Disjoint sum of partially ordered sets

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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
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Disjoint sum of partially ordered sets. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article