For duals of order-theoretic lattices, see Duality (order theory).
Algebraic structure → Group theory Group theory
Basic notions
Subgroup
Normal subgroup
Quotient group
(Semi-)direct product
Group homomorphisms
kernel
image
direct sum
wreath product
simple
finite
infinite
continuous
multiplicative
additive
cyclic
abelian
dihedral
nilpotent
solvable
action
Glossary of group theory
List of group theory topics
Finite groups
Cyclic group Zn
Symmetric group Sn
Alternating group An
Dihedral group Dn
Quaternion group Q
Cauchy's theorem
Lagrange's theorem
Sylow theorems
Hall's theorem
p-group
Elementary abelian group
Frobenius group
Schur multiplier
Classification of finite simple groups
cyclic
alternating
Lie type
sporadic
Discrete groups
Lattices
Integers ()
Free group
Modular groups
PSL(2, )
SL(2, )
Arithmetic group
Lattice
Hyperbolic group
Topological and Lie groups
Solenoid
Circle
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Euclidean E(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
Symplectic Sp(n)
G2
F4
E6
E7
E8
Lorentz
Poincaré
Conformal
Diffeomorphism
Loop
Infinite dimensional Lie group
O(∞)
SU(∞)
Sp(∞)
Algebraic groups
Linear algebraic group
Reductive group
Abelian variety
Elliptic curve
v
t
e
In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice is the reciprocal of the geometry of , a perspective which underlies many of its uses.
Dual lattices have many applications inside of lattice theory, theoretical computer science, cryptography and mathematics more broadly. For instance, it is used in the statement of the Poisson summation formula, transference theorems provide connections between the geometry of a lattice and that of its dual, and many lattice algorithms exploit the dual lattice.
For an article with emphasis on the physics / chemistry applications, see Reciprocal lattice. This article focuses on the mathematical notion of a dual lattice.
theory of lattices, the duallattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the duallattice of a...
In physics, the reciprocal lattice emerges from the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in...
In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins. The model is notable...
of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, Modular law a ≤ b implies a ∨ (x ∧...
discoveries. Simple duality in two dimensions implies that all fully triangulated lattices (e.g., the triangular, union jack, cross dual, martini dual and asanoha...
matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the...
of the lattice is 1). Equivalently, Γ8 is self-dual, meaning it is equal to its duallattice. It is even, meaning that the norm of any lattice vector...
mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley...
above condition is equivalent to its dual: x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) for all x, y, and z in L. In every lattice, if one defines the order relation...
the dual ordering ≥. Semilattices are employed to construct other order structures, or in conjunction with other completeness properties. A lattice is...
mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice that satisfies at...
mechanical description of an elementary vibrational motion in which a lattice of atoms or molecules uniformly oscillates at a single frequency. In classical...
action, lattice gauge theory can be shown to be exactly dual to spin foam models. Hamiltonian lattice gauge theory Lattice field theory Lattice QCD Quantum...
compactified on an even, self-duallattice (a discrete subgroup of a linear space). There are two possible even self-duallattices in 16 dimensions, and it...
viewing from the duallattice, each frustrated edge must be "covered" by a 1x2 rectangle, such that the rectangles span the entire lattice and do not overlap...
distributive lattices via ordered topologies: Priestley's representation theorem for distributive lattices. Many other Stone-type dualities could be added...
≡ ... ≡ xn (mod n). This is the lattice A n − 1 ∗ {\displaystyle A_{n-1}^{*}} , the duallattice of the root lattice A n − 1 {\displaystyle A_{n-1}} ...
Global Information
