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Created page with "In mathematics, a '''lattice''' is one of the fundamental algebraic structures used in abstract algebra. == Description == It consists of a p..."

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In [[mathematics]], a '''lattice''' is one of the fundamental [[Algebraic structure|algebraic structures]] used in [[abstract algebra]].

== Description ==

It consists of a partially ordered set in which every two elements have a unique [[Infimum and supremum|supremum]] (also called a least upper bound or join) and a unique [[Infimum and supremum|infimum]] (also called a greatest lower bound or meet).

An example is given by the [[Natural number|natural numbers]], partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.

Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities.

Since the two definitions are equivalent, lattice theory draws on both [[order theory]] and [[universal algebra]].

Semilattices include lattices, which in turn include [[Heyting algebra|Heyting]] and [[Boolean algebra (structure)|Boolean algebras]]. These "lattice-like" structures all admit [[Order theory|order-theoretic]] as well as algebraic descriptions.

== See also ==

* [[0,1-simple lattice]]
* [[Eulerian lattice]]
* [[Ideal and filter]] (dual notions)
* [[Join and meet]]
* [[Map of lattices]]
* [[Orthocomplemented lattice]]
* [[Skew lattice]] (generalization to non-commutative join and meet)
* [[Post's lattice]]
* [[Tamari lattice]]
* [[Total order]]
* [[Young–Fibonacci lattice]]

=== Applications that use lattice theory ===

Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.

* [[Abstract interpretation]]
* [[Algebraizations of first-order logic]]
* [[Analogical modeling]]
* [[Bloom filter]]
* [[Closure operator]]
* [[Domain theory]]
* [[Formal concept analysis]] and [[lattice miner]] (theory and tool)
* [[Fuzzy set theory]]
* [[Information flow]]
* [[Invariant subspace]]
* [[Knowledge space]]
* [[Lattice of subgroups]]
* [[Median graph]]
* [[Multiple inheritance]]
* [[Ordinal optimization]]
* [[Quantum logic]]
* [[Ontology (computer science)]]
* [[Pointless topology]]
* [[Regular language learning]]
* [[Semantics of programming languages]]
* [[Spectral space]]
* [[Subsumption lattice]]

== External links ==

* [https://en.wikipedia.org/wiki/Lattice_(order) Lattice (order)] @ Wikipedia.org

[[Category:Computer science]]
[[Category:Computing]]
[[Category:Mathematics]]

Lattice (order) - Revision history

Karl Jones: Created page with "In mathematics, a '''lattice''' is one of the fundamental algebraic structures used in abstract algebra. == Description == It consists of a p..."