In mathematics, a hereditary property is a property of an object, that inherits to all its subobjects, where the term subobject depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory.
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In topology
In topology, a topological property is said to be hereditary if whenever a topological space has that property, then so does every subspace of it. If the latter is true only for closed subspaces, then the property is called weakly hereditary or closed-hereditary.
For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary. Connectivity is not weakly hereditary.
If P is a property of a topological space X and every subspace also has property P, then X is said to be "hereditarily P".
In graph theory
In graph theory, a hereditary property is a property of a graph which also holds for (is "inherited" by) its induced subgraphs. Alternately, a hereditary property is preserved by the removal of vertices. A graph class
In some cases, the term "hereditary" has been defined with reference to graph minors, but this is more properly called a minor-hereditary property. The Robertson–Seymour theorem implies that a minor-hereditary property may be characterized in terms of a finite set of forbidden minors.
The term "hereditary" has been also used for graph properties that are closed with respect to taking subgraphs. In such a case, properties, that are closed with respect to taking induced subgraphs, are called induced-hereditary. This approach is used by the members of the scientific society Hereditarnia Club. The language of hereditary properties and induced-hereditary properties provides a powerful tool for study of structural properties of various types of generalized colourings. The most important result from this area is the Unique Factorisation Theorem.
Monotone property
There is no consensus for the meaning of "monotone property" in graph theory. Examples of definitions are:
The complementary property of a property that is preserved by the removal of edges is preserved under the addition of edges. Hence some authors avoid this ambiguity by saying a property A is monotone if A or AC (the complement of A) is monotone. Some authors choose to resolve this by using the term increasing monotone for properties preserved under the addition of some object, and decreasing monotone for those preserved under the removal of the same object.
In model theory
In model theory and universal algebra, a class K of structures of a given signature is said to have the hereditary property if every substructure of a structure in K is again in K. A variant of this definition is used in connection with Fraïssé's theorem: A class K of finitely generated structures has the hereditary property if every finitely generated substructure is again in K. See age.
In matroid theory
In a matroid, every subset of an independent set is again independent. This is also sometimes called the hereditary property.
In set theory
Recursive definitions using the adjective "hereditary" are often encountered in set theory.
A set is said to be hereditary (or pure) if all of its elements are hereditary sets. It is vacuously true that the empty set is a hereditary set, and thus the set
A couple of notions are defined analogously:
Based on the above, it follows that in ZFC a more general notion can be defined for any predicate
If we instantiate in the above schema