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Construction analogous to that of a dual vector space
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
Dual_lattice
Fourier transform of a real-space lattice, important in solid-state physics
the dual of physical space considered as a vector space. In other words, the reciprocal lattice is the sublattice which is dual to the direct lattice. The
Reciprocal_lattice
Integral lattice of determinant 1 or –1
dual lattice is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual.
Unimodular_lattice
Set whose pairs have minima and maxima
subset and filter (dual notions) Skew lattice (generalization to non-commutative join and meet) Eulerian lattice Post's lattice – Lattice in universal algebra
Lattice_(order)
Model in statistical mechanics
\{\sigma \}} , a polygon is associated to the lattice by drawing a line on the edge of the dual lattice if the spins separated by the edge are unlike
Square_lattice_Ising_model
mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley
Duality theory for distributive lattices
Duality_theory_for_distributive_lattices
Type of lattice in mathematical order theory
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 ∧
Modular_lattice
Lattice in 8-dimensional space with special properties
of the lattice is 1). Equivalently, Γ8 is self-dual, meaning it is equal to its dual lattice. It is even, meaning that the norm of any lattice vector
E8_lattice
Periodic set of points
Coordinate-wise addition or subtraction of two points in the lattice produces another lattice point. The lattice points are all separated by some minimum distance
Lattice_(group)
Join-meet algebra on matroid flats
matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the
Geometric_lattice
General concept and operation in mathematics
Dual abelian variety Dual basis Dual (category theory) Dual code Dual lattice Dual norm Dual numbers, a certain associative algebra; the term "dual"
Duality_(mathematics)
Special type of lattice
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
Distributive_lattice
Term in the mathematical area of order theory
notions which are self-dual include: Being a (complete) lattice Monotonicity of functions Distributivity of lattices, i.e. the lattices for which ∀x,y,z: x
Duality_(order_theory)
Threshold of percolation theory models
discoveries. Simple duality in two dimensions implies that all fully triangulated lattices (e.g., the triangular, union jack, cross dual, martini dual and asanoha
Percolation_threshold
Equation in Fourier analysis
{f}}(\lambda ')e^{2\pi i\lambda 'x}} where Λ ′ {\displaystyle \Lambda '} is the dual lattice to Λ {\displaystyle \Lambda } . (Note that the Fourier series on the
Poisson_summation_formula
Banach space with a compatible structure of a lattice
of non-lattice Banach spaces are now known; James' space is one such. The continuous dual space of a Banach lattice is equal to its order dual. Every
Banach_lattice
Lattice in universal algebra
subset of either the monotone, affine, self-dual, truth-preserving, or false-preserving functions. Post's lattice consists of 9 named clones, two countably
Post's_lattice
Partial order with joins
the dual ordering ≥. Semilattices are employed to construct other order structures, or in conjunction with other completeness properties. A lattice is
Semilattice
Algebraic structure modeling logical operations
In mathematics, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties
Boolean_algebra_(structure)
Symmetry in statistical physics
The Kramers–Wannier duality is a symmetry in statistical physics. It relates the free energy of a two-dimensional square-lattice Ising model at a low temperature
Kramers–Wannier_duality
Distance-regular graph with 56 vertices
Grishukhin, V. P. (2011), "Delone and Voronoĭ polytopes of the root lattice E7 and the dual lattice E7*", Trudy Matematicheskogo Instituta imeni V. A. Steklova
Gosset_graph
this lattice of norm 4 (the shortest nonzero vectors in this lattice). The Coxeter–Todd lattice can be made into a 6-dimensional lattice self dual over
Coxeter–Todd_lattice
Lattice formed by all integer partitions
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative
Young's_lattice
Group controlling representation theory
root datum (X*, Δ,X*, Δv), where X* is the lattice of characters of a maximal torus, X* the dual lattice (given by the 1-parameter subgroups), Δ the
Langlands_dual_group
Partially ordered set in which all subsets have both a supremum and infimum
complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally complete lattice satisfies
Complete_lattice
geometric lattice and corresponds to a matroid of finite rank. Semimodular lattices are also known as upper semimodular lattices; the dual notion is that
Semimodular_lattice
Geometric arrangements of points, foundational to Lie theory
called the dual root system (or sometimes inverse root system). By direct calculation, α∨∨ = α, so that Φ is the dual root system of Φ∨. The lattice in E spanned
Root_system
Relationship between certain categories
