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| In [[mathematics]], a '''ternary ring''' is an [[algebraic structure]] <math>(R,T)</math> consisting of a non-empty set <math>R</math> and a ternary mapping <math>T \colon R^3 \to R \,</math>, and a '''planar ternary ring''' (PTR) or '''ternary field''' is special type of ternary ring used by {{harvtxt|Hall|1943}} to construct [[projective plane]]s by means of coordinates. A planar ternary ring is not a [[Ring (mathematics)|ring]] in the traditional sense.
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| ==Definition==
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| A '''planar ternary ring''' is a structure <math>(R,T)</math> where <math>R</math> is a nonempty set, containing at least two distinct elements, called 0 and 1, and <math>T\colon R^3\to R \,</math> a mapping which satisfies these five axioms:
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| # <math>T(a,0,b)=T(0,a,b)=b\quad \forall a,b \in R</math>;
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| # <math>T(1,a,0)=T(a,1,0)=a\quad \forall a \in R</math>;
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| # <math>\forall a,b,c,d \in R, a\neq c</math>, there is a unique <math>x\in R</math> such that : <math>T(x,a,b)=T(x,c,d) \,</math>;
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| # <math>\forall a,b,c \in R</math>, there is a unique <math>x \in R</math>, such that <math>T(a,b,x)=c \,</math>; and
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| # <math>\forall a,b,c,d \in R, a\neq c</math>, the equations <math>T(a,x,y)=b,T(c,x,y)=d \,</math> have a unique solution <math>(x,y)\in R^2</math>.
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| When <math>R</math> is finite, the third and fifth axioms are equivalent in the presence of the fourth.<ref>{{harvnb|Hughes|Piper|1973|loc=p. 118, Theorem 5.4}}</ref>
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| No other pair (0', 1') in <math>R^2</math> can be found such that <math>T</math> still satisfies the first two axioms.
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| ==Binary operations==
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| ===Addition===
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| Define <math>a\oplus b=T(a,1,b)</math>.<ref>In the literature there are two versions of this definition. This is the form used by {{harvtxt|Hall|1959|loc=p. 355}}, {{harvtxt|Albert|Sandler|1968|loc=p. 50}},{{harvtxt|Stevenson|1972|loc=p. 274}} and {{harvtxt|Dembowski|1968|loc=p. 128}}, while <math>a \oplus b = T(1,a,b)</math> is used by {{harvtxt|Hughes|Piper|1973|loc=p. 117}} and {{harvtxt|Pickert|1975|loc=p. 38}}. The difference comes from the alternative ways these authors coordinatize the plane.</ref> The structure <math>(R,\oplus)</math> is a [[Loop (algebra)|loop]] with [[identity element]] 0.
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| ===Multiplication===
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| Define <math>a\otimes b=T(a,b,0)</math>. The set <math>R_{0} = R \setminus \{0\} \,</math> is closed under this multiplication. The structure <math>(R_{0},\otimes)</math> is also a loop, with identity element 1.
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| ===Linear PTR===
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| A planar ternary ring <math>(R,T)</math> is said to be ''linear'' if <math>T(a,b,c)=(a\otimes b)\oplus c\quad \forall a,b,c \in R</math>.
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| For example, the planar ternary ring associated to a [[quasifield]] is (by construction) linear.{{cn|date=July 2013}}
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| ==Connection with projective planes==
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| Given a planar ternary ring <math>(R,T)</math>, one can construct a [[projective plane]] with point set ''P'' and line set ''L'' as follows:<ref>R. H. Bruck, ''Recent Advances in the Foundations of Euclidean Plane Geometry'', (1955) Appendix I.</ref><ref>{{harvnb|Hall|1943|loc=p.247 Theorem 5.4}}</ref> (Note that <math>\infty</math> is an extra symbol not in <math>R</math>.)
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| Let
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| * <math>P=\{(a,b)|a,b\in R\}\cup \{(a)|a \in R \}\cup \{(\infty)\}</math>, and
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| * <math>L=\{[a,b]|a,b \in R\}\cup\{[a]|a \in R \}\cup \{[\infty]\}</math>.
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| Then define, <math>\forall a,b,c,d \in R</math>, the [[incidence relation]] <math>I</math> in this way:
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| :<math>((a,b),[c,d])\in I \Longleftrightarrow T(a,c,d)=b</math>
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| :<math>((a,b),[c])\in I \Longleftrightarrow a=c</math>
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| :<math> ((a,b),[\infty])\notin I</math>
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| :<math>((a), [c,d])\in I \Longleftrightarrow a=c</math>
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| :<math>((a), [c])\notin I</math>
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| :<math>((a),[\infty])\in I</math>
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| :<math>(((\infty),[c,d])\notin I</math>
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| :<math>((\infty),[a])\in I</math> | |
| :<math>((\infty),[\infty])\in I</math>
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| Every projective plane can be constructed in this way, starting with an appropriate planar ternary ring. However, two nonisomorphic planar ternary rings can lead to the construction of isomorphic projective planes.
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| Conversely, given any finite projective plane π, by chosing an (ordered) set of four points, labelled ''o'', ''e'', ''u'', and ''v'', no three of which lie on the same line, coordinates can be introduced in π so that these special points are given the coordinates: ''o'' = (0,0), ''e'' = (1,1), ''v'' = (<math>\infty</math>) and ''u'' = (0).<ref>This can be done in several ways. A short description of the method used by {{harvtxt|Hall|1943}} can be found in {{harvtxt|Dembowski|1968|loc=p. 127}}.</ref> The ternary operation is now defined on the (finite) coordinate symbols by ''y'' = T(''x'',''a'',''b'') if and only if the point (''x'',''y'') lies on the line which joins (''a'') with (0,''b''). The axioms defining a projective plane are used to show that this gives a planar ternary ring.
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| Linearity of the PTR is equivalent to a geometric condition holding in the associated projective plane.<ref>{{harvnb|Dembowski|1968|loc=p. 129}}</ref>
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| ==Related algebraic structures==
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| PTR's which satisfy additional algebraic conditions are given other names. These names are not uniformly applied in the literature. The following listing of names and properