In mathematics, a **complete lattice** is a partially ordered set in which *all* subsets have both a supremum (join) and an infimum (meet). Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra.

Complete lattices must not be confused with complete partial orders (*cpo*s), which constitute a strictly more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (*locales*).

Read more about Complete Lattice: Formal Definition, Examples, Morphisms of Complete Lattices, Representation, Further Results

### Other articles related to "complete lattice, lattice, complete, complete lattices, lattices":

... If S is a partially ordered set, a completion of S means a

**complete lattice**L with an order-embedding of S into L ... The notion of a

**complete lattice**means that every subset of elements of L has a unique least upper bound and a unique greatest lower bound this generalizes ... S is embedded in this

**lattice**by mapping each element x to the lower set of elements that are less than or equal to x ...

... Nowadays, the term "

**complete**semilattice" has no generally accepted meaning, and various inconsistent definitions exist ... this immediately leads to partial orders that are in fact

**complete lattices**... Nevertheless, the literature on occasion still takes

**complete**join- or meet-semilattices to be

**complete lattices**...

**Complete Lattice**- Further Results

... there are some other statements that can be made about

**complete lattices**, or that take a particularly simple form in this case ... theorem, which states that the set of fixed points of a monotone function on a

**complete lattice**is again a

**complete lattice**...

... of all lower sets of a poset X, ordered by subset inclusion, yields a

**complete lattice**D(X) (the downset-

**lattice**) ... now shows that e has a lower adjoint if and only if X is a

**complete lattice**... situation occurs whenever this supremum map is also an upper adjoint in this case the

**complete lattice**X is constructively completely distributive ...

... of a superintuitionistic logic L ordered by inclusion forms a

**complete lattice**, denoted ExtL ... Similarly, the set of normal extensions of a modal logic M is a

**complete lattice**NExtM ... ρM, τL, and σL can be considered as mappings between the

**lattices**ExtIPC and NExtS4 It is easy to see that all three are monotone, and is the identity function on ExtIPC ...

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