distributive lattices via ordered topologies: Priestley's representation theorem for distributive lattices. Many other Stone-type dualities could be added
Stone_duality
Geometric construct
periodic lattice. At the ground state, each plaquette of the spin model must contain exactly one frustrated interaction. Therefore, viewing from the dual lattice
Domino_tiling
Theory of quantum gauge fields on a lattice
action, lattice gauge theory can be shown to be exactly dual to spin foam models. Hamiltonian lattice gauge theory Lattice field theory Lattice QCD Wilson
Lattice_gauge_theory
{\displaystyle X^{*}} is the lattice of characters of the maximal torus, X ∗ {\displaystyle X_{*}} is the dual lattice (given by the 1-parameter subgroups)
Root_datum
Equivalence of distributive lattices and set families
distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations
Birkhoff's representation theorem
Birkhoff's_representation_theorem
Quasiparticle of mechanical vibrations
vibrations in a lattice of atoms. Phonons have both wave and particle-like properties, in a way related to the wave–particle duality of quantum mechanics
Phonon
(triangular) face. Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B4, D4, or F4 lattice. Hexadecachoric
16-cell_honeycomb
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
L\setminus \{0\}} . Furthermore, such lattice can be self-dual. Even though Minkowski's theorem guarantees a short lattice vector within a certain magnitude
Minkowski's_theorem
Polytope whose vertices represent permutations
{n}}.\end{aligned}}} This is the lattice A n − 1 ∗ {\displaystyle A_{n-1}^{*}} , the dual lattice of the root lattice A n − 1 {\displaystyle A_{n-1}}
Permutohedron
lattice is a locally convex vector lattice. The strong dual of a normed lattice is a Banach lattice with respect to the dual norm and canonical order. If it
Normed_vector_lattice
The lattice spanned by the following matrix is isomorphic to the above. Indeed, the following generator matrix can be obtained as the dual lattice (up
Barnes–Wall_lattice
Linear error-correcting code
the E7 lattice and, in fact, can be used to construct it, or more precisely, its dual lattice E7∗ (a similar construction for E7 uses the dual code [7
Hamming(7,4)
Kind of complex manifold
\rangle \subseteq \mathbb {Z} \}} called the dual lattice of Λ {\displaystyle \Lambda } . Then, we can form the dual complex torus X ^ ≅ Ω ¯ / Λ ^ {\displaystyle
Complex_torus
Mathematical structure
symmetric bilinear form with orthonormal basis vi, the map sending a lattice to its dual lattice gives an automorphism whose square is the identity, giving the
Building_(mathematics)
Microprocessor RISC soft core
LatticeMico32 is a 32-bit microprocessor reduced instruction set computer (RISC) soft core from Lattice Semiconductor optimized for field-programmable
LatticeMico32
Type of uniform space-filling tessellation
cubic, the union of two 5-cube honeycombs in dual positions. ∪ ∪ ∪ = ∪ . The kissing number of the D* 5 lattice is 10 (2n for n≥5) and its Voronoi tessellation
5-demicubic_honeycomb
Physics concept of subatomic structure
compactified on an even, self-dual lattice (a discrete subgroup of a linear space). There are two possible even self-dual lattices in 16 dimensions, and it
Heterotic_string_theory
Polyhedron associated with another by swapping vertices for faces
rhombicosidodecahedron with original lattice, both on common midsphere. Dual of the symmetrohedron i,3,5,*,v with original lattice. Self-dual compound of two rhombic
Dual_polyhedron
Method of deriving an ontology
opposition is a dual weak complementation. A (bounded) lattice such as a concept algebra, which is equipped with a weak complementation and a dual weak complementation
Formal_concept_analysis
Bound lattice in which every element has a complement
the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every
Complemented_lattice
algebra, a skew lattice is an algebraic structure that is a non-commutative generalization of a lattice. While the term skew lattice can be used to refer
Skew_lattice
Every subgroup of a cyclic group is cyclic, and if finite, its order divides its parent's
infinite case, the lattice of subgroups of a cyclic group is isomorphic to the dual of a divisibility lattice. In the finite case, the lattice of subgroups
Subgroups_of_cyclic_groups
six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated
5-simplex_honeycomb
a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices. Completely distributive complete lattices (also called
Distributivity_(order_theory)
category of Heyting algebras is dually equivalent to the category of Heyting spaces. Duality theory for distributive lattices Esakia, Leo (1974). "Topological
Esakia_duality
In mathematics, vector space of linear forms
contravariance of vectors Dual module Dual norm Duality (mathematics) Duality (projective geometry) Pontryagin duality Reciprocal lattice – dual space basis, in
Dual_space
Correspondence between properties of a category and its opposite
or of duality applied to lattices. Limits and colimits are dual notions. Fibrations and cofibrations are examples of dual notions in algebraic topology
