# Category: Lattice theory

Formal concept analysis
In information science, formal concept analysis (FCA) is a principled way of deriving a concept hierarchy or formal ontology from a collection of objects and their properties. Each concept in the hier
Geometric lattice
In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finitene
Dominance order
In discrete mathematics, dominance order (synonyms: dominance ordering, majorization order, natural ordering) is a partial order on the set of partitions of a positive integer n that plays an importan
Spectral space
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent t
Absorption law
In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations. Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if: a ¤ (a ⁂ b)
Finite lattice representation problem
In mathematics, the finite lattice representation problem, or finite congruence lattice problem, asks whether every finite lattice is isomorphic to the congruence lattice of some finite algebra.
Young's lattice
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 substitutional analysis, developed the
Semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilatt
Complemented lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b
Join and meet
In mathematics, specifically order theory, the join of a subset of a partially ordered set is the supremum (least upper bound) of denoted and similarly, the meet of is the infimum (greatest lower boun
Young–Fibonacci lattice
In mathematics, the Young–Fibonacci graph and Young–Fibonacci lattice, named after Alfred Young and Leonardo Fibonacci, are two closely related structures involving sequences of the digits 1 and 2. An
Skew lattice
In abstract 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 to any non-commutative generalizat
Lattice (order)
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
Duality theory for distributive lattices
In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pair
Modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, Modular lawa ≤ b implies a ∨ (x ∧ b) = (a ∨ x) ∧ b where x, a, b are
Median graph
In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a, b, and c have a unique median: a vertex m(a,b,c) that belongs to shortest paths betwe
Continuous geometry
In mathematics, continuous geometry is an analogue of complex projective geometry introduced by von Neumann , where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can
0,1-simple lattice
In lattice theory, a bounded lattice L is called a 0,1-simple lattice if nonconstant lattice homomorphisms of L preserve the identity of its top and bottom elements. That is, if L is 0,1-simple and ƒ
Dedekind number
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) counts the number of monotone boolean f
Union-closed sets conjecture
In combinatorics, the union-closed sets conjecture is a problem, posed by Péter Frankl in 1979 and is still open. A family of sets is said to be union-closed if the union of any two sets from the fami
Esakia duality
In mathematics, Esakia duality is the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an order-topological representation of Heytin
Lattice-based access control
In computer security, lattice-based access control (LBAC) is a complex access control model based on the interaction between any combination of objects (such as resources, computers, and applications)
Lattice of subgroups
In mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the partial order relation being set inclusion.In this lattice, the join of two subgroups
Stone algebra
In mathematics, a Stone algebra, or Stone lattice, is a pseudo-complemented distributive lattice such that a* ∨ a** = 1. They were introduced by and named after Marshall Harvey Stone. Boolean algebras
Tonnetz
In musical tuning and harmony, the Tonnetz (German for 'tone network') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representation
Lattice of stable matchings
In mathematics, economics, and computer science, the lattice of stable matchings is a distributive lattice whose elements are stable matchings. For a given instance of the stable matching problem, thi
Distributive 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 such structures are collections of sets for which
Semimodular lattice
In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition: Semimodular lawa ∧ b <: a implies b <: a ∨ b. The notation a <: b means
Antimatroid
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remain
Map of lattices
The concept of a lattice arises in order theory, a branch of mathematics. The Hasse diagram below depicts the inclusion relationships among some important subclasses of lattices.
Congruence lattice problem
In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilwo
De Morgan algebra
In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that: * (A, ∨, ∧, 0, 1) is a bounded distributive
Residuated lattice
In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\z and z/y, loosely analogous to division or implic
Free lattice
In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property.
Dual lattice
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 geom
Tamari lattice
In mathematics, a Tamari lattice, introduced by Dov Tamari, is a partially ordered set in which the elements consist of different ways of grouping a sequence of objects into pairs using parentheses; f
Slim lattice
In lattice theory, a mathematical discipline, a finite lattice is slim if no three join-irreducible elements form an antichain. Every slim lattice is . A finite planar semimodular lattice is slim if a
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is know
Metric lattice
In the mathematical study of order, a metric lattice L is a lattice that admits a positive valuation: a function v∈L→ℝ satisfying, for any a,b∈L, and
Pseudocomplement
In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement. In a lattice L with bottom element 0, an element x ∈ L is said to have a pseudocompl
Heyting algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped w
Division lattice
The division lattice is an infinite complete bounded distributive lattice whose elements are the natural numbers ordered by divisibility. Its least element is 1, which divides all natural numbers, whi
Maximal semilattice quotient
In abstract algebra, a branch of mathematics, a maximal semilattice quotient is a commutative monoid derived from another commutative monoid by making certain elements equivalent to each other. Every
Subsumption lattice
A subsumption lattice is a mathematical structure used in the theoretical background of automated theorem proving and other symbolic computation applications.
Lattice Miner
Lattice Miner is a formal concept analysis software tool for the construction, visualization and manipulation of concept lattices. It allows the generation of formal concepts and association rules as
Tolerance relation
In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexive symmetric relation that is compatible with all operations of the structure. Thus a tolerance is l