Disjoint sum of partially ordered sets

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

2020 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
How to Cite This Entry:
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