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 non-empty, 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 $⟨A⟩$. Note that $⟨A⟩$ 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 $⟨A⟩=\downarrow A^{\prime }$.

Dually, an order filter (or simply a 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 $F$). 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	2013-03-22 17:01:14
Last modified on	2013-03-22 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