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: ↓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 ↓x for some x∈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∈P, ↓x is the ideal generated by x. Also, if P is an upper semilattice
, then for any A⊆P, 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 filter is a filter of the form ↑x for some x∈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∈I, then a∨b∈I (condition for being an upper subsemilattice)
-
2.
if a∈I and b≤a, then b∈I
Then the two definitions are equivalent: if P is an upper semilattice, then I⊆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∨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∈I, then there is a c∈I that is an upper bound of a and b. Since I is lower, and a∨b≤c, we have a∨b∈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≤a∈I, then b=a∧b∈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∈I, then a∧b∈I as well since a∧b≤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∈I iff a∨b∈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 |