latticeembedded

Latticeembedded is a term used to describe the embedding of one lattice into another lattice, preserving the lattice operations. In mathematics, a lattice is a partially ordered set in which any two elements have a greatest lower bound (meet) and a least upper bound (join), usually denoted ∧ and ∨. A lattice embedding is an injective function f from a lattice L into a lattice M that preserves both operations: for all a,b in L, f(a ∨ b) = f(a) ∨ f(b) and f(a ∧ b) = f(a) ∧ f(b). Equivalently, a lattice embedding is an order-embedding, meaning a ≤ b in L if and only if f(a) ≤ f(b) in M.

A lattice embedding need not be onto; its image is a sublattice of M that is lattice-isomorphic to L. Common examples include the inclusion of a sublattice, such as the even integers under gcd and lcm embedded into the full integers; and the induced embedding of a finite Boolean algebra P(X) into P(Y) when X is mapped injectively into Y, via S ↦ f[S], which preserves union and intersection.

Lattice embeddings play a role in several areas. In universal algebra, they help study sublattices and substructures. In domain theory and type systems, embeddings model the refinement of information. In formal concept analysis, concept lattices embed into product lattices to compare different contexts.