Gilbreath's conjecture

In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed point theorem for partially ordered sets. It states that if X is a non-empty chain complete poset, and

$f:X\to X$

such that

$f(x)\geq x$ for all $x,$

then f has a fixed point. Such a function f is called inflationary or progressive.

Special case of a finite poset

If the poset X is finite then the statement of the theorem has a clear interpretation that leads to the proof. The sequence of successive iterates,

$x_{n+1}=f(x_{n}),n=0,1,2,\ldots ,$

where x₀ is any element of X, is monotone increasing. By the finiteness of X, it stabilizes:

$x_{n}=x_{\infty },\quad$ for n sufficiently large.

It follows that x_∞ is a fixed point of f.

Proof of the theorem

Pick some $y\in X$. Define a function K recursively on the ordinals as follows:

$\,K(0)=y$

$\,K(\alpha +1)=f(K(\alpha )).$

If $\beta$ is a limit ordinal, then by construction

$\{K(\alpha )\ :\ \alpha <\beta \}$

is a chain in X. Define

$K(\beta )=\sup\{K(\alpha )\ :\ \alpha <\beta \}.$

This is now an increasing function from the ordinals into X. It cannot be strictly increasing, as if it were we would have an injective function from the ordinals into a set, violating Hartogs' lemma. Therefore the function must be eventually constant, so for some

$\alpha ,\ \ K(\alpha +1)=K(\alpha );$

that is,

$\,f(K(\alpha ))=K(\alpha ).$

So letting

$\,x=K(\alpha ),$

we have our desired fixed point. Q.E.D.

Applications

The Bourbaki–Witt theorem has various important applications. One of the most common is in the proof that the axiom of choice implies Zorn's lemma. We first prove it for the case where X is chain complete and has no maximal element. Let g be a choice function on

$P(X)-\{\varnothing \}.$

Define a function

$f:X\to X$

by

$f(x)=g(\{y\ :\ y>x\}).$

This is allowed as, by assumption, the set is non-empty. Then f(x) > x, so f is an inflationary function with no fixed point, contradicting the theorem.

This special case of Zorn's lemma is then used to prove the Hausdorff maximality principle, that every poset has a maximal chain, which is easily seen to be equivalent to Zorn's Lemma.

Bourbaki–Witt has other applications. In particular in computer science, it is used in the theory of computable functions. It is also used to define recursive data types, e.g. linked lists, in domain theory.

References

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  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
  
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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