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In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
Let be a locally compact group and a discrete subgroup (this means that there exists a neighbourhood of the identity element of such that = {}).Then is called a lattice in if in addition there exists a Borel measure on the quotient space / which is finite (i.e. (/) < +) and -invariant (meaning that for any and any open subset / the equality () = is satisfied).
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
In group theory, the correspondence theorem [1] [2] [3] [4] [5] [6] [7] [8] (also the lattice theorem, [9] and variously and ambiguously the third and fourth ...
A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms ...
Lattice-theoretic information about the lattice of subgroups can sometimes be used to infer information about the original group, an idea that goes back to the work of Øystein Ore (1937, 1938). For instance, as Ore proved , a group is locally cyclic if and only if its lattice of subgroups is distributive .
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The E 8 lattice and the Leech lattice are two famous examples.
3.3 Using group theory. 4 Velocity and effective mass. ... Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is ...