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'order ideal'
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| Title of object: |
order ideal |
| Canonical Name: |
OrderIdeal |
| Type: |
Definition |
| Created on: |
2007-04-30 20:58:54 |
| Modified on: |
2007-05-01 00:23:51 |
| Classification: |
msc:06A06, msc:06A12 |
| Defines: |
order filter, semilattice ideal, semilattice filter, subsemilattice |
| Synonyms: |
order ideal=filter
order ideal=ideal |
Preamble:
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Content:
Let $P$ be a poset. A subset $I$ of $P$ is said to be an \emph{order ideal} if
\begin{itemize}
\item $I$ is a lower set: $\down I=I$, and
\item $I$ is a directed set: $I$ is non-empty, and every pair of elements in $I$ has an upper bound in $I$.
\end{itemize}
An order ideal is also called an ideal for short.
Dually, an \emph{order filter} (or simply a \emph{filter}) in $P$ is a non-empty subset $F$ which is both an upper set and a filtered set (every pair of elements in $F$ has a lower bound in $F$).
\textbf{Remark}. This is a generalization of the notion of a \PMlinkname{filter}{Filter} in a set. In fact, both ideals and filters are generalizations of ideals and filters in semilattices and lattices.
For example, a subset $I$ in an upper semilattice $P$ is a \emph{semilattice ideal} if
\begin{enumerate}
\item
if $a,b\in I$, then $a\vee b\in I$ (condition for being an upper subsemilattice)
\item
if $a\in I$ and $b\le a$, then $b\in I$
\end{enumerate}
Then the two definitions are equivalent: if $P$ is an upper semilattice, then $I\subseteq P$ is a semilattice ideal iff $I$ is an order ideal of $P$: if $I$ is a semilattice ideal, then $I$ is clearly a lower and directed (since $a\vee b$ is an upper bound of $a$ and $b$); if $I$ is an order ideal, then condition 2 of a semilattice ideal is satisfied. If $a,b\in I$, then there is a $c\in I$ that is an upper bound of $a$ and $b$. Since $I$ is lower, and $a\vee b\le c$, we have $a\vee b\in I$.
Going one step further, we see that if $P$ is a lattice, then a lattice ideal is exactly an order ideal: if $I$ is a lattice ideal, then it is clearly an upper subsemilattice, and if $b\le a\in I$, then $b=a\wedge b\in I$ also, so that $I$ is a semilattice ideal. On the other hand, if $I$ is a semilattice ideal, then $I$ is an upper subsemilattice, as well as a lower subsemilattice, for if $a\in I$, then $a\wedge b\in I$ as well since $a\wedge b\le a$. This shows that $I$ is a lattice ideal.
Dually, we can define a \emph{filter} in a lower semilattice, which is equivalent to an order filter of the underly poset. Going one step futher, we also see that a lattice filter in a lattice is an order filter of the underlying poset.
\textbf{Remark}. An alternative but equivalent characterization of a semilattice ideal $I$ in an upper semilattice $P$ is the following: $a,b\in I$ iff $a\vee b\in I$. |
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