# 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). 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 (cpos), 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":

Dedekind–Mac Neille Completion - Order Embeddings and Lattice Completions
... 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 ...
Complete Semilattices
... 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 ...
Completeness (order Theory) - Completeness in Terms of Adjunctions
... 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 ...
Modal Companion - Blok–Esakia Isomorphism
... 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 ...

### Famous quotes containing the word complete:

For us to go to Italy and to penetrate into Italy is like a most fascinating act of self-discovery—back, back down the old ways of time. Strange and wonderful chords awake in us, and vibrate again after many hundreds of years of complete forgetfulness.
—D.H. (David Herbert)