Priestley|2002|pp=[https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA17 17–18]}} {{nowrap|1=''Z'' = ''X'' ⊕ ''Y''}}, defined on the union of the underlying sets ''X'' and ''Y'' by the order {{nowrap|''a'' ≤<sub>''Z''</sub> ''b''}} if and only if:
* ''a'', ''b'' ∈ ''X'' with ''a'' ≤<sub>''X''</sub> ''b'', or
* ''a'', ''b'' ∈ ''Y'' with ''a'' ≤<sub>''Y''</sub> ''b'', or
* ''a'' ∈ ''X'' and ''b'' ∈ ''Y''.

If two posets are [[well-ordered]], then so is their ordinal sum.<ref>{{cite book|author=P. R. Halmos|title=Naive Set Theory|url=https://archive.org/details/naivesettheory0000halm_r4g0|url-access=registration|year=1974|publisher=Springer |isbn=978-1-4757-1645-0|page=[https://archive.org/details/naivesettheory0000halm_r4g0/page/82 82]}}</ref>

[[Series-parallel partial order]]s are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is the [[disjoint union]] of two partially ordered sets, with no order relation between elements of one set and elements of the other set.