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In [[mathematics]], the '''bicyclic semigroup''' is an algebraic object important for the structure theory of [[semigroup]]s. Although it is in fact a [[monoid]], it is usually referred to as simply a semigroup.
== you and I shot four people at the same time ==


==History==
Roared.<br><br>'Tang Zhen Gu Zhu, glaciers Guzhu, and the Xiao Yan little friend, you and I shot four people at the same time, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 腕時計 メンズ casio] to help fight the holy bones stardom solve this old devil, how?' Old Devil roar heard [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 腕時計 チタン] stardom , the day the demon Phoenix family man also knows not look black robe drama, the moment the attention is turned to Tang Zhen, glaciers, such as strength reached a five-star and four stars statue strong, but so was a bit surprised people is that he this opening times, even the strength of the stars just a statue of Xiao Yan [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html casio 腕時計 説明書] also called on to.<br><br>Xiao Yan fingers moved, glances at the old man a black robe, this old guy is obviously worried about too leisurely, the remaining spare capacity too much against them, so he [http://nrcil.net/sitemap.xml http://nrcil.net/sitemap.xml] wanted to pull into this that the most dangerous Battle circle.<br><br>'ah.' Xiao Yan [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html casio 腕時計 edifice] Shen between 'Yin', Tang Zhen and ice Venerable'd nodded slightly, seeing this, Xiao Yan could only nod, eyes looked around, and said: 'As the six with
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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 (mathematician)|David Rees]], discovered it independently (without publication) at some point before 1943.
== the other one is bound to be hit. ==


==Construction==
Xiao Yan is taking into account the two difficult position, [http://nrcil.net/sitemap.xml http://nrcil.net/sitemap.xml] if only anti Xiao [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html casio 腕時計] Yan a, the other one is bound to be hit.<br><br>in numerous road gaze, as sculptural Xiao Yan, the two Quanfeng getting to that Yi Sha, stature and shoved shocked, everyone is stunned immediately saw two illusory shadow flotation [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 gps] feet on both sides , while two unreal foot shadow, while Cliff and Lin Xiu Qing and Liu fist collision, but like turning [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ 腕時計 電波 ソーラー] to [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 デジタル] the substance of like, an instant burst of extremely terror forces.<br><br>first trick!<br><br>'bang!'<br><br>sounded muffled muffled presence, fury Effort at that moment, such as a flood vent out, so that was also a direct cliff and Liu Qing Lin Xiu figure Baotui nearly ten steps, every foot of the fall, will be in left deep marks on the floor in the foot.<br><br>'This guy, good speed and strength of terror,' Liu Qing stabilize stature, heart an idea just flashed, muddy
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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 <math>\mathcal{C}</math>; and most recent papers have tended to use ''B''. This article will use the modern style throughout.
== is sharp with a little overbearing ==


===From a free semigroup===
Warm sun shining on the peaks sit cross-legged black robe young body, and seems to feel the outside world weather changes, quivering young eyes closed for a moment, fingertips lingering hint of silver 'color' lightning, but also quietly into body, and finally disappear.<br><br>with silver 'color' lightning disappeared, the young black robe eyelashes 'gross' trembling increasingly violent, after a moment, as if finally broke loose in general, suddenly opened, suddenly, like real silver 'color' lightning, is sharp with a little overbearing, violent [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html casio 腕時計 レディース] 'shoot' out of its eyes, and reached full distance [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 gps] Cunxu long!<br><br>eyes 'shoot' the silver 'color' lightning lasted just one moment of time is dissipated abruptly, and with silver 'color' lightning disappeared, and that the black eye is once again caught in peace.<br><br>hand printed statement dispersed, Xiao [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html 腕時計 casio] Yan slightly [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 時計 カシオ] raised his head, looked exudes [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオの時計] the warmth of the sun on the sky Yaori chest
 
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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.
<ul>
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''<sup>''a''</sup> ''p''<sup>''b''</sup>, for some [[natural number]]s ''a'' and ''b''. The composition operation simplifies to
  <li>[http://www.jlywx.com/plus/feedback.php?aid=29 http://www.jlywx.com/plus/feedback.php?aid=29]</li>
: (''q''<sup>''a''</sup> ''p''<sup>''b''</sup>) (''q''<sup>''c''</sup> ''p''<sup>''d''</sup>) = ''q''<sup>''a'' &minus; ''b'' + max{''b'', ''c''}</sup> ''p''<sup>''d'' &minus; ''c'' + max{''b'', ''c''}</sup>.
 
