Hereditary property

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.

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.^[1] 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.^[2] Alternately, a hereditary property is preserved by the removal of vertices. A graph class $\mathcal{G}$ is said hereditary if it is closed under induced subgraphs. Examples of hereditary graph classes are independent graphs (graphs with no edges), which is a special case (with c = 1) of being c-colorable for some number c, being forests, planar, complete, complete multipartite etc.

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.^[3] 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.^[4]

Monotone property

There is no consensus for the meaning of "monotone property" in graph theory. Examples of definitions are:

• Preserved by the removal of edges.^[5]
• Preserved by the removal of edges and vertices (i.e., the property should hold for all subgraphs).^[2]
• Preserved by the addition of edges and vertices (i.e., the property should hold for all supergraphs).^[6]
• Preserved by the addition of edges.^[7] (This meaning is used in the statement of the Aanderaa–Karp–Rosenberg conjecture.)

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 A^C (the complement of A) is monotone.^[8] 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 $\{\varnothing\}$ containing only the empty set $\varnothing$ is a hereditary set, and recursively so is $\{\varnothing, \{\varnothing \}\}$, for example. In formulations of set theory that are intended to be interpreted in the von Neumann universe or to express the content of Zermelo–Fraenkel set theory, all sets are hereditary, because the only sort of object that is even a candidate to be an element of a set is another set. Thus the notion of hereditary set is interesting only in a context in which there may be urelements.

A couple of notions are defined analogously:

• A hereditarily finite set is defined as a finite set consisting of zero or more hereditarily finite sets. Equivalently, a set is hereditarily finite if and only if its transitive closure is finite.
• A hereditarily countable set is a countable set of hereditarily countable sets. Assuming the axiom of countable choice, then a set is hereditarily countable if and only if its transitive closure is countable.

Based on the above, it follows that in ZFC a more general notion can be defined for any predicate $\Phi(x)$. A set x is said to have hereditarily the property $\Phi(x)$ if x itself and all members of its transitive closure satisfy $\Phi(y)$, i.e. $x\cup \mathop{\rm tc}(x)\subseteq \{y|\Phi(y)\}$. Equivalently, x hereditarily satisfies $\Phi(x)$ iff it is a member of a transitive subset of $\{y|\Phi(y)\}$.^[9][10] A property (of a set) is thus said to be hereditary if is inherited by every subset. For example, being well-ordered is a hereditary property, and so it being finite.^[11]

If we instantiate in the above schema $\Phi(x)$ with "x has cardinality less than κ", we obtain the more general notion of a set being hereditarily of cardinality less than κ, usually denoted by $H_\kappa \!$^[12] or $H(\kappa) \!$. We regain the two simple notions we introduced above as $H(\omega)$ being the set of hereditarily finite sets and $H(\omega_1)$ being the set of hereditarily countable sets.^[13] ($\omega_1$ is the first uncountable ordinal.)

References

1. • Goreham, Anthony, "Sequential Convergence in Topological Spaces
2. Alon, Noga; Shapira, Asaf (2008). "Every monotone graph property is testable" (PDF). SIAM Journal on Computing 38 (2): 505–522. doi:10.1137/050633445.
3. M. Borowiecki, I. Broere, M. Frick, P. Mihók, G. Semanišin :A Survey of Hereditary Properties of Graphs, Discussiones Mathematicae – Graph Theory 17 (1997) 5–50.
4. A. Farrugia, Peter Mihók, R. Bruce Richter, Gabriel Semanišin: Factorizations and characterizations of induced-hereditary and compositive properties, Journal of Graph Theory 49(1): 11-27 (2005).
5. King, R. (1990), A lower bound for the recognition of digraph properties, Combinatorica, vol 10, 53–59
6. http://www.cs.ucsc.edu/~optas/papers/k-col-threshold.pdf
7. Spinrad, J. (2003), Efficient Graph Representations, AMS Bookstore, ISBN 0-8218-2815-0, p9.
8. Ashish Goel; Sanatan Rai; Bhaskar Krishnamachari (2003). "Monotone properties of random geometric graphs have sharp thresholds". Annals of Applied Probability 15 (4): 2535–2552. arXiv:math.PR/0310232. doi:10.1214/105051605000000575.
9. Azriel Levy (2002), Basic set theory, p. 82
10. Thomas Forster (2003), Logic, induction and sets, p. 197
11. Judith Roitman (1990), Introduction to modern set theory, p. 10
12. Levy (2002), p. 137
13. Kenneth Kunen (1983), Set theory, p. 131