Dual_(category_theory)
algebra is a metric lattice; any finitely-additive measure on its Stone dual gives a valuation. Every metric lattice is a modular lattice, c.f. lower picture
Metric_lattice
Partially ordered vector space, ordered as a lattice
Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces
Riesz_space
order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering. The order dual of a vector lattice is
Order dual (functional analysis)
Order_dual_(functional_analysis)
Concept in order theory
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
Join_and_meet
absorbing sets. The strong dual of a locally convex vector lattice X {\displaystyle X} is an order complete locally convex vector lattice (under its canonical
Locally_convex_vector_lattice
Existence of certain infima or suprema of a given poset
special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist. The motivation
Completeness_(order_theory)
Geometric figure
4 lattice is the union of five A4 lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is
5-cell_honeycomb
Matroid with no linear representation
This means that the dual lattice of the geometric lattice of the Vámos matroid cannot be order-embedded into another geometric lattice of the same rank.
Vámos_matroid
Tiling of a plane by regular hexagons and equilateral triangles
this pattern has been taken up in physics, where it is called a kagome lattice. It occurs also in the crystal structures of certain minerals. Conway calls
Trihexagonal_tiling
Algebraic structure used in logic
lattices, although the latter term may denote the dual definition, or have a slightly more general meaning. A Heyting algebra H is a bounded lattice such
Heyting_algebra
Algebra used in 2D conformal field theories and string theory
must satisfy (s, λ) ∈ Z for all λ ∈ Λ, i.e., s lies in the dual lattice. If the even lattice Λ is generated by its "root vectors" (those satisfying (α
Vertex_operator_algebra
Subset of incomparable elements
inclusion, the antichains are called Sperner families and their lattice is a free distributive lattice, with a Dedekind number of elements. More generally, counting
Antichain
Matroid with graph forests as independent sets
n} -element set. Since the lattices of flats of matroids are exactly the geometric lattices, this implies that the lattice of partitions is also geometric
Graphic_matroid
Quasiregular space-filling tesselation
the union of two cubic honeycombs in dual positions. ∪ ∪ ∪ = dual of = ∪ . The kissing number of the D* 3 lattice is 8 and its Voronoi tessellation is
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
Lattice field theory with only spatial discretization
from the edges (the electric contribution). Hamiltonian lattice gauge theory is exactly dual to a theory of spin networks. This involves using the Peter–Weyl
Hamiltonian lattice gauge theory
Hamiltonian_lattice_gauge_theory
Partial differential equations whose solutions are instantons
is a similar duality between instantons invariant under dual lattices inside R 4 {\displaystyle \mathbb {R} ^{4}} , instantons on dual four-dimensional
Yang–Mills_equations
Nonempty, upper-bounded, downward-closed subset
P\setminus I} is a filter (which is then also prime, in the dual sense). For a complete lattice the further notion of a completely prime ideal is meaningful
Ideal_(order_theory)
Glossary of terms used in branch of mathematics
various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics
Glossary_of_order_theory
Concept in combinatorial mathematics
Unlike the lattice of all partitions of the set, the lattice of all noncrossing partitions is self-dual, i.e., it is order-isomorphic to the lattice that results
Noncrossing_partition
Cluster update algorithm
the edge opposite to it. If we were to track the closed bonds in the dual lattice, by drawing a straight/bent line inside each plaquette such that it intersects
KBD_algorithm
Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices. Various different characterizations
Completely distributive lattice
Completely_distributive_lattice
Mathematical concept
rather the topology generated by the union. Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element
Comparison_of_topologies
Subset of a preorder that contains all larger elements
{\displaystyle X} ordered with the inclusion relation is a complete lattice, the upper set lattice. Every upper set Y {\displaystyle Y} of a finite partially ordered
Upper_and_lower_sets
uniform dimension for modular lattices such that the hollow dimension of a module was the uniform dimension of its dual lattice of submodules. It is always
Uniform_module
Mathematical model of ferromagnetism in statistical mechanics
of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing
Ising_model
International hardware and software company
AMI, A Lattice Company, which was formerly known as American Megatrends, Inc., is an international hardware and software company, specializing in PC hardware
American_Megatrends
Amusement ride manufacturer
The Skycoaster® at Elitch Gardens in Denver, Colorado is a Dual Lattice 173-foot Skycoaster.
