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In [[mathematics]], an '''automatic semigroup''' is a finitely generated [[semigroup]] equipped with several [[regular languages]] over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator.


Formally, let <math>S</math> be a semigroup and <math>A</math> be a finite set of generators.  Then an ''automatic structure'' for <math>S</math> with respect to <math>A</math> consists of a regular language <math>L</math> over <math>A</math> such that every element of <math>S</math> has at least one representative in <math>L</math> and such that for each <math>a \in A \cup \{\varepsilon\}</math>, the relation consisting of pairs <math>(u,v)</math> with <math>ua = v</math> is regular.


The concept of an automatic semigroup was generalized from [[automatic group]]s by Campbell et al. (2001)
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By obtaining the assistance of a skilled, knowledgable legal professional, you tremendously increase the probability of obtaining the monetary compensation you want and deserve. Hopefully this article has offered the instruments to assist you do so.
 
Unlike automatic groups (see Epstein et al. 1992), a semigroup may have an automatic structure with respect to one generating set, but not with respect to another. However, if an automatic semigroup has an identity, then it has an automatic structure with respect to any generating set (Duncan et al. 1999).
 
==Decision problems==
 
Like automatic groups, automatic semigroups have [[word problem for groups|word problem]] solvable in quadratic time. Kambites & Otto (2006) showed that it is undecidable whether an element of an automatic monoid possesses a right inverse.
 
Cain (2006) proved that both cancellativity and left-cancellativity are undecidable for automatic semigroups. On the other hand, right-cancellativity is decidable for automatic semigroups (Silva & Steinberg 2004).
 
==Geometric characterization==
 
Automatic structures for groups have an elegant geometric characterization called the ''fellow traveller property'' (Epstein et al. 1992, ch. 2). Automatic structures for semigroups ''possess'' the fellow traveller property but are not in general characterized by it (Campbell et al. 2001). However, the characterization can be generalized to certain '[[Group (mathematics)|group]]-like' classes of semigroups, notably [[completely simple semigroup]]s (Campbell et al. 2002) and group-embeddable semigroups (Cain et al. 2006).
 
==Examples of automatic semigroups==
*[[Bicyclic monoid]]
*Finitely generated subsemigroups of a [[free semigroup]]
 
== References ==
 
*{{citation
| last1 = Cain | first1 = Alan J.
| title = Cancellativity is undecidable for automatic semigroups
| journal = Quarterly Journal of Mathematics
| volume = 57 | year = 2006 | issue = 3 | pages = 285–295
| doi = 10.1093/qmath/hai023}}
 
*{{citation
| last1 = Cain | first1 = Alan J.
| last2 = Robertson | first2 = Edmund F.
| last3 = Ruskuc | first3 = Nik
| title = Subsemigroups of groups: presentations, Malcev presentations, and automatic structures
| journal = Journal of Group Theory
| volume = 9 | year = 2006 | issue = 3 | pages = 397–426
| doi = 10.1515/JGT.2006.027}}.
 
*{{citation
| last1 = Campbell | first1 = Colin M.
| last2 = Robertson | first2 = Edmund F.
| last3 = Ruskuc | first3 = Nik
| last4 = Thomas | first4 = Richard M.
| title = Automatic semigroups
| journal = Theoretical Computer Science
| volume = 250 | year = 2001 | issue = 1–2 | pages = 365–391
| doi = 10.1016/S0304-3975(99)00151-6}}.
 
*{{citation
| last1 = Campbell | first1 = Colin M.
| last2 = Robertson | first2 = Edmund F.
| last3 = Ruskuc | first3 = Nik
| last4 = Thomas | first4 = Richard M.
| title = Automatic completely simple semigroups
| journal = Acta Mathematica Hungarica
| volume = 95 | year = 2002 | issue = 3 | pages = 201–215
| doi = 10.1023/A:1015632704977}}.
 
*{{citation
  | last1 = Duncan | first1 = A. J.
| last2 = Robertson | first2 = E. F.
| last3 = Ruskuc | first3 = N.
| title = Automatic monoids and change of generators
| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| volume = 127 | year = 1999 | issue = 3 | pages = 403–409
| doi = 10.1017/S0305004199003722}}.
 
*{{citation
| last1 = Epstein | first1 = David B. A. | authorlink1 = David B. A. Epstein
| last2 = Cannon | first2 = James W.
| last3 = Holt | first3 = Derek F.
| last4 = Levy | first4 = Silvio V. F.
| last5 = Paterson | first5 = Michael S.
| last6 = Thurston | first6 = William P. | authorlink6 = William Thurston
| title = Word Processing in Groups
| publisher = Jones and Bartlett Publishers | location = Boston, MA | year = 1992 | isbn = 0-86720-244-0}}.
 
