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| In [[mathematics]], a '''quasifield''' is an [[algebraic structure]] <math>(Q,+,\cdot)</math> where + and <math>\cdot</math> are [[binary operation]]s on Q, much like a [[division ring]], but with some weaker conditions.
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| ==Definition==
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| A quasifield <math>(Q,+,\cdot)</math> is a structure, where + and <math> \cdot \,</math> are binary operations on Q, satisfying these axioms :
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| * <math>(Q,+) \,</math> is a [[Group (mathematics)|group]]
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| * <math>(Q_{0},\cdot)</math> is a [[Quasigroup|loop]], where <math>Q_{0} = Q \setminus \{0\} \,</math>
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| * <math>a \cdot (b+c)=a \cdot b+a \cdot c \quad\forall a,b,c \in Q</math> (left [[distributivity]])
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| * <math>a \cdot x=b \cdot x+c</math> has exactly one solution <math>\forall a,b,c \in Q, a\neq b</math>
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| Strictly speaking, this is the definition of a ''left'' quasifield. A ''right'' quasifield is similarly defined, but satisfies right distributivity instead. A quasifield satisfying both distributive laws is called a '''[[semifield]]''', in the sense in which the term is used in [[projective geometry]].
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| Although not assumed, one can prove that the axioms imply that the additive group <math>(Q,+)</math> is [[Abelian group|abelian]]. Thus, when referring to an ''abelian quasifield'', one means that <math>(Q_0, \cdot)</math> is abelian.
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| ==Kernel==
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| The kernel K of a quasifield Q is the set of all elements c such that :
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| * <math>a \cdot(b \cdot c)=(a \cdot b) \cdot c\quad \forall a,b\in Q</math>
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| * <math>(a+b) \cdot c=(a \cdot c)+(b \cdot c)\quad \forall a,b\in Q</math>
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| Restricting the binary operations + and <math>\cdot</math> to K, one can shown that <math>(K,+,\cdot)</math> is a [[division ring]].
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| One can now make a vector space of Q over K, with the following scalar multiplication :
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| <math>v \otimes l = v \cdot l\quad \forall v\in Q,l\in K</math>
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| As a finite division ring is a finite field by [[Wedderburn's little theorem|Wedderburn's theorem]], the order of the kernel of a finite quasifield is a [[prime power]]. The vector space construction implies that the order of any finite quasifield must also be a prime power.
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| ==Examples==
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| All division rings, and thus all fields, are quasifields.
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| The smallest quasifields are abelian and unique. They are the finite fields of orders up to and including eight. The smallest quasifields which aren't division rings are the 4 non-abelian quasifields of order 9; they are presented in {{harvtxt|Hall, Jr.|1959}} and {{harvtxt|Weibel|2007}}.
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| ==Projective planes==
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| {{main|Projective plane#Generalized coordinates|l1=Projective plane}}
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| Given a quasifield <math>Q</math>, we define a ternary map <math>\scriptstyle T\colon Q\times Q\times Q\to Q \,</math> by
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| <math>T(a,b,c)=a \cdot b+c \quad \forall a,b,c\in Q</math>
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| One can then verify that <math>(Q,T)</math> satisfies the axioms of a [[planar ternary ring]]. Associated to <math>(Q,T)</math> is its corresponding [[projective plane]]. The projective planes constructed this way are characterized as follows;
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| the details of this relationship are given in {{harvtxt|Hall, Jr.|1959}}.
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| A projective plane is a [[translation plane]] with respect to the line at infinity if and only if any (or all) of its associated planar ternary rings are right quasifields. It is called a ''shear plane'' if any (or all) of its ternary rings are left quasifields.
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| The plane does not uniquely determine the ring; all 4 nonabelian quasifields of order 9 are ternary rings for the unique non-Desarguesian translation plane of order 9. These differ in the [[Complete quadrangle|fundamental quadrilateral]] used to construct the plane (see Weibel 2007).
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| ==History==
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| Quasifields were called "Veblen-Wedderburn systems" in the literature before 1975, since they were first studied in the
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| 1907 paper (Veblen-Wedderburn 1907) by [[Oswald_Veblen|O. Veblen]] and [[Joseph_Wedderburn|J. Wedderburn]]. Surveys of quasifields and their applications to [[projective plane]]s may be found in {{harvtxt|Hall, Jr.|1959}} and {{harvtxt|Weibel|2007}}.
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| ==References==
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| * {{citation | first1=Marshall | last1=Hall, Jr. | title=Theory of Groups | publisher=Macmillan | year=1959 | id={{MathSciNet|id=103215}} | lccn=595035 }}.
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| * {{Citation|last1=Veblen|first1=O.|last2=Wedderburn|first2=J.H.M.|title=Non-Desarguesian and non-Pascalian geometries|journal=Transactions of the American Mathematical Society|year=1907|volume=8|pages=379–388}}
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| * {{Citation | last1=Weibel | first1=Charles | title=Survey of Non-Desarguesian Planes | url=http://www.ams.org/notices/200710/ | year=2007 | journal= Notices of the AMS | volume= 54 | issue=10 | pages=1294–1303}}
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| == See also ==
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| * [[Near-field (mathematics)|Near-field]]
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| * [[Semifield]]
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| * [[Alternative division ring]]
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| * [[Hall plane#Algebraic construction via Hall systems|Hall systems]] (Hall planes)
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| * [[Moufang plane]]
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| == External links ==
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| * [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/quasi.html Quasifields] by Hauke Klein.
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| [[Category:Non-associative algebra]]
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| [[Category:Projective geometry]]
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