isomorphismclosed

Isomorphismclosed, often written isomorphism-closed (and rarely as isomorphismclosed), is a term used in model theory and universal algebra to describe a quality of a class of mathematical structures. A class K of structures in a given language is isomorphism-closed if whenever M is in K and N is isomorphic to M, then N also lies in K. Equivalently, the property is determined by the isomorphism type of a structure, not by any particular presentation or labeling of its elements.

This notion captures the idea that isomorphism preserves the truth of structural properties. If a structure has a feature that is invariant under relabeling of the elements, then every isomorphic copy will also have that feature, and hence the class is closed under isomorphism.

Examples: The class of finite graphs is isomorphism-closed, since isomorphic graphs share the same finiteness. The class of graphs containing a Hamiltonian cycle is isomorphism-closed, as the property is preserved by isomorphism. Conversely, if one decorates structures with a distinguished element or a labeled topology, the class may fail to be isomorphism-closed because isomorphisms can move or permute the distinguished feature.

In model theory, elementary classes, i.e., models of a theory, are automatically isomorphism-closed because first-order sentences are preserved under isomorphism. Related concepts include invariants under isomorphism and closure under other constructions such as substructures or products.

See also: isomorphism, model theory, elementary class, invariant, category theory.