Disjoint sum of partially ordered sets
2010 Mathematics Subject Classification: Primary: 06A [MSN][ZBL]
disjoint sum of posets
Let $P$ and $Q$ be two partially ordered sets.
The disjoint sum $P+Q$ of $P$ and $Q$ is the disjoint union of the sets $P$ and $Q$ with the original ordering on $P$ and $Q$ and no other comparable pairs. A poset is disconnected if it is (isomorphic to) the disjoint sum of two sub-posets. Otherwise it is connected. The maximal connected sub-posets are called components.
The disjoint sum is the direct sum in the category of posets and order-preserving mappings. The direct product in this category is the Cartesian product $P\times Q$ with partial ordering
$$(p,q)\geq(p',q')\Leftrightarrow p\geq p',q\geq q'.$$
|[a1]||W.T. Trotter, "Partially ordered sets" R.L. Graham (ed.) M. Grötschel (ed.) L. Lovász (ed.) , Handbook of Combinatorics , I , North-Holland (1995) pp. 433–480|
Disjoint sum of partially ordered sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjoint_sum_of_partially_ordered_sets&oldid=35409