Nonextendability

Nonextendability is a term used in various branches of mathematics to describe when a structure defined on a subset cannot be extended to a larger domain, while preserving a prescribed property or structure. The precise meaning depends on the context and the property being preserved.

In analysis and complex analysis, nonextendability often refers to the inability to analytically continue a function beyond its domain. A function may be defined on a region D but have no analytic continuation to any larger region. A classic phenomenon is the existence of natural boundaries, such as in certain lacunary power series whose unit circle is a natural boundary, making analytic continuation across the boundary impossible.

In graph theory and combinatorics, nonextendability frequently describes maximal or nonextendable substructures. For instance, a matching that cannot be enlarged by adding another edge without violating the matching condition is called nonextendable; equivalently, a maximal matching cannot be extended to a larger matching. Similarly, a subgraph might be nonextendable with respect to a property like connectivity or planarity.

In model theory and related areas, nonextendability can refer to the failure to extend an embedding or an elementary substructure to a larger model while preserving the defining properties of the theory, depending on the exact framework used.

Because nonextendability depends on the chosen property and allowed extensions, precise definitions must specify the domain, the ambient structure, and the property to be preserved. In practice, the term signals a fundamental obstruction to continuation or enlargement within a given mathematical setting.