nLab hereditary property

Idea

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.

Definition

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

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 $M$-hereditary (for various classes $M$ of monomorphisms, e.g., the class of regular monomorphisms) if, whenever it holds for an object $X$, then it also holds for $M$-subobjects of $X$.

Examples

In topology

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

• The separation axioms abbreviated as $T_0$, $T_1$, $T_2$, $T_3$, and $T_{3 \frac1{2}}$ are hereditary.

• The second-countability axiom is hereditary, as is the first-countability axiom.

• Metrizability is hereditary.

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

• Compactness is weakly hereditary.

• Paracompactness is weakly hereditary.

• Normality is weakly hereditary.

In graph theory

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

• The property of being a forest is hereditary.

• The property of being acyclic is hereditary.

• The property of being planar is hereditary.

In algebra

The following examples are well-known.

• In the category of groups, the property of being a free group is hereditary.

• In the category of abelian groups, the property of being free abelian is hereditary.

• More generally, in the category of modules over a PID, the property of being free is hereditary.

• In the category of modules over any commutative ring, being torsionfree is hereditary.

• In the category of modules over a Noetherian ring, being a finitely generated module is hereditary.