**Lattices**

Further interesting examples for Galois connections are described in the article on completeness properties. It turns out that the usual functions and are adjoints in two suitable Galois connections. The same is true for the mappings from the one element set that point out the least and greatest elements of a partial order. Going further, even complete lattices can be characterized by the existence of suitable adjoints. These considerations give some impression of the ubiquity of Galois connections in order theory.

Read more about this topic: Galois Connection, Examples, Order Theory

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

Chabauty Topology

... The intuitive idea may be seen in the case of the set of all

... The intuitive idea may be seen in the case of the set of all

**lattices**in a Euclidean space E ... limiting cases or degenerating a certain sequence of**lattices**... One can find linear subspaces or discrete groups that are**lattices**in a subspace, depending on how one takes a limit ...List Of First-order Theories -

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**Lattices**...

**Lattices**can be considered either as special sorts of partially ordered sets, with a signature consisting of one binary relation symbol ≤, or as algebraic structures with a signature ... For two binary operations the axioms for a**lattice**are Commutative laws Associative laws Absorption laws For one relation ≤ the axioms are Axioms stating ≤ is a partial order, as above ... of c=a∨b) First order properties include (distributive**lattices**) (modular**lattices**) Completeness is not a first order property of**lattice**...Smith–Minkowski–Siegel Mass Formula - Examples - Dimension

... This implies that there are more than 80 million even unimodular

*n*= 32... This implies that there are more than 80 million even unimodular

**lattices**of dimension 32, as each has automorphism group of order at least 2 so contributes at most 1/2 to the mass ... King (2003) showed that there are more than a billion such**lattices**... In higher dimensions the mass, and hence the number of**lattices**, increases very rapidly ...Lattice (discrete Subgroup) -

... Arithmetic

*S*-arithmetic**Lattices**... Arithmetic

**lattices**admit an important generalization, known as the S-arithmetic**lattices**... diagonally embedded subgroup This is a**lattice**in the product of algebraic groups over different local fields, both real and p-adic ... Under fairly general assumptions, this construction indeed produces a**lattice**...Lattice-based Cryptography - History

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**Lattices**were first studied by mathematicians Joseph Louis Lagrange and Carl Friedrich Gauss ...**Lattices**have been used recently in computer algorithms and in cryptanalysis ... In 1996, Miklós Ajtai showed in a seminal result the use of**lattices**as a cryptography primitive ...Main Site Subjects

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