In mathematics, a lattice can be either of two things: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

• 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 In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In the mathematical discipline known as order theory, a Hasse diagram (pronounced HAHS uh, named after Helmut Hasse (1898–1979)) is a simple picture of a finite partially ordered set, forming a drawing of the transitive reduction of the partial order. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... See lattice for other meanings of this term, both within and without mathematics. ... In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1. ... In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech ( 16 (1964), 657--682). ... In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Niemeier. ... A translation slides an object by a vector a: Ta(p) = p + a. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... The ordinary meaning of lattice is the basis for several technical usages A cherry lattice pastry A mathematical lattice that is a type of partially ordered set. ...