Difference between revisions of "Ordered sum"
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P = \coprod_{\alpha \in L} P_\alpha | P = \coprod_{\alpha \in L} P_\alpha | ||
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| − | 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 trivially ordered, 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 $x \in X$, $y \in Y$. | + | 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==== | ||
Latest revision as of 19:36, 6 December 2014
2020 Mathematics Subject Classification: Primary: 06A [MSN][ZBL]
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) |
Ordered sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_sum&oldid=35417