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{{DISPLAYTITLE:''n''-ary group}}
In [[mathematics]], an '''''n''-ary group''' (also '''''n''-group''', '''polyadic group''' or '''multiary group''') is a generalization of a [[group (mathematics)|group]] to a set ''G'' with a [[arity|''n''-ary operation]] instead of a binary operation.<ref name="oldandnew">{{Citation| last=Dudek |first=W.A. |title=On some old and new problems in ''n''-ary groups |url=http://www.quasigroups.eu/contents/contents8.php?m=trzeci |journal=Quasigroups and Related Systems |year=2001 |volume=8 |pages= 15–36}}.</ref> The [[axioms]] for an ''n''-ary group are defined in such a way as to reduce to those of a group in the case {{nowrap|''n'' {{=}} 2}}.


==Axioms==
===Associativity===
The easiest axiom to generalize is the associative law. Ternary associativity is {{nowrap|(''abc'')''de''&nbsp;{{=}}&nbsp;''a''(''bcd'')''e''&nbsp;{{=}}&nbsp;''ab''(''cde'')}}, i.e. the string ''abcde'' with any three adjacent elements bracketed. ''n''-ary associativity is a string of length ''n''+(''n''-1) with any ''n'' adjacent  elements bracketed. A set ''G'' with a closed ''n''-ary operation is an '''''n''-ary [[Magma (algebra)|groupoid]]'''. If the operation is associative then it is an '''''n''-ary semigroup.'''


===Inverses / Unique Solutions===
Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. It is used by around 25% of all new websites, and there are more than 27 thousand plugins currently available. The effect is to promote older posts by moving them back onto the front page and into the rss feed. If you're using Wordpress and want to make your blog a "dofollow" blog, meaning that links from your blog pass on the benefits of Google pagerank, you can install one of the many dofollow plugins available. In the event you loved this short article and you would like to receive details with regards to [http://yo.io/backup_plugin_551257 wordpress backup] please visit our own web-page. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website. <br><br>Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. Wordpress have every reason with it which promote wordpress development. This is the reason for the increased risk of Down Syndrome babies in women over age 35. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Word - Press makes it possible to successfully and manage your website. <br><br>The entrepreneurs can easily captivate their readers by using these versatile themes. Note:  at a first glance WP Mobile Pro  themes do not appear to be glamorous or fancy. You've got invested a great cope of time developing and producing up the topic substance. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. If you've hosted your Word - Press website on a shared hosting server then it'll be easier for you to confirm the restricted access to your site files. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Instead, you can easily just include it with our bodies integration field in e - Panel. s ability to use different themes and skins known as Word - Press Templates or Themes. However, there are a few other Wordpress plugins also for its development which requires adding files in your Wordpress setup. You should stay away from plugins that are full of flaws and bugs. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and  full user registration plug-ins.
The inverse axiom is generalized as follows: in the case of binary operations the existence of an inverse means {{nowrap|''ax'' {{=}} ''b''}} has a unique solution for ''x'', and likewise {{nowrap|''xa'' {{=}} ''b''}} has a unique solution. In the ternary case we generalize this to {{nowrap|''abx'' {{=}} ''c''}}, {{nowrap|''axb'' {{=}} ''c''}} and {{nowrap|''xab'' {{=}} ''c''}} each having unique solutions, and the ''n''-ary case follows a similar pattern of existence of unique solutions and we get an '''''n''-ary quasigroup.'''
 
===Definition of ''n-ary''-group===
An '''''n''-ary group''' is an ''n''-ary semigroup which is also an ''n''-ary quasigroup.
 
===Identity / Neutral elements===
In the 2-ary case, i.e. for an ordinary group, the existence of an identity element is a consequence of the associativity and inverse axioms, however in n-ary groups for n ≥ 3 there can be zero, one, or many identity elements.  
 
An ''n''-ary groupoid (''G'',&nbsp;''ƒ'') with ''ƒ'' = (''x''<sub>1</sub> ◦ ''x''<sub>2</sub> ◦ . . . ◦ ''x''<sub>''n''</sub>), where (''G'',&nbsp;◦) is a group is called reducible or derived from the group (''G'',&nbsp;◦). In 1928 Dornte published the first main results: An ''n''-ary groupoid which is reducible is an ''n''-ary group, however for all ''n''&nbsp;>&nbsp;2 there exist ''n''-ary groups which are not reducible. In some ''n''-ary groups there exists an element ''e'' (called an ''n''-ary identity or neutral element) such that any string of ''n''-elements consisting of all ''e''<nowiki>'</nowiki>s, apart from one place, is mapped to the element at that place. E.g., in a quaternary group with identity ''e'', ''eeae''&nbsp;=&nbsp;''a'' for every ''a''.
 
An ''n''-ary group containing a neutral element is reducible. Thus, an ''n''-ary group that is not reducible does not contain such elements. There exist ''n''-ary groups with more than one neutral element. If the set of all neutral elements of an ''n''-ary group is non-empty it forms an ''n''-ary subgroup.<ref>Wiesław A. Dudek, [http://arxiv.org/abs/0704.2749 Remarks to Głazek's results on ''n''-ary groups], ''Discussiones Mathematicae. General Algebra and Applications'' 27 (2007), 199–233.</ref>
 
Some authors include an identity in the definition of an ''n''-ary group but as mentioned above such ''n''-ary operations are just repeated binary operations. Groups with intrinsically ''n''-ary operations do not have an identity element.<ref>Wiesław A. Dudek and Kazimierz Głazek, [http://arxiv.org/abs/math/0510185v1 Around the Hosszú-Gluskin theorem for ''n''-ary groups], ''Discrete Mathematics'' 308 (2008), 486&ndash;4876.</ref>
 
===Weaker axioms===
 
The axioms of associativity and unique solutions in the definition of an ''n''-ary group are stronger than they need to be. Under the assumption of ''n''-ary associativity it suffices to postulate the existence of the solution of equations with the unknown at the start or end of the string, or at one place other than the ends; e.g., in the 6-ary case, ''xabcde''=''f'' and ''abcdex''=''f'', or an expression like ''abxcde''=''f''. Then it can be proved that the equation has a unique solution for ''x'' in any place in the string.<ref>E. L. Post, [http://www.ams.org/journals/tran/1940-048-02/S0002-9947-1940-0002894-7/home.html Polyadic groups], ''Transactions of the American Mathematical Society'' 48 (1940), 208&ndash;350.</ref> The associativity axiom can also be given in a weaker form - see page 17 of "On some old and new problems in ''n''-ary groups".<ref name="oldandnew"/>
 
==Example==
The following is an example of a three element ternary group, one of four such groups<ref>http://home.comcast.net/~tamivox/dave/math/tern_quasi/assoc1234.html</ref>
 
:<math>aaa = a, aab = b, aac = c, aba = c, abb = a, abc = b, aca = b, acb = c, acc = a,</math>
:<math>baa = b, bab = c, bac = a, bba = a, bbb = b, bbc = c, bca = c, bcb = a, bcc = b,</math>
:<math>caa = c, cab = a, cac = b, cba = b, cbb = c, cbc = a, cca = a, ccb = b, ccc = c.</math>
 
==See also==
*[[Universal algebra]]
 
==References==
{{reflist}}
* S. A. Rusakov: Some applications of n-ary group theory, (Russian), Belaruskaya navuka, Minsk 1998.
 
[[Category:Algebraic structures]]

Latest revision as of 22:51, 13 September 2014


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