 
  <li>[http://www.cgiw.cn/plus/feedback.php?aid=74 http://www.cgiw.cn/plus/feedback.php?aid=74]</li>
===From ordered pairs===
 
 
  <li>[http://www.iera.org.cn/plus/feedback.php?aid=1164 http://www.iera.org.cn/plus/feedback.php?aid=1164]</li>
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<ref>Hollings (2007), p. 332</ref>
</ul>
:(''a'', ''b'') (''c'', ''d'') = (''a'' &minus; ''b'' + max{''b'', ''c''}, ''d'' &minus; ''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 [[Isomorphism|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&mdash;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''<sup>''k'' + ''h''</sup> = ''a''<sup>''h''</sup> for some ''h'' and ''k''. Then ''a''<sup>''k''</sup> = ''e'', and
:''b'' = ''e'' ''b'' = ''a''<sup>''k''</sup> ''b'' = ''a''<sup>''k'' - 1</sup> ''e'' = ''a''<sup>''k'' - 1</sup>,
so
:''b'' ''a'' = ''a''<sup>''k''</sup> = ''e'',
which is not allowed&mdash;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'' &minus; 1 otherwise.
These three functions have the required properties, so the semigroup they generate is ''B''.<ref name=Lot459>{{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=Cambridge University Press | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 | page=459 }}</ref>
 
==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 [[idempotent]]s 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 (ring theory)|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''.<ref>Howie p.60</ref>
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 [[Green's relations|egg-box diagram]] for ''B'' is infinitely large; the upper left corner begins:
{| cellpadding=6
| (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 (order)|lattice]] under min and max operations), another set with appropriate properties could appear instead, and the "+", "&minus;" 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 tree]]s and [[associative algebra]]s.
 
==See also==
*[[Four-spiral semigroup]]
*[[Special classes of semigroups]]
 
== Notes ==
<references />
 
==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].
* {{cite journal |last1=Hollings |first1=C.D. |last2= |first2= |year=2007 |title=Some First Tantalizing Steps into Semigroup Theory |journal=Mathematics Magazine |volume=80 |issue= |pages=331–344 |publisher=Mathematical Association of America |doi= |jstor=27643058 }}
 
[[Category:Semigroup theory]]

Revision as of 04:04, 28 February 2014

you and I shot four people at the same time

Roared.

'Tang Zhen Gu Zhu, glaciers Guzhu, and the Xiao Yan little friend, you and I shot four people at the same time, 腕時計 メンズ casio to help fight the holy bones stardom solve this old devil, how?' Old Devil roar heard カシオ 腕時計 チタン stardom , the day the demon Phoenix family man also knows not look black robe drama, the moment the attention is turned to Tang Zhen, glaciers, such as strength reached a five-star and four stars statue strong, but so was a bit surprised people is that he this opening times, even the strength of the stars just a statue of Xiao Yan casio 腕時計 説明書 also called on to.

Xiao Yan fingers moved, glances at the old man a black robe, this old guy is obviously worried about too leisurely, the remaining spare capacity too much against them, so he http://nrcil.net/sitemap.xml wanted to pull into this that the most dangerous Battle circle.

'ah.' Xiao Yan casio 腕時計 edifice Shen between 'Yin', Tang Zhen and ice Venerable'd nodded slightly, seeing this, Xiao Yan could only nod, eyes looked around, and said: 'As the six with 相关的主题文章：

the other one is bound to be hit.

Xiao Yan is taking into account the two difficult position, http://nrcil.net/sitemap.xml if only anti Xiao casio 腕時計 Yan a, the other one is bound to be hit.

in numerous road gaze, as sculptural Xiao Yan, the two Quanfeng getting to that Yi Sha, stature and shoved shocked, everyone is stunned immediately saw two illusory shadow flotation カシオ 腕時計 gps feet on both sides , while two unreal foot shadow, while Cliff and Lin Xiu Qing and Liu fist collision, but like turning カシオ 腕時計 電波 ソーラー to casio 腕時計 デジタル the substance of like, an instant burst of extremely terror forces.

first trick!

'bang!'

sounded muffled muffled presence, fury Effort at that moment, such as a flood vent out, so that was also a direct cliff and Liu Qing Lin Xiu figure Baotui nearly ten steps, every foot of the fall, will be in left deep marks on the floor in the foot.

'This guy, good speed and strength of terror,' Liu Qing stabilize stature, heart an idea just flashed, muddy 相关的主题文章：

is sharp with a little overbearing

Warm sun shining on the peaks sit cross-legged black robe young body, and seems to feel the outside world weather changes, quivering young eyes closed for a moment, fingertips lingering hint of silver 'color' lightning, but also quietly into body, and finally disappear.

with silver 'color' lightning disappeared, the young black robe eyelashes 'gross' trembling increasingly violent, after a moment, as if finally broke loose in general, suddenly opened, suddenly, like real silver 'color' lightning, is sharp with a little overbearing, violent casio 腕時計 レディース 'shoot' out of its eyes, and reached full distance カシオ 腕時計 gps Cunxu long!

eyes 'shoot' the silver 'color' lightning lasted just one moment of time is dissipated abruptly, and with silver 'color' lightning disappeared, and that the black eye is once again caught in peace.

hand printed statement dispersed, Xiao 腕時計 casio Yan slightly 時計 カシオ raised his head, looked exudes カシオの時計 the warmth of the sun on the sky Yaori chest 相关的主题文章：

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