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''of partially ordered sets''
 
''of partially ordered sets''
  
An operation which associates with a system of disjoint partially ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701701.png" />, where the index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701702.png" /> is also partially ordered, a new [[Partially ordered set|partially ordered set]]
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An operation which associates with a system of disjoint partially ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701701.png" />, where the index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701702.png" /> is also partially ordered, a new [[partially ordered set]]
  
 
<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/o/o070/o070170/o0701703.png" /></td> </tr></table>
 
<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/o/o070/o070170/o0701703.png" /></td> </tr></table>
  
the elements of which are the elements of the set-theoretical union of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701704.png" />, with order defined as follows. On the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701705.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701706.png" /> if and only if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701708.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701709.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017012.png" />. Important particular cases of ordered sums are the cardinal and ordinal sums. The first of these is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017013.png" /> is trivially ordered, i.e. each of its elements is comparable only to itself, and the second when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017014.png" /> is a [[Totally ordered set|totally ordered set]]. Thus, in the cardinal sum of two disjoint partially ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017016.png" /> the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017017.png" /> retains its meaning in the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017019.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017021.png" /> are incomparable; in the ordinal sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017023.png" /> the order relation is again preserved in the components and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017026.png" />.
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the elements of which are the elements of the set-theoretical [[disjoint union]] of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701704.png" />, with order defined as follows. On the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701705.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701706.png" /> if and only if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701708.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o0701709.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017012.png" />. Important particular cases of ordered sums are the cardinal and ordinal sums. The first of these is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017013.png" /> is trivially ordered, i.e. each of its elements is comparable only to itself, and the second when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017014.png" /> is a [[Totally ordered set|totally ordered set]]. Thus, in the cardinal sum of two disjoint partially ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017016.png" /> the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017017.png" /> retains its meaning in the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017019.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017021.png" /> are incomparable; in the ordinal sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017023.png" /> the order relation is again preserved in the components and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070170/o07017026.png" />.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR>
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</table>

Revision as of 18:57, 6 December 2014

of partially ordered sets

An operation which associates with a system of disjoint partially ordered sets , where the index set is also partially ordered, a new partially ordered set

the elements of which are the elements of the set-theoretical disjoint union of the sets , with order defined as follows. On the set one has if and only if either and in or , and . Important particular cases of ordered sums are the cardinal and ordinal sums. The first of these is obtained when is trivially ordered, i.e. each of its elements is comparable only to itself, and the second when is a totally ordered set. Thus, in the cardinal sum of two disjoint partially ordered sets and the relation retains its meaning in the components and , while and are incomparable; in the ordinal sum of and the order relation is again preserved in the components and for all , .

References

[1] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2] L.A. Skornyakov, "Elements of lattice theory" , Hindushtan Publ. Comp. (1977) (Translated from Russian)
How to Cite This Entry:
Ordered sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_sum&oldid=14587
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article