View source for Disjoint sum of partially ordered sets

''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]]).

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.

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

<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>

====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>