Difference between revisions of "Ordered sum"
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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 [[ | + | 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" />. | + | 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> | + | <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> |
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) |
Ordered sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_sum&oldid=35416