partialextension

Partialextension is a concept used in mathematics and theoretical computer science to describe an object that extends another object only partially, in the sense that new elements or relationships may be added while preserving the existing structure on the original domain. A partialextension of a structure A consists of a larger structure B together with an inclusion map from A to B such that the interpretation of symbols agrees on A, but where some relations or functions on B are only defined for a subset of the new elements or are allowed to be undefined on certain inputs.

In model-theoretic terms, A is a substructure of B and B may enlarge the domain and interpretation while not forcing a full extension of all properties to the new part. The extension is called partial when certain properties, axioms, or definability constraints are not required to hold universally over B, or when some function symbols remain partially defined. The notion captures scenarios where growth is controlled and compatible with existing truths, rather than complete and uniform expansion.

Typical constructions involve partial graphs, where new vertices and edges are added while existing connections are kept intact; partial functions, where a new mapping domain is restricted; and partial orders, where new elements are introduced with order relations only partially completed. These ideas facilitate incremental development and local compatibility across larger structures.

Applications include incremental database design, where schemas are extended while preserving existing data constraints; staged programming languages, where environments grow with partial bindings; and model construction in logic, where larger models are built by gradually adding elements that preserve previously established truths. See also extension, substructure, embedding, partial function, definability.