lenclosure

Lenclosure is a term used in some mathematical discussions to denote the least superset of a given set that satisfies a designated property L. Given a space X and a subset A ⊆ X, and a property L that can be verified for subsets of X, the lenclosure of A, denoted lencl(A), is the intersection of all subsets S ⊇ A with property L. If no such S exists, lencl(A) is undefined. If A already has L, lencl(A) = A.

The concept is deliberately abstract and dependent on the chosen property L. In practice, lenclosure is a generalization of the idea of a closure operator, where the standard topological closure corresponds to the case where L is the property of being closed. Other specializations yield familiar constructions: if L is convexity, lenclosure becomes the convex hull; if L specifies a certain algebraic or combinatorial structure, lenclosure yields the smallest enclosure of A with that structure. Because L can vary, lenclosure can capture a wide range of “smallest enclosing” notions across different mathematical contexts.

Construction and computation of lenclosure typically involve finding the intersection of all L-satisfying supersets of A, or performing an iterative enlargement of A until the resulting set satisfies L. The practicality of this approach depends on how readily L can be verified and on the availability of effective algorithms for the particular space X and property L.

Lenclosure is not a universally standardized term; its precise meaning is defined by the chosen property L and the setting in which it is used. Related concepts include closure, hull, and various hull operators used in geometry, topology, and algebra.