order ideal
Order Ideals and Filters
Let $P$ be a poset. A subset $I$ of $P$ is said to be an order ideal if

•
$I$ is a lower set: $\downarrow I=I$, and

•
$I$ is a directed set^{}: $I$ is nonempty, and every pair of elements in $I$ has an upper bound^{} in $I$.
An order ideal is also called an ideal for short. An ideal is said to be principal if it has the form $\downarrow x$ for some $x\in P$.
Given a subset $A$ of a poset $P$, we say that $B$ is the ideal generated by^{} $A$ if $B$ is the smallest order ideal (of $P$) containing $A$. $B$ is denoted by $\u27e8A\u27e9$. Note that $\u27e8A\u27e9$ exists iff $A$ is a directed set. In particular, for any $x\in P$, $\downarrow x$ is the ideal generated by $x$. Also, if $P$ is an upper semilattice^{}, then for any $A\subseteq P$, let ${A}^{\prime}$ be the set of finite joins of elements of $A$, then ${A}^{\prime}$ is a directed set, and $\u27e8A\u27e9=\downarrow {A}^{\prime}$.
Dually, an order filter (or simply a filter) in $P$ is a nonempty subset $F$ which is both an upper set and a filtered set (every pair of elements in $F$ has a lower bound in $F$). A principal filter^{} is a filter of the form $\uparrow x$ for some $x\in P$.
Remark. This is a generalization^{} of the notion of a filter (http://planetmath.org/Filter) in a set. In fact, both ideals and filters are generalizations of ideals and filters in semilattices and lattices.
Examples in a Semilattice
A subset $I$ in an upper semilattice $P$ is a semilattice ideal if

1.
if $a,b\in I$, then $a\vee b\in I$ (condition for being an upper subsemilattice)

2.
if $a\in I$ and $b\le a$, then $b\in I$
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 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.
Remark. An alternative but equivalent characterization of a semilattice ideal $I$ in an upper semilattice $P$ is the following: $a,b\in I$ iff $a\vee b\in I$.
Title  order ideal 
Canonical name  OrderIdeal 
Date of creation  20130322 17:01:14 
Last modified on  20130322 17:01:14 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06A06 
Classification  msc 06A12 
Synonym  filter 
Synonym  ideal 
Related topic  Filter 
Related topic  LatticeFilter 
Related topic  LatticeIdeal 
Defines  order filter 
Defines  semilattice ideal 
Defines  semilattice filter 
Defines  subsemilattice 
Defines  principal ideal^{} 
Defines  principal filter 