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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 $\langle A\rangle$. Note that $\langle A\rangle$ 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 $\langle A\rangle=\downarrow\!\!A^{{\prime}}$.
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 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 $b\leq a$, 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 $a\vee b\leq c$, 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 $b\leq a\in I$, 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 $a\wedge b\leq a$. 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 $a\vee b\in I$.
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