Formal definition
In a partially ordered set (''P'',≤) an element ''c'' 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 everyExamples
* The most basic example is obtained by considering the power set of some set ''A'', ordered by subset inclusion. Within this complete lattice, the compact elements are exactly the finite subsets of ''A''. This justifies the name "finite element". * The term "compact" is inspired by the definition of (topologically) compact subsets of aAlgebraic posets
A poset in which every element is the supremum of the compact elements below it is called an ''algebraic poset''. Such posets that are dcpos are much used in domain theory. As an important special case, an ''algebraic lattice'' is a complete lattice ''L'' where every element ''x'' of ''L'' is the supremum of the compact elements below ''x''. A typical example (which served as the motivation for the name "algebraic") is the following: For any algebra ''A'' (for example, a group, a ring, a field, a lattice, etc.; or even a mere set without any operations), let Sub(''A'') be the set of all substructures of ''A'', i.e., of all subsets of ''A'' which are closed under all operations of ''A'' (group addition, ring addition and multiplication, etc.). Here the notion of substructure includes the empty substructure in case the algebra ''A'' has no nullary operations. Then: * The set Sub(''A''), ordered by set inclusion, is a lattice. * The greatest element of Sub(''A'') is the set ''A'' itself. * For any ''S'', ''T'' in Sub(''A''), the greatest lower bound of ''S'' and ''T'' is the set theoretic intersection of ''S'' and ''T''; the smallest upper bound is the subalgebra generated by the union of ''S'' and ''T''. * The set Sub(''A'') is even a complete lattice. The greatest lower bound of any family of substructures is their intersection (or ''A'' if the family is empty). * The compact elements of Sub(''A'') are exactly the finitely generated substructures of ''A''. * Every substructure is the union of its finitely generated substructures; hence Sub(''A'') is an algebraic lattice. Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub(''A'') for some algebra ''A''. There is another algebraic lattice that plays an important role in universal algebra: For every algebra ''A'' we let Con(''A'') be the set of allApplications
Compact elements are important in computer science in the semantic approach called 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. 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 compact elements is often smaller than the original poset—the examples above illustrate this.Literature
See the literature given for order theory and domain theory. Order theory