 nary group

In mathematics, an nary group (also ngroup, polyadic group or multiary group) is a generalization of a group to a set G with a nary operation instead of a binary operation.^{[1]} The axioms for an nary group are defined in such a way as to reduce to those of a group in the case n = 2.
Contents
Axioms
Associativity
The easiest axiom to generalize is the associative law. Ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. nary associativity is a string of length n+(n1) with any n adjacent elements bracketed. A set G with a closed nary operation is an nary groupoid. If the operation is associative then it is an nary semigroup.
Inverses / Unique Solutions
The inverse axiom is generalized as follows: in the case of binary operations the existence of an inverse means ax = b has a unique solution for x, and likewise xa = b has a unique solution. In the ternary case we generalize this to abx = c, axb = c and xab = c each having unique solutions, and the nary case follows a similar pattern of existence of unique solutions and we get an nary quasigroup.
Definition of narygroup
An nary group is an nary semigroup which is also an nary quasigroup.
Identity / Neutral elements
In the 2ary case, i.e. for an ordinary group, the existence of an identity element is a consequence of the associativity and inverse axioms, however in nary groups for n ≥ 3 there can be zero, one, or many identity elements.
An nary groupoid (G, ƒ) with ƒ = (x_{1} ◦ x_{2} ◦ . . . ◦ x_{n}), where (G, ◦) is a group is called reducible or derived from the group (G, ◦). In 1928 Dornte published the first main results: An nary groupoid which is reducible is an nary group, however for all n > 2 there exist nary groups which are not reducible. In some nary groups there exists an element e (called an nary identity or neutral element) such that any string of nelements consisting of all e's, apart from one place, is mapped to the element at that place. E.g., in a quaternary group with identity e, eeae = a for every a.
An nary group containing a neutral element is reducible. Thus, an nary group that is not reducible does not contain such elements. There exist nary groups with more than one neutral element. If the set of all neutral elements of an nary group is nonempty it forms an nary subgroup.^{[2]}
Some authors include an identity in the definition of an nary group but as mentioned above such nary operations are just repeated binary operations. Groups with intrinsically nary operations do not have an identity element.^{[3]}
Weaker axioms
The axioms of associativity and unique solutions in the definition of an nary group are stronger than they need to be. Under the assumption of nary 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 6ary 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.^{[4]} The associativity axiom can also be given in a weaker form  see page 17 of "On some old and new problems in nary groups".^{[1]}
See also
References
 ^ ^{a} ^{b} Dudek, W.A. (2001), "On some old and new problems in nary groups", Quasigroups and Related Systems 8: 15–36, http://www.quasigroups.eu/contents/contents8.php?m=trzeci.
 ^ Wiesław A. Dudek, Remarks to Głazek's results on nary groups, Discussiones Mathematicae. General Algebra and Applications 27 (2007), 199–233.
 ^ Wiesław A. Dudek and Kazimierz Głazek, Around the HosszúGluskin theorem for nary groups, Discrete Mathematics 308 (2008), 486–4876.
 ^ E. L. Post, Polyadic groups, Transactions of the American Mathematical Society 48 (1940), 208–350.
 S. A. Rusakov: Some applications of nary group theory, (Russian), Belaruskaya navuka, Minsk 1998.
Categories: Algebraic structures
Wikimedia Foundation. 2010.
Look at other dictionaries:
ARY Group — The ARY Group or Abdul Razzaq Yaqoob Group is a Dubai based holding company founded by a Pakistani businessman, Haji Abdul Razzak Yaqoob (ARY). Haji Abdul Razzak is the Chairman of the Group. The ARY Group is into various sectors, which include:* … Wikipedia
ARY Digital — Launched 2000 Network ARY Digital Network Owned by ARY Group Country Pakistan Sister channel(s) ARY News ARY Musik ARY Qtv … Wikipedia
ARY Television Network — is a subsidiary of the ARY Group. The ARY Group of companies is a Dubai based holding company founded by a Pakistani businessman, Haji Abdul Razzak Yaqoob (ARY). The channels owned or affiliated with ARY Television Network include:* ARY Digital… … Wikipedia
ARY — may stand for: Abdul Razzak Yaqoob, a Pakistani expatriate businessman Andre Romelle Young, real name of Dr. Dre ARY may also refer to: ARY Digital, a Pakistani television network ARY Group, a Dubai based holding company ARY Television Network, a … Wikipedia
Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines … Wikipedia
ARY One World — Infobox TV channel name = ARY One World logofile = ARY One World.jpg logoalt = ARY one world launch = 2004 available = Worldwide broadcast area = Australia, Europe, Asia, India, Middle East, United Kingdom, United States owner = ARY Group country … Wikipedia
List of programs broadcast by ARY Digital — ARY Digital, a venture of the ARY Group, is a popular Pakistani television network available in Pakistan, the Middle East, North America, Europe and Australia. The network caters to the needs of South Asians, particularly the Pakistani diaspora.… … Wikipedia
Link group — In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Bachelor s thesis, (Milnor 1954). Contents 1 Definition 2 Examples 3 … Wikipedia
QTV (ARY) — Infobox TV channel name = Quran TV logofile = logoalt = logo2 = launch = 2004 closed date = picture format = share = share as of = share source = network = owner = Business Recorder Group slogan = The Only Islamic Channel Round the Clock country … Wikipedia
hon·or·ary — /ˈɑːnəˌreri, Brit ˈɑːnərəri/ adj 1 : given as a sign of honor or achievement He was awarded an honorary degree/title. 2 always used before a noun 2 a : regarded as one of a group although not officially elected or included He s an honorary member … Useful english dictionary