Compact element

Summary

A compact element of a partially ordered set is an element with this property. Whenever it lies below the supremum of a directed set, it already lies below some member of that set. In a complete lattice this means: whenever it lies below a join, it lies below the join of finitely many of the joined elements. Compact elements play the role of finite or finitely generated objects, and they are the building blocks of algebraic lattices and of Scott domains.