latticesdepending

Latticesdepending is a term used to describe a pattern in lattice theory in which a family of lattices varies with external input or parameters. Although not a standard, widely used term, it serves to denote constructions where lattice structure changes across a parameter space and is studied in a coordinated way.

Definition and formalism: Let P be a poset that indexes a family of lattices {L_p : p in P}. For p ≤ q in P there are lattice homomorphisms φ_{p,q}: L_p → L_q that satisfy φ_{p,p} = id and φ_{q,r} ∘ φ_{p,q} = φ_{p,r} for all p ≤ q ≤ r. This setup is equivalently a functor F: P → Lat, sending each parameter to a lattice and each relation to a compatible lattice homomorphism, capturing the notion that the lattice content depends on the parameter.

Examples: One example is a directed system of Boolean lattices given by the power sets of a family of sets S_p with maps induced by inclusions. Another example comes from topology: for a family of spaces X_p indexed by p, the lattices L_p = Open(X_p) form a parameter-dependent system via maps Open(X_p) → Open(X_q) induced by continuous mappings when p ≤ q. A further example is forming, inside a fixed lattice L, sublattices L_p defined by a parameter constraint that filters elements of L.

Properties and constructions: The diagram {L_p} can admit colimits (direct limits) in Lat, yielding a lattice that represents the cumulative structure across parameters. Finite-lattice cases often allow explicit computation of inductive limits, while infinite cases may require additional conditions to preserve lattice operations.

Applications: The concept is relevant in domain theory for parameterized computation models, in sheaf-like contexts where lattice-valued data vary over a base, and in formal concept analysis when concept lattices are indexed by changing attribute sets. It provides a language for discussing how order-theoretic structures respond to external variation.

See also: lattice theory; presheaf; diagram (category theory); direct limit; parametric families.