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Compact elements are important in domain theory, where they are considered as a kind of primitive element: the information represented by compact elements cannot be obtained by any approximation that does not already contain this knowledge. Especially compact elements cannot be approximated by elements strictly below them. On the other hand, it may happen that all non-compact elements can be obtained as directed suprema of compact elements. This is a desirable situation, since the set of compacts is often smaller than the original poset -- the examples below illustrate this. Posets that can be recovered from their set of compact elements are called algebraic poset s.

1 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 orders or complete lattices 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 realIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may unit interval [0,1] shows.

2 Examples

• The most basic example is obtained by considering the powerset of some set, ordered by subset inclusionIf X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes X; Y ⊇ X. Every set Y is a subset of itself. A subset of Y which is not equa. Within this complete lattice, the compact elements are exactly the finite sets. This justifies the name "finite element".

• The term "compact" is explained by considering the complete lattices of open setIn topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements ofs of some topological spaceTopological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies. Within this order, the compact elements are just the compact setIn mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. Heine-Borel Theorem In R n a compact set is both closed and bounded. Note that a set within any colles. Indeed, the condition for compactness in join-semilattices translates immediately to the corresponding definition.