lintersection

Lintersection is a term encountered in some mathematical writings to denote a generalized or localization of the set-theoretic intersection. It is not a standard, widely adopted concept, and its precise meaning varies by author or field. Broadly, lintersection refers to intersecting a collection of sets along a specified indexing structure, such as a lattice or poset, rather than over a simple finite index.

Definition: Let L be a lattice (or poset) with index set I, and let {A_i} be a family of subsets of a common universe U indexed by I. For a subcollection J⊆I, the lintersection over J is defined as ⋂_{i∈J} A_i, assuming the intersection is taken within U. If the family has additional compatibility conditions among indices, the lintersection may be taken with these constraints in mind, yielding a localized or directed intersection.

Properties and examples: If J1⊆J2⊆I, then lintersection over J2 is contained in lintersection over J1; i.e., the operation is anti-monotone with respect to inclusion of the index set. Simple cases mirror ordinary intersections: for a finite subcollection of open sets in a topology, the lintersection is the usual intersection; for a chain of subspaces, it is their intersection.

Notes: The term is nonstandard and usage varies. When encountered, check the source for the exact definition, as lintersection can be defined differently in lattice-theoretic, topological, or sheaf-theoretic contexts. Related concepts include intersection, lattice, poset, and directed systems.