order ideal

Order Ideals and Filters

Let P be a poset. A subset I of P is said to be an order ideal if

An order ideal is also called an ideal for short. An ideal is said to be principal if it has the form x for some xP.

Given a subset A of a poset P, we say that B is the ideal generated byPlanetmathPlanetmath 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 xP, x is the ideal generated by x. Also, if P is an upper semilatticePlanetmathPlanetmath, then for any AP, let A be the set of finite joins of elements of A, then A is a directed set, and A=A.

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 filterPlanetmathPlanetmath is a filter of the form x for some xP.

Remark. This is a generalizationPlanetmathPlanetmath 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. 1.

    if a,bI, then abI (condition for being an upper subsemilattice)

  2. 2.

    if aI and ba, then bI

Then the two definitions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath: if P is an upper semilattice, then IP 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 ab 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,bI, then there is a cI that is an upper bound of a and b. Since I is lower, and abc, we have abI.

Going one step further, we see that if P is a latticeMathworldPlanetmath, then a lattice ideal is exactly an order ideal: if I is a lattice ideal, then it is clearly an upper subsemilattice, and if baI, then b=abI 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 aI, then abI as well since aba. This shows that I is a lattice ideal.

Dually, we can define a filter in a lower semilatticePlanetmathPlanetmath, 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,bI iff abI.

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 idealMathworldPlanetmathPlanetmath
Defines principal filter