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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.

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 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 AC (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 {} containing only the empty set is a hereditary set, and recursively so is {,{}}, 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:

Based on the above, it follows that in ZFC a more general notion can be defined for any predicate Φ(x). A set x is said to have hereditarily the property Φ(y) if x itself and all members of its transitive closure satisfy Φ(y), i.e. xtc(x){y|Φ(y)}. Equivalently, x hereditarily satisfies Φ(y) iff it is a member of a transitive subset of {y|Φ(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 Φ(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 Hκ[12] or H(κ). We regain the two simple notions we introduced above as H(ω) being the set of hereditarily finite sets and H(ω1) being the set of hereditarily countable sets.[13] (ω1 is the first uncountable ordinal.)

References

  1. *Goreham, Anthony, "Sequential Convergence in Topological Spaces
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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  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. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  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

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