Lattice

In colloquial usage, a lattice is a structure of crossed laths with open spaces left between them. The term is used in various technical senses, all of which have some geometrical relation to the dictionary definition.

• In one mathematical usage, a lattice is a partially ordered set (poset) in which any two elements have a supremum and an infimum. The Hasse diagrams of these posets look (in some simple cases) like the aforementioned lattices.

• In another mathematical usage, a lattice is a discrete subgroup of Rⁿ that spans Rⁿ as a real vector space. The elements of a lattice are regularly spaced, reminiscent of the intersection points of a lath lattice. See also unimodular lattice, Leech lattice, Niemeier lattice, lattice point problems.
  • This concept is used in materials science, in which a lattice is a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal.
  • It also occurs in computational physics, in which a lattice is an n-dimensional geometrical structure of sites, connected by bonds, which represent positions which may be occupied by atoms, molecules, electrons, spins, etc. For an article dealing with the formal representation of such structures see Lattice Geometries. Different types of lattice models are used in physics, mathematics, and finance.

• In digital signal processing, lattice filters are electronic filters with a special recursive structure.

• In music lattices are often used to model pitch space, especially just intonation. See: TonalSoft.com: Lattices Text and diagrams © 1998 by Joseph L. Monzo.

• Lattice Corporation made some of the first compilers for PCs.

• Lattice Semiconductor [1] is a company which makes FPGAs and related electronics products.

• In baking, various pastries and pies can be made with a lattice pattern, and special cutters are available to facilitate the creation of such patterns.