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criteria for a poset to be a complete lattice
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(Theorem)
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Proposition. Let be a poset. Then the following are equivalent.
is a complete lattice.
- for every subset
of , exists.
- for every finite subset
of and every directed set of , and exist.
Proof. 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 :
for all 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 . 
Remarks.
- 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.
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Cross-references: closure system, infimum, iff, lower bounds, empty set, arbitrary meet, proper subset, upper bound, bounded, least upper bound, Zorn's lemma, maximal element, chain, well-defined, clear, implications, directed set, finite, subset, complete lattice, the following are equivalent, poset, proposition
There are 5 references to this entry.
This is version 4 of criteria for a poset to be a complete lattice, born on 2007-01-27, modified 2007-07-27.
Object id is 8832, canonical name is CriteriaForAPosetToBeACompleteLattice.
Accessed 762 times total.
Classification:
| AMS MSC: | 06B23 (Order, lattices, ordered algebraic structures :: Lattices :: Complete lattices, completions) | | | 03G10 (Mathematical logic and foundations :: Algebraic logic :: Lattices and related structures) |
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Pending Errata and Addenda
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