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FORMAL DEFINITION


For some partially ordered set (''P'',≤) an element ''c'' of ''P'' is called ''compact'' (or ''finite'') if it satisfies one of the following equivalent conditions:
  • For every Directed Subset ''D'' of ''P'', if ''D'' has a supremum sup ''D'' and ''c'' ≤ sup ''D'' then ''c'' ≤ ''d'' for some element ''d'' of ''D''.

  • For every Ideal ''I'' of ''P'', if ''I'' has a supremum sup ''I'' and ''c'' ≤ sup ''I'' than ''c'' is an element of ''I''.

  • The element ''c'' is Way Below itself, i.e. ''c'' << ''c''


If the poset ''P'' additionally is a Join-semilattice (i.e. if it has binary suprema) then these conditions are equivalent to the following statement:
  • For every subset ''S'' of ''P'', if ''S'' has a supremum sup ''S'' and ''c'' ≤ sup ''S'', then ''c'' ≤ sup ''T'' for some finite subset ''T'' of ''S''.


Using the definitions of the involved concepts these equivalences are easily verified. For the case of the join-semilattices note that any set can be turned into a directed set with the same supremum by closing under finite (non-empty) suprema.

When considering Directed Complete Partial Order s or Complete Lattice s the additional requirements that the specified suprema exist can of course be dropped. Note also that a join-semilattice which is directed complete is almost a complete lattice (possibly lacking a Least Element ) -- see Completeness (order Theory) for details.

If it exists, the least element of a poset is always compact. It may well be that this is the only compact element, as the example of the Real unit interval {Link without Title} shows.


EXAMPLES


  • The most basic example is obtained by considering the Powerset of some set, ordered by Subset Inclusion . Within this complete lattice, the compact elements are exactly the Finite Set s. This justifies the name "finite element".


  • The term "compact" is explained by considering the complete lattices of Open Set s of some Topological Space . Within this order, the compact elements are just the Compact Set s. Indeed, the condition for compactness in join-semilattices translates immediately to the corresponding definition.



LITERATURE


Compact elements are standard. See the literature given for Order Theory and Domain Theory .