Shannon's source coding theorem

In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup.

History

The first published description of this object was given by Evgenii Lyapin in 1953. Alfred H. Clifford and Gordon Preston claim that one of them, working with David Rees, discovered it independently (without publication) at some point before 1943.

Construction

There are at least three standard ways of constructing the bicyclic semigroup, and various notations for referring to it. Lyapin called it P; Clifford and Preston used $\mathcal{𝒞}$; and most recent papers have tended to use B. This article will use the modern style throughout.

From a free semigroup

The bicyclic semigroup is the free semigroup on two generators p and q, under the relation p q = 1. That is, each semigroup element is a string of those two letters, with the proviso that the subsequence "p q" does not appear. The semigroup operation is concatenation of strings, which is clearly associative. It can then be shown that all elements of B in fact have the form q^a p^b, for some natural numbers a and b. The composition operation simplifies to

(q^a p^b) (q^c p^d) = q^{a − b + max{b, c}} p^{d − c + max{b, c}}.

From ordered pairs

The way in which these exponents are constrained suggests that the "p and q structure" can be discarded, leaving only operations on the "a and b" part. So B is the semigroup of pairs of natural numbers (including zero), with operation

(a, b) (c, d) = (a − b + max{b, c}, d − c + max{b, c}).

This is sufficient to define B so that it is the same object as in the original construction. Just as p and q generated B originally, with the empty string as the monoid identity, this new construction of B has generators (1, 0) and (0, 1), with identity (0, 0).

From functions

It can be shown that any semigroup S generated by elements e, a, and b satisfying the statements below is isomorphic to the bicyclic semigroup.

• a e = e a = a
• b e = e b = b
• a b = e
• b a ≠ e

It is not entirely obvious that this should be the case—perhaps the hardest task is understanding that S must be infinite. To see this, suppose that a (say) does not have infinite order, so a^{k + h} = a^h for some h and k. Then a^k = e, and

b = e b = a^k b = a^{k - 1} e = a^{k - 1},

so

b a = a^k = e,

which is not allowed—so there are infinitely many distinct powers of a. The full proof is given in Clifford and Preston's book.

Note that the two definitions given above both satisfy these properties. A third way of deriving B uses two appropriately-chosen functions to yield the bicyclic semigroup as a monoid of transformations of the natural numbers. Let α, β, and ι be elements of the transformation semigroup on the natural numbers, where

• ι(n) = n
• α(n) = n + 1
• β(n) = 0 if n = 0, and n − 1 otherwise.

These three functions have the required properties, so the semigroup they generate is B.

Properties

The bicyclic semigroup has the property that the image of any morphism φ from B to another semigroup S is either cyclic, or it is an isomorphic copy of B. The elements φ(a), φ(b) and φ(e) of S will always satisfy the conditions above (because φ is a morphism) with the possible exception that φ(b) φ(a) might turn out to be φ(e). If this is not true, then φ(B) is isomorphic to B; otherwise, it is the cyclic semigroup generated by φ(a). In practice, this means that the bicyclic semigroup can be found in many different contexts.

The idempotents of B are all pairs (x, x), where x is any natural number (using the ordered pair characterisation of B). Since these commute, and B is regular (for every x there is a y such that x y x = x), the bicyclic semigroup is an inverse semigroup. (This means that each element x of B has a unique inverse y, in the "weak" semigroup sense that x y x = x and y x y = y.)

Every ideal of B is principal: the left and right principal ideals of (m, n) are

• (m, n) B = {(s, t) : s ≥ m} and
• B (m, n) = {(s, t) : t ≥ n}.

Each of these contains infinitely many others, so B does not have minimal left or right ideals.

In terms of Green's relations, B has only one D-class (it is bisimple), and hence has only one J-class (it is simple). The L and R relations are given by

• (a, b) R (c, d) if and only if a = c; and
• (a, b) L (c, d) if and only if b = d.

This implies that two elements are H-related if and only if they are identical. Consequently, the only subgroups of B are infinitely many copies of the trivial group, each corresponding to one of the idempotents.

The egg-box diagram for B is infinitely large; the upper left corner begins:

(0, 0)	(1, 0)	(2, 0)	...
(0, 1)	(1, 1)	(2, 1)	...
(0, 2)	(1, 2)	(2, 2)	...
...	...	...	...

Each entry represents a singleton H-class; the rows are the R-classes and the columns are L-classes. The idempotents of B appear down the diagonal, in accordance with the fact that in a regular semigroup with commuting idempotents, each L-class and each R-class must contain exactly one idempotent.

The bicyclic semigroup is the "simplest" example of a bisimple inverse semigroup with identity; there are many others. Where the definition of B from ordered pairs used the class of natural numbers (which is not only an additive semigroup, but also a commutative lattice under min and max operations), another set with appropriate properties could appear instead, and the "+", "−" and "max" operations modified accordingly.

Relation to combinatorics

The bicyclic monoid occurs in combinatorics, as the syntactic monoid of the Dyck language. The Dyck language is the set of all strings of balanced pairs of parentheses, and thus finds common applications in defining binary trees and associative algebras.

See also

• Four-spiral semigroup
• Special classes of semigroups

References

• The algebraic theory of semigroups, A. H. Clifford and G. B. Preston. American Mathematical Society, 1961 (volume 1), 1967 (volume 2).
• Semigroups: an introduction to the structure theory, Pierre Antoine Grillet. Marcel Dekker, Inc., 1995.
• Canonical form of elements of an associative system given by defining relations, Evgenii Sergeevich Lyapin, Leningrad Gos. Ped. Inst. Uch. Zap. 89 (1953), pages 45–54 [Russian].
• 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