Namespaces
Variants
Actions

Difference between revisions of "Ordered sum"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(link)
 
(2 intermediate revisions by the same user not shown)
Line 1: Line 1:
''of partially ordered sets''
+
{{TEX|done}}{{MSC|06A}}
  
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]]
+
''of [[partially ordered set]]s''
  
<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>
+
An operation which associates with a system of disjoint partially ordered sets $\{P_\alpha : \alpha \in L \}$, where the index set $L$ is also partially ordered, a new [[partially ordered set]]
 
+
$$
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" />.
+
P = \coprod_{\alpha \in L} P_\alpha
 +
$$
 +
the underlying set being the [[disjoint union]] of the sets $\{P_\alpha : \alpha \in L \}$, with order defined as follows. On the set $P$ one has $a \le b$ if and only if either $a,b \in P_\alpha$ and $a \le b$ in $P_\alpha$, for some $\alpha$, or $a \in P_\alpha$, $b \in P_\beta$ and $\alpha < \beta$ in $L$. Important particular cases of ordered sums are the ''cardinal'' and ''ordinal'' sums. The first of these is obtained when $L$ is a [[trivially ordered set]], i.e. each of its elements is comparable only to itself, and the second when $L$ is a [[totally ordered set]]. Thus, in the cardinal sum of two disjoint partially ordered sets $X$ and $Y$ the relation $x \le y$ retains its meaning in the components $X$ and $Y$, while $x \in X$ and $y \in Y$ are incomparable; in the ordinal sum of $X$ and $Y$ the order relation is again preserved in the components and $x < y$ for all $x \in X$, $y \in Y$.
  
 
====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>

Latest revision as of 19:36, 6 December 2014

2020 Mathematics Subject Classification: Primary: 06A [MSN][ZBL]

of partially ordered sets

An operation which associates with a system of disjoint partially ordered sets $\{P_\alpha : \alpha \in L \}$, where the index set $L$ is also partially ordered, a new partially ordered set $$ P = \coprod_{\alpha \in L} P_\alpha $$ the underlying set being the disjoint union of the sets $\{P_\alpha : \alpha \in L \}$, with order defined as follows. On the set $P$ one has $a \le b$ if and only if either $a,b \in P_\alpha$ and $a \le b$ in $P_\alpha$, for some $\alpha$, or $a \in P_\alpha$, $b \in P_\beta$ and $\alpha < \beta$ in $L$. Important particular cases of ordered sums are the cardinal and ordinal sums. The first of these is obtained when $L$ is a trivially ordered set, i.e. each of its elements is comparable only to itself, and the second when $L$ is a totally ordered set. Thus, in the cardinal sum of two disjoint partially ordered sets $X$ and $Y$ the relation $x \le y$ retains its meaning in the components $X$ and $Y$, while $x \in X$ and $y \in Y$ are incomparable; in the ordinal sum of $X$ and $Y$ the order relation is again preserved in the components and $x < y$ for all $x \in X$, $y \in Y$.

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