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Difference between revisions of "Disjoint sum of partially ordered sets"

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''disjoint sum of posets''
 
''disjoint sum of posets''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202202.png" /> be two partially ordered sets (cf. [[Partially ordered set|Partially ordered set]]).
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Let $P$ and $Q$ be two partially ordered sets (cf. [[Partially ordered set|Partially ordered set]]).
  
The disjoint sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202203.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202205.png" /> is the disjoint union of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202207.png" /> with the original ordering on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d1202209.png" /> 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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d12022010.png" /> with partial ordering
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120220/d12022011.png" /></td> </tr></table>
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$$(p,q)\geq(p',q')\Leftrightarrow p\geq p',q\geq q'.$$
  
 
====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=15649
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article