Ride_Entertainment_Group
Regular tiling of the plane
can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice. + + = dual of = The vertices of the triangular tiling are the
Triangular_tiling
Special subset of a partially ordered set
elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal. Special
Filter_(mathematics)
Theorem in order and lattice theory
lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following: Let (L, ≤) be a complete lattice and
Knaster–Tarski_theorem
Two-port electrical wave filter
A symmetrical lattice is a two-port electrical wave filter in which diagonally-crossed shunt elements are present – a configuration which sets it apart
Lattice_network
Topological vector lattice
Fréchet lattice is a topological vector lattice that is also a Fréchet space. Fréchet lattices are important in the theory of topological vector lattices. Every
Fréchet_lattice
Uniform 7-Honeycomb
of all four 7-demicubic lattices: It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions. ∪ ∪ ∪ = ∪ . The
7-demicubic_honeycomb
In mathematics, an algebraic structure
In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y that admits operations x\z
Residuated_lattice
52-dimensional exceptional simple Lie group
The F4 lattice is a four-dimensional body-centered cubic lattice (i.e. the union of two hypercubic lattices, each lying in the center
F4_(mathematics)
lattice is the union of four A7 lattices, which is identical to the E7* lattice (or E2 7). ∪ ∪ ∪ = + = dual of . The A* 7 lattice (also called A8 7) is the
7-simplex_honeycomb
honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of . This honeycomb is one of 45 unique uniform
8-simplex_honeycomb
Dense arrangement of congruent spheres in an infinite, regular arrangement
arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is
Close-packing of equal spheres
Close-packing_of_equal_spheres
Linear algebra concept
of V {\displaystyle V} ), the dual set of B {\displaystyle B} is a set B ∗ {\displaystyle B^{*}} of vectors in the dual space V ∗ {\displaystyle V^{*}}
Dual_basis
Graph representing edges of another graph
line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the θ-obrazom, as well as the
Line_graph
Scattering from arrays of atoms
scattering of waves from a large crystal lattice. It describes how the superposition of wave fronts scattered by lattice planes leads to a strict relation between
Bragg's_law
High symmetry orientation of a crystal
rule of crystallographic "dual vector spaces in 3D", e.g. reciprocal lattices, is that the condition for a direct lattice vector [uvw] (or zone axis)
Zone_axis
Tiling of the plane with 60° rhombi
rhomb are in the ratio 1:√3. This is the dual tiling of the trihexagonal tiling or kagome lattice. As the dual to a uniform tiling, it is one of eleven
Rhombille_tiling
Graded lattice with modular maximal chain
left and (dual) right modular as an element of the lattice of subgroups. Richard Stanley noticed in the 1970s that certain geometric lattices, such as
Supersolvable_lattice
dual of an AM-space with unit is an AL-space. The reason for the name abstract L-space is because every AL-space is isomorphic (as a Banach lattice)
Abstract_L-space
Mathematical model of magnetism
model is a quantum version of the classical Ising model. It features a lattice with nearest neighbour interactions determined by the alignment or anti-alignment
Transverse-field_Ising_model
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