View source for Disjoint sum of partially ordered sets

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''disjoint sum of posets''

Let $P$ and $Q$ be two partially ordered sets (cf. [[Partially ordered set|Partially ordered set]]).

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

$$(p,q)\geq(p',q')\Leftrightarrow p\geq p',q\geq q'.$$

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR></table>

[[Category:Order, lattices, ordered algebraic structures]]