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Special type of lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of
Distributive_lattice
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
Set whose pairs have minima and maxima
of lattices can be generalized to semilattices, and some notable subclasses of lattices are Heyting algebras, Boolean algebras, distributive lattices, and
Lattice_(order)
Property involving two mathematical operations
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x ⋅ ( y + z
Distributive_property
completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets. Formally, a complete lattice L is said
Completely distributive lattice
Completely_distributive_lattice
Bound lattice in which every element has a complement
called an orthomodular lattice. In bounded distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation
Complemented_lattice
Algebraic structure used in logic
are distributive lattices. Every Boolean algebra is a Heyting algebra when a → b is defined as ¬a ∨ b, as is every complete distributive lattice satisfying
Heyting_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
Algebra whose elements are stable matchings
mathematics, economics, and computer science, a lattice of stable matchings is a distributive lattice whose elements are all the solutions to a given
Lattice_of_stable_matchings
concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the
Distributivity_(order_theory)
duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces
Duality theory for distributive lattices
Duality_theory_for_distributive_lattices
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)
System of logic lacking the excluded middle law
structure A = (A, ∨, ∧, 0, 1, ¬) such that: (A, ∨, ∧, 0, 1) is a bounded distributive lattice, and ¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x
De_Morgan_algebra
Correlation-type inequality for four functions on a finite distributive lattice
is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilistic
Ahlswede–Daykin_inequality
Mathematical system of orderings or sets
case of greedoids and of semimodular lattices, and as a generalization of partial orders and of distributive lattices. Antimatroids are equivalent, by complementation
Antimatroid
continuous distributive lattice on the points of the polytope. Every face of a distributive polytope is itself a distributive polytope. The distributive polytopes
Distributive_polytope
Set with operations obeying given axioms
lattice: a lattice in which arbitrary meet and joins exist. Bounded lattice: a lattice with a greatest element and least element. Distributive lattice: a lattice
Algebraic_structure
Graph with a median for each three vertices
graphs arise naturally in the study of ordered sets and discrete distributive lattices, and have an extensive literature". In phylogenetics, the Buneman
Median_graph
finite distributive lattices, the upper sets of any partially ordered set form a finite distributive lattice, and every finite distributive lattice can be
Order_polytope
Important problem in lattice theory
congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem
Congruence_lattice_problem
Algebraic ring that need not have additive negative elements
inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean
Semiring
Every subgroup of a cyclic group is cyclic, and if finite, its order divides its parent's
lattices of subgroups are distributive. More generally, a finitely generated group is cyclic if and only if its lattice of subgroups is distributive and
Subgroups_of_cyclic_groups
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
Category where each homset contains at most one morphism
defined as a small skeletal thin category, a distributive lattice as a small skeletal thin distributive category, a Heyting algebra as a small skeletal
Thin_category
Nonempty, upper-bounded, downward-closed subset
is no proper filter that is a strict superset. When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse
Ideal_(order_theory)
variety; indeed, so do pseudocomplemented semilattices. Every finite distributive lattice is pseudocomplemented. Every Stone algebra is pseudocomplemented
Pseudocomplement
Mathematical structure
sub-poset of the subsumption lattice, and is itself a lattice. This lattice, too, includes N5 and the minimal non-distributive lattice M3 as sublattices (see
Subsumption_lattice
Subset of a preorder that contains all larger elements
representation theorem asserts that every finite distributive lattice arises (up to isomorphism) in this way as the lattice of lower sets of a unique finite poset
Upper_and_lower_sets
Partial order with joins
semilattice necessarily be bounded.) A totally ordered set is a distributive lattice, hence in particular a meet-semilattice and join-semilattice: any
Semilattice
Technical treatment of Boolean algebras
lattice basis, it suffices to define a Boolean algebra as a distributive lattice satisfying x∧¬x = 0 and x∨¬x = 1, called a complemented distributive
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
Combinatorial sequence of numbers
{\displaystyle n} -element set, the number of elements in a free distributive lattice with n {\displaystyle n} generators, and one more than the number
Dedekind_number
Type of lattice in mathematical order theory
contains a copy of N5 as a sublattice. Every distributive lattice is modular. Conversely, a lattice is distributive if and only if x ∨ ( y ∧ z ) ≥ ( x ∨ y )
Modular_lattice
Algebraic manipulation of "true" and "false"
axiomatization of Boolean algebra, such as the axioms for a complemented distributive lattice, a sufficient condition for an algebraic structure of this kind to
Boolean_algebra
Correlation inequality
{\displaystyle X} be a finite distributive lattice, and μ a nonnegative function on it, that is assumed to satisfy the (FKG) lattice condition (sometimes a function
FKG_inequality
Ordered topological space with special properties
them. Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality ("Priestley duality") between
Priestley_space
Order whose elements are all comparable
Systems. Pergamon Press. George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN 0-7167-0442-0 Halmos
Total_order
Family of graphs based on the Fibonacci sequence
Hamming distance, independent sets of vertices in path graphs, or via distributive lattices. Like the hypercube graph, the vertices of the Fibonacci cube of
Fibonacci_cube
Relationship between certain categories
DLat01 of bounded distributive lattices. Hence, DLat01 is dual to CohSp—one obtains Stone's representation theorem for distributive lattices. When restricting
Stone_duality
Lattice formed by all integer partitions
by intersections and unions, it is a distributive lattice. If a partition p covers k elements of Young's lattice for some k then it is covered by k + 1
Young's_lattice
In mathematics, an algebraic structure
necessary distributivity of • over ∨ does not in general entail distributivity of ∧ over ∨, that is, a residuated lattice need not be a distributive lattice. However
Residuated_lattice
Lattice whose elements are the subgroups of a given group
finite lattice is isomorphic to a sublattice of the subgroup lattice of some finite group (Schmidt 1994, p. 9). Every finite distributive lattice is also
Lattice_of_subgroups
Structure-preserving correspondence between node-link graphs
composed leads to rich algebraic structures: a preorder on graphs, a distributive lattice, and a category (one for undirected graphs and one for directed graphs)
Graph_homomorphism
element of S has a largest element. We say that θ is distributive, if it is a join, in the congruence lattice Con S of S, of monomial join-congruences of S.
Distributive_homomorphism
Structure on sequences of digits 1 and 2
further observes, the Young–Fibonacci lattice is modular. Fomin (1988) incorrectly claims that it is distributive; however, the sublattice formed by the
Young–Fibonacci_lattice
Order-preserving mathematical function
analysis (second ed.). Grätzer, George (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman. ISBN 0-7167-0442-0. Pemberton,
Monotonic_function
Discrete math concept
the lattice Ln, such as the minimal height and the maximal covering number, and classified the intervals of small length. While Ln is not distributive for
Dominance_order
Concept in mathematics
relationships among some important subclasses of lattices. 1. A boolean algebra is a complemented distributive lattice. (def) 2. A boolean algebra is a heyting
Map_of_lattices
\rangle } also suffice. In a Boolean algebra, or more generally a distributive lattice, the median function ⟨ x , y , z ⟩ = ( x ∨ y ) ∧ ( y ∨ z ) ∧ ( z
Median_algebra
1990 book on mathematical order theory
complete lattices. The fourth of the introductory chapters concerns special classes of lattices, including modular lattices, distributive lattices, and Boolean
Introduction to Lattices and Order
Introduction_to_Lattices_and_Order
Integer that divides another integer
into a partially ordered set that is a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation
Divisor
Overview of and topical guide to algebraic structures
the distributive law; in the case of lattices, they are linked by the absorption law. Ringoids also tend to have numerical models, while lattices tend
Outline of algebraic structures
Outline_of_algebraic_structures
Largest integer that divides given integers
lcm(0, 0) = 0 because then the natural numbers become a complete distributive lattice with GCD as meet and LCM as join operation. This extension of the
Greatest_common_divisor
Partially ordered set with alternatingly-related elements
The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation
Fence_(mathematics)
(order theory) Dense order Distributivity (order theory) Modular lattice Distributive lattice Completely distributive lattice Ascending chain condition
List_of_order_theory_topics
The division lattice is an infinite complete bounded distributive lattice whose elements are the natural numbers ordered by divisibility. Its least element
Division_lattice
Natural number
antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families.)". The On-Line Encyclopedia
168_(number)
Graph representing connectivity between cliques of another graph
given a stronger structure as a distributive lattice, and in this case the simplex graph is the graph of the lattice. As is true for median graphs more
Simplex_graph
Existence of certain infima or suprema of a given poset
case the complete lattice X is constructively completely distributive. See also the articles on complete distributivity and distributivity (order theory)
Completeness_(order_theory)
lattices are symmetric and can be shown to form a variety. Unlike lattices, they need not be distributive, and conversely. Distributive skew lattices
Skew_lattice
Smallest complete lattice containing a partial order
(complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x. The result is a distributive lattice and is used
Dedekind–MacNeille_completion
orbits. Any distributive lattice L is subdirectly representable as a subalgebra of a direct power of the two-element distributive lattice. This can be
Subdirect_product
Proof that every structure with certain properties is isomorphic to another structure
representation theorem for distributive lattices, states that every distributive lattice is isomorphic to a sublattice of the power set lattice of some set. Another
Representation_theorem
Ideals in a Boolean algebra can be extended to prime ideals
ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article focuses on prime
Boolean_prime_ideal_theorem
Mathematical structure with greatest common divisors
that the operations of GCD and LCM make the quotient R/~ into a distributive lattice, where "~" denotes the equivalence relation of being associate elements
GCD_domain
Branch of mathematics
This condition is called distributivity and gives rise to distributive lattices. There are some other important distributivity laws which are discussed
Order_theory
Partially ordered set equipped with a rank function
fixed N The Boolean lattice of finite subsets of a set ordered by inclusion (number of elements of the subset) Any distributive lattice of finite lower sets
Graded_poset
British mathematician
her application of these spaces in the representation theory of distributive lattices. Priestley, Hilary A. (2003). Introduction to Complex Analysis (2nd ed
Hilary_Priestley
Generalized topological space
"logical" structures such as semilattices, distributive lattices, complete and completely distributive lattices, Boolean algebras, complete atomic Boolean
Chu_space
Mathematical relation inside orderings
covering relation of a Tamari lattice is the skeleton of an associahedron. The covering relation of any finite distributive lattice forms a median graph. On
Covering_relation
Space homeomorphic to some ring spectrum
to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic (as a bounded lattice) to the lattice K ∘ {\displaystyle \circ } (X)
Spectral_space
Family closed under unions and relative complements
respectively. Conversely, every distributive lattice is isomorphic to a ring of sets; in the case of finite distributive lattices, this is Birkhoff's representation
Ring_of_sets
In mathematics, an Ockham algebra is a bounded distributive lattice L {\displaystyle L} with a dual endomorphism, that is, an operation ∼ : L → L {\displaystyle
Ockham_algebra
Topics referred to by the same term
the cubic metre, a unit of volume M3, the minimal modular, but non-distributive lattice in mathematical order theory ATC code M03, Muscle relaxants, a subgroup
M3
Glossary of terms used in branch of mathematics
that are not already complete lattices. Completely distributive lattice. A complete lattice is completely distributive if arbitrary joins distribute over
Glossary_of_order_theory
Natural number
games. 10001 = strobogrammatic number 10006 = number of unlabelled distributive lattices with 19 nodes. 10007 = smallest five-digit prime number, twin prime
10,000
Topics referred to by the same term
manufacture of populated, printed circuit boards Ockham algebra, bounded distributive lattice with a dual endomorphism Ockham Awards, annual awards by The Skeptic
Ockham
Left adjoint to a forgetful functor to sets
group free Kleene algebra free lattice free Boolean algebra free distributive lattice free Heyting algebra free modular lattice free Lie algebra free magma
Free_object
Mathematical group that can be generated as the set of powers of a single element
fraction. A group is locally cyclic if and only if its lattice of subgroups is a distributive lattice. A cyclically ordered group is a group together with
Cyclic_group
Term in the mathematical area of order theory
self-dual include: Being a (complete) lattice Monotonicity of functions Distributivity of lattices, i.e. the lattices for which ∀x,y,z: x ∧ (y ∨ z) = (x
Duality_(order_theory)
Concept in education theory
structure: A quasi-ordinal knowledge space can be associated with a distributive lattice under set union and set intersection. The name "quasi-ordinal" arises
Knowledge_space
Submodule of a mathematical ring
ideals of a given ring forms a complete modular lattice. The lattice is not, in general, a distributive lattice. The three operations of intersection, sum
Ideal_(ring_theory)
Norwegian mathematician
implicative lattice (now also called a Skolem lattice) is distributive and, as a partial converse, that every finite distributive lattice is implicative
Thoralf_Skolem
On graphs with given symmetry groups
distributive lattice, it follows that every finite group can be realized as the symmetries of a distributive lattice, and of the graph of the lattice
Frucht's_theorem
Pairing where no unchosen pair prefers each other over their choice
the stable marriage problem can be given the structure of a finite distributive lattice, and this structure leads to efficient algorithms for several problems
Stable_matching_problem
Topics referred to by the same term
homomorphism, subalgebra and product Birkhoff's representation theorem for distributive lattices Birkhoff's theorem (equational logic), stating that syntactic and
Birkhoff's_theorem
Algebraic structure
{\displaystyle x\land \bigvee _{s\in S}s=\bigvee _{s\in S}(x\land s).} P is a distributive lattice, i.e., for all x, y and z in P, we have x ∧ ( y ∨ z ) = ( x ∧ y )
Complete_Heyting_algebra
Result of partitioning the elements of an algebraic structure using a congruence relation
conditions for varieties having congruence lattices that are distributive (thus called congruence-distributive varieties), while in 1969 Alan Day did the
Quotient_(universal_algebra)
matching polytope, of defining a continuous distributive lattice is analogous to the defining property of a distributive polytope, a polytope in which coordinatewise
Stable_matching_polytope
modules over a principal ideal domain Fundamental theorem of finite distributive lattices Fundamental theorem of Galois theory Fundamental theorem of geometric
List of theorems called fundamental
List_of_theorems_called_fundamental
Assignment of colors to edges of a graph
labelings, the set of regular labelings of a fixed graph forms a distributive lattice that may be used to quickly list all geometric structures based on
Edge_coloring
Boolean algebra
A B ¯ {\displaystyle A{\overline {AB}}=A{\overline {B}}} (2 is a distributive lattice) Where concatenation = OR, 1 = true, and 0 = false, or concatenation
Two-element_Boolean_algebra
Mathematical concept
has at least three elements, the lattice of topologies on X {\displaystyle X} is not modular, and hence not distributive either. Initial topology, the coarsest
Comparison_of_topologies
Affine subspace of a Euclidean space
distinct points or by two distinct planes. However, the lattice of all flats is not a distributive lattice. If two lines ℓ1 and ℓ2 intersect, then ℓ1 ∩ ℓ2 is
Flat_(geometry)
Mathematical graph with at least one median per triple of vertices
median graphs are related to distributive lattices in the same way that modular graphs are related to modular lattices. However, the modular graphs also
Modular_graph
Graded lattice with modular maximal chain
In mathematics, a supersolvable lattice is a graded lattice that has a maximal chain of elements, each of which obeys a certain modularity relationship
Supersolvable_lattice
Mathematical set with an ordering
set can be generalized to a wide class of partial orders, called distributive lattices; see Birkhoff's representation theorem. Sequence A001035 in OEIS
Partially_ordered_set
Mathematical function
The space of seminorms on X {\displaystyle X} is generally not a distributive lattice with respect to the above operations. For example, over R 2 {\displaystyle
Seminorm
In mathematics, vector subspace
{\displaystyle \neg } ), makes the lattice of subspaces a (possibly infinite) orthocomplemented lattice (although not a distributive lattice).[citation needed] In
Linear_subspace
Mathematics theorem
MR 0150049 Yamamoto, Koichi (1954), "Logarithmic order of free distributive lattice", Journal of the Mathematical Society of Japan, 6 (3–4): 343–353
Lubell–Yamamoto–Meshalkin inequality
Lubell–Yamamoto–Meshalkin_inequality
Topics referred to by the same term
dictionary. N5 or N-5 may refer to: N5, the minimal non-modular and non-distributive lattice in mathematical order theory N5, abbreviation for the 5 nanometer
N5
travel, tourism, insurance
DISTRIBUTIVE LATTICE
DISTRIBUTIVE LATTICE
Boy/Male
Hindu, Indian
Distribute Love
Surname or Lastname
English
English : habitational name from places in Cumbria and Hertfordshire named Corney, from Old English corn ‘grain’ or corn, a metathesized form of cron, cran ‘crane’ + ēg ‘island’. It seems possible, from the distribution of early forms, that it may also derive from a lost place in Lancashire.
Girl/Female
Indian, Sikh
Distributing Happiness
Surname or Lastname
English
English : habitational name from a place named in Old English with hÄlig ‘holy’ + Old English feld ‘open country’. This may be Holyfield in Essex (which belonged to Waltham Abbey), but the present-day distribution of the name (mainly in the Midlands and Wales) suggests that another source may be involved.
Surname or Lastname
English
English : possibly a habitational name from either of two places named Charton, in Devon and Kent, the latter being the more likely source, to judge by the current distribution of the surname.French (Normandy and Champagne) : reduced form of Char(r)eton, denoting a carter, from a derivative of Old French charette ‘cart’.
Surname or Lastname
English (Devon)
English (Devon) : unexplained. Reaney and Wilson suggest that this may be from an Anglo-Scandinavian personal name Tukka, but the distribution in England makes a Scandinavian connection unlikely.
Surname or Lastname
English (Cambridge)
English (Cambridge) : unexplained; perhaps a habitational name from a lost or unidentified place. There are two places in England called Warland, in Durham and West Yorkshire, but the distribution of the modern surname suggests that a different souce is most probably involved.
Surname or Lastname
English
English : apparently a habitational name from places named Rushford in Devon, Norfolk, and Warwickshire. However, in view of the present-day distribution of the surname, a more likely source is Ryshworth in Bingley, West Yorkshire, which was earlier called Rushford (from Old English rysc ‘rushes’ + ford ‘ford’).
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : apparently a habitational name from a lost or unidentified minor place in West Yorkshire, probably in the parish of Halifax, to judge by the distribution of early occurrences of the surname.
Surname or Lastname
English (Lincolnshire)
English (Lincolnshire) : unexplained. Black identified this as a Scottish name of Pictish origin. However, the modern distribution of the surname, almost exclusively in Lincolnshire and adjoining counties, suggests a more localized eastern English origin.
Boy/Male
Hindu
Distribute Love, Well wisher
Boy/Male
Hindu
Distribute Love, Well wisher
Boy/Male
Indian, Modern
Distribute the Knowledge
Boy/Male
Tamil
Hetarth | ஹேதாரà¯à®¤Â
Distribute Love, Well wisher
Hetarth | ஹேதாரà¯à®¤Â
Boy/Male
Arabic, British, Islamic, Malaysian, Muslim, Pakistani, Tamil, Urdu
Distribution
Boy/Male
Tamil
Hitarth | ஹிதாரà¯à®¤Â
Distribute Love, Well wisher
Hitarth | ஹிதாரà¯à®¤Â
Surname or Lastname
English
English : of uncertain origin. Reaney suggests that it may be habitational name from Wincheap Street in Canterbury, but this origin is not supported by the present-day distribution of the surname, which is heavily concentrated in northeastern England.
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from a place so called, perhaps Forshaw Heath in Solihull, Warwickshire, although the modern distribution is much further north.
Surname or Lastname
English (chiefly West Midlands)
English (chiefly West Midlands) : habitational name from any of the various places so called, from Old English sūð ‘south’ + halh ‘nook’, ‘recess’. The distribution of the surname in Britain makes a Midlands origin likely: places called Southall in Doverdale, Worcestershire, and Billingsley, Shropshire, are possible sources.
Surname or Lastname
English
English : unexplained; perhaps a habitational name from a lost or unidentified place. It has been suggested that it might be an altered form of Scottish Ballantine, but the distribution and variants (including Blanding) make it more probable that it is an altered form of a French original.
DISTRIBUTIVE LATTICE
DISTRIBUTIVE LATTICE
DISTRIBUTIVE LATTICE
DISTRIBUTIVE LATTICE
DISTRIBUTIVE LATTICE
DISTRIBUTIVE LATTICE
DISTRIBUTIVE LATTICE
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