Hlattice

Hlattice is a term used in mathematics to describe a lattice equipped with an additional unary operator that acts as a closure or hull on the lattice elements. The notion appears in various sources with slightly different specifics, but the common idea is to augment a lattice with an operator that “closes” elements to a larger, canonical form.

Formally, let L = (L, ∨, ∧) be a bounded lattice and H: L → L a unary operator. In many treatments, H is required to be a closure operator, meaning it satisfies: x ≤ H(x) for all x, monotonicity: if x ≤ y then H(x) ≤ H(y), and idempotence: H(H(x)) = H(x). The elements that are fixed by H (those with H(x) = x) are called closed elements, and they form a sublattice of L. If H is instead a dual notion such as an interior or hull operator, the same general framework applies with appropriate dual properties.

Examples help illustrate the idea. On the lattice of subsets of a set S, with H given by the closure operator induced by a topology on S, H is the usual topological closure and closed sets are those equal to their closure. On the lattice of subspaces of a vector space V, fixing a subspace W and defining H(U) = U + W yields a closure operator whose closed subspaces are exactly those that contain W.

Hlattices connect to closure systems, Moore families, and areas like formal concept analysis and domain theory, where closure-like operations model data completion, inference, or expansion within a lattice framework. See also closure operator, hull operator, Moore family, lattice theory.