*{{citation
| last1 = Kambites | first1 = Mark
| last2 = Otto
| first2 = F
| title = Uniform decision problems for automatic semigroups
| journal = Journal of Algebra
| volume = 303 | year = 2006 | issue = 2 | pages = 789–809
| doi = 10.1016/j.jalgebra.2005.11.028}}
 
*{{citation
| last1 = Silva | first1 = P.V.
| last2 = Steinberg | first2 = B.
| title = A geometric characterization of automatic monoids
| journal = Quarterly Journal of Mathematics
| volume = 55 | issue = 3 | year = 2004 | pages = 333–356
| doi = 10.1093/qmath/hah006}}
 
[[Category:Semigroup theory]]
[[Category:Computability theory]]

Latest revision as of 14:22, 6 June 2014


The upheaval and lose faith that will result from a persona damage can often last a long time , and it is important to protect the payment you are entitled to if some thing this sad comes about in your life. The best thing you can do less than such situations is to discover a excellent personal injury legal representative you never know what must be done to triumph in such issues. Keep reading to find out how.

Compile paperwork, varieties and statements relevant to your healthcare expenses. This will consist of medical doctor notes, statements and following damage treatment directions. Help save e-mail through your medical doctor concerning your trauma.

Getting along with your legal representative is essential, but there is however some thing you will need to look for. Opt for legal counsel who really listens to what you say. Should they at any time seem to not hearing, don't select that lawyer. While they may believe they understand all this, typically people who have that notion are the types who understand the minimum.

Should you be working with a injury situation, you need to check out all your doctor's meetings and get them nicely reported. If you want to come forth with an optimistic judgment, you've got to have confirmation that you simply were wounded and confirmation that you http://youtube.com/watch?v=iLpYNP_oSfo are currently hoping to get much better too. If you don't, a legal court won't trust you.

Be sure you request the proper concerns if you talk to an individual injuries legal representative. Make sure the lawyer or attorney is skilled in personal injury circumstances and is also competent in your state. Ask what kind of document the legal representative has. You need a profitable attorney, obviously, and something who may have many years of practical experience.

Whilst getting to courtroom rapidly helps to the scenario, you will probably find your lawyer is simply too busy. Locate a legal representative who seems to be offered immediately so that the odds are loaded within your love. Of course, this is among numerous variables you need to think about when hiring a legal representative.

When choosing the individual injuries attorney that suits your needs, take advantage of the cost-free appointment that is made available from most trustworthy law offices. Request the important points of how you will certainly be charged for that services you will need and ensure being fully honest using the legal representative about the specifics of your situation.

Be completely sincere with your legal representative. Your legal professional is sure by a discretion clause, so go ahead and inform them whatever you know, even though you consider it may well damage your scenario. Withholding information and facts is actually unhealthy for your case than everything you could inform your legal representative. An attorney must know each of the information available to become efficient.

Point any individual on the opposite side in the scenario to talk to your legal representative if they try to contact you. This can be to actually do not say something that could impact your situation adversely, as well as promise http://youtube.com/watch?v=iLpYNP_oSfo that your particular attorney knows specifically what is going on. Remember this tip to ensure that your case has the greatest final result.

Employ a nearby accidental injuries attorney. It can be appealing to call that 1-800 number you discovered on the Television with this or that attorney, but you'll most likely get yourself a lawyer who life not anywhere close to you. When the situation is on the line, you'll want so that you can talk encounter-to-encounter with the lawyers, so nearby is best.

Most accidental injury attorneys signify their clients using a contingency contract. A contingency contract ensures that the lawyer doesn't receive money unless of course he wins your situation. Most contingency contracts declare that the attorney will get a portion of your respective settlement in case the circumstance is received. To shield yourself, completely look at the contingency contract and decide on the portion the legal representative is eligible for if he wins your circumstance.

Using a legitimate recommendation service is a doubtful way of finding a legal professional. Even though some solutions are extremely strict in who they allow within their fold, other folks will take anybody who pays off the every year payment. Discover what credentials needs to be fulfilled or proven by any assistance you are looking for.

Ensure your individual trauma attorney takes care of your location of require. In case you have a workers comp claim, that is certainly very different from an automobile incident state or even a malpractice state. Numerous attorneys are skilled, which means they are aware the legal guidelines around quite distinct varieties of circumstances. Be sure your lawyer is extremely knowledgeable about his section of experience and truly has got the information required to make the greatest situation.

If the unthinkable happens and you also endure a personal damage due to acts or omissions of another, you are certain to concern yourself with the influence the event could have on your ability to offer your family's requires. By obtaining the assistance of a skilled, knowledgable legal professional, you tremendously increase the probability of obtaining the monetary compensation you want and deserve. Hopefully this article has offered the instruments to assist you do so.