sublattices

A sublattice of a lattice L is a subset S of L that is closed under the lattice operations and inherits the same order. Concretely, if a and b are elements of S, then both the meet a ∧ b and the join a ∨ b lie in S, and the induced structure (S, ∧, ∨) is itself a lattice.

It is not required that S contain the least element or greatest element of L; if both are included, S is a sublattice with the same bounds as L. A sublattice S is called proper when S is strictly contained in L.

Examples help illustrate the concept. In the divisors lattice of a positive integer n, ordered by divisibility, the meet is gcd and the join is lcm. Any subset of divisors closed under gcd and lcm forms a sublattice. In particular, divisors of n that involve only a fixed subset of prime factors form a sublattice. Another standard example is the lattice of subspaces of a finite-dimensional vector space V, ordered by inclusion, with meet given by intersection and join by the sum; any collection of subspaces closed under intersection and sum is a sublattice, and the sublattice of all subspaces contained in a fixed subspace W is itself a sublattice.

Generation and embeddings. Given a subset A of L, the sublattice generated by A is the smallest sublattice containing A. It can be constructed by closing A under finite meets and joins, equivalently by taking intersections of all sublattices containing A.

Sublattices preserve the algebraic structure of L under the inherited operations, and they play a central role in the study of lattice properties, substructure lattices, and lattice homomorphisms.