Difference between revisions of "Disjoint sum of partially ordered sets"
From Encyclopedia of Mathematics
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''disjoint sum of posets'' | ''disjoint sum of posets'' | ||
| − | Let | + | Let $P$ and $Q$ be two partially ordered sets (cf. [[Partially ordered set|Partially ordered set]]). |
| − | The disjoint sum | + | 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 | + | 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==== | ====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> | <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> | ||
Revision as of 22:01, 14 April 2014
disjoint sum of posets
Let $P$ and $Q$ be two partially ordered sets (cf. 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
| [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: http://encyclopediaofmath.org/index.php?title=Disjoint_sum_of_partially_ordered_sets&oldid=31701
Disjoint sum of partially ordered sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjoint_sum_of_partially_ordered_sets&oldid=31701
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article