criteria for a poset to be a complete lattice
Implications are clear. We will show
If , then by definition. So assume be a non-empty subset of . Let be the set of all finite subsets of and . By assumption, is well-defined and . Next, let be the set of all directed subsets of , and . By assumption again, is well-defined and . Now, every chain in has a maximal element in (since a chain is a directed set), itself has a maximal element by Zorn’s Lemma. We will show that is the least upper bound of elments of . It is clear that each is bounded above by (). If is an upper bound of elements of , then it is an upper bound of elements of , and hence an upper bound of elements of , which means .
By assumption exists (), so that . Now suppose is a proper subset of . We want to show that exists. If , then by definition of an arbitrary meet over the empty set. So assume . Let be the set of lower bounds of : and let , the least upper bound of . exists by assumption. Since is a set of upper bounds of , for all . This means that is a lower bound of elements of , or . If is any lower bound of elements of , then , since is bounded above by (). This shows that exists and is equal to . ∎
Dually, a poset is a complete lattice iff every subset has an infimum iff infimum exists for every finite subset and every directed subset.
The above proposition shows, for example, that every closure system is a complete lattice.
|Title||criteria for a poset to be a complete lattice|
|Date of creation||2013-03-22 16:37:53|
|Last modified on||2013-03-22 16:37:53|
|Last modified by||CWoo (3771)|