Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a supremum (also called a least upper bound or join) and an infimum (also called a greatest lower bound or meet).
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.
| Algebraic structures |
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Group-like structures
Semigroup and Monoid Quasigroup and Loop Abelian group |
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Ring-like structures
Semiring Near-ring Ring Commutative ring Integral domain Field |
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Lattice-like structures
Semilattice Lattice Map of lattices |
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Module-like structures
Group with operators Module Vector space |
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Algebra-like structures
Algebra Associative algebra Non-associative algebra Graded algebra Bialgebra |
Read more about Lattice (order): Lattices As Posets, Connection Between The Two Definitions, Examples, Morphisms of Lattices, Properties of Lattices, Sublattices, Free Lattices, Important Lattice-theoretic Notions