Lattice (order)
In mathematics, a lattice is one of the fundamental algebraic structures used in abstract algebra.
Description
It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
An example is given by the 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 and Boolean algebras. These "lattice-like" structures all admit 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
- Lattice (order) @ Wikipedia.org
