hereditary property



A property of a mathematical structure is hereditary if every substructure also satisfies that property. The idea is that substructures “inherit” the property from the structure.


A general model-theoretic definition is as follows: a sentence SS (thought of as a property which may or may not hold for a structure) of a language LL is hereditary if, whenever SS is true for a structure XX of LL, then it is also true for every embedded substructure? of XX.

This general definition admits variants, some of which are described below. In category theory, a categorical property (a sentence expressed in the language of category theory) may be said to be MM-hereditary (for various classes MM of monomorphisms, e.g., the class of regular monomorphisms) if, whenever it holds for an object XX, then it also holds for MM-subobjects of XX.


In topology

In general topology, the default meaning of “hereditary” is that if the property holds for a topological space XX, then it holds also for subspaces of XX. (Note that subspaces are equivalent to regular subobjects in TopTop.) Examples:

A property is weakly hereditary or closed hereditary if it is inherited by closed subspaces. (Note that in the category HausHaus of Hausdorff spaces, closed subspaces are equivalent to regular subobjects.) Examples:

In graph theory

In graph theory, the default meaning of “hereditary” is that the property be inherited by induced subgraphs. (If G=(V,E)G = (V, E) is a simple graph and SVS \subseteq V, then the subgraph induced by this inclusion is where every edge of GG whose incident vertices lie in SS is an edge of the subgraph. Induced subgraphs are equivalent to regular subobjects in the quasitopos of simple graphs.) Examples:

In algebra

The following examples are well-known.