Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete.

Complete lattices appear in many applications in mathematics and computer science. Both order theory and universal algebra study them as a special class of lattices.

Complete lattices must not be confused with complete partial orders (CPOs), a more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales).^{[citation needed]}

A complete lattice is a partially ordered set (L, ≤) such that every subset A of L has both a greatest lower bound (the infimum, or meet) and a least upper bound (the supremum, or join) in (L, ≤).

The meet is denoted by ${\displaystyle \bigwedge A}$, and the join by ${\displaystyle \bigvee A}$.

In the special case where A is the empty set, the meet of A is the greatest element of L. Likewise, the join of the empty set is the least element of L. Then, complete lattices form a special class of bounded lattices.

A sublattice M of a complete lattice L is called a complete sublattice of L if for every subset A of M the elements ${\displaystyle \bigwedge A}$ and ${\displaystyle \bigvee A}$, as defined in L, are actually in M.

If the above requirement is lessened to require only non-empty meet and joins to be in M, the sublattice M is called a closed sublattice of L.