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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.
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).
Lattice (group), a repeating arrangement of points Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure; Lattice (module), a module over a ring that is embedded in a vector space over a field; Lattice graph, a graph that can be drawn within a repeating arrangement of points
The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2 1 is a 180° (twofold) rotation followed by a translation of 1 / 2 of the lattice vector. 3 1 is a 120° (threefold) rotation followed by a translation of 1 / 3 of ...
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 .
a lattice (group), a discrete subgroup of R n and its generalizations; a lattice ordered group, a group that with a partial ordering that is a lattice order
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).
The smallest group of particles in material that constitutes this repeating pattern is the unit cell of the structure. The unit cell completely reflects symmetry and structure of the entire crystal, which is built up by repetitive translation of unit cell along its principal axes. The translation vectors define the nodes of Bravais lattice.