Lattice (mathematics)
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In mathematics, a lattice can be either of two things:
- In one usage, a lattice is a partially ordered set (poset) in which any two elements have a supremum and an infimum—see lattice (order). The Hasse diagrams of these posets look (in some simple cases) like the lattices of ordinary language. These lattices can also be defined as algebraic structures.
- In another usage, a Euclidean lattice is a discrete subgroup of
that spans as a real vector space, or a translate of a discrete subgroup—see lattice (group). The elements of a lattice are regularly spaced, reminiscent of the intersection points of a lath lattice. See also below, and unimodular lattice, Leech lattice, Niemeier lattice, lattice point problems.
- Euclidean lattices arise from translational symmetry. For each point in we have a set of corresponding points, called a lattice, that consists of all translates of the original point. If the original point is the origin of coordinates, this lattice is a discrete subgroup of . In any case, it is a translate of a discrete subgroup.
- More generally, a lattice in a locally compact group is a discrete subgroup which is also cofinite, meaning here that the quotient of the group by the lattice has finite measure. (Note this makes sense because every locally compact group has a Haar measure which is unique up to scaling.) An important example of a lattice is the modular group
.
See also: lattice
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