Order Ideals and Filters
Let be a poset. A subset of is said to be an order ideal if
is a lower set: , and
An order ideal is also called an ideal for short. An ideal is said to be principal if it has the form for some .
Given a subset of a poset , we say that is the ideal generated by if is the smallest order ideal (of ) containing . is denoted by . Note that exists iff is a directed set. In particular, for any , is the ideal generated by . Also, if is an upper semilattice, then for any , let be the set of finite joins of elements of , then is a directed set, and .
Dually, an order filter (or simply a filter) in is a non-empty subset which is both an upper set and a filtered set (every pair of elements in has a lower bound in ). A principal filter is a filter of the form for some .
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 in an upper semilattice is a semilattice ideal if
if , then (condition for being an upper subsemilattice)
if and , then
Then the two definitions are equivalent: if is an upper semilattice, then is a semilattice ideal iff is an order ideal of : if is a semilattice ideal, then is clearly a lower and directed (since is an upper bound of and ); if is an order ideal, then condition 2 of a semilattice ideal is satisfied. If , then there is a that is an upper bound of and . Since is lower, and , we have .
Going one step further, we see that if is a lattice, then a lattice ideal is exactly an order ideal: if is a lattice ideal, then it is clearly an upper subsemilattice, and if , then also, so that is a semilattice ideal. On the other hand, if is a semilattice ideal, then is an upper subsemilattice, as well as a lower subsemilattice, for if , then as well since . This shows that 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 in an upper semilattice is the following: iff .
|Date of creation||2013-03-22 17:01:14|
|Last modified on||2013-03-22 17:01:14|
|Last modified by||CWoo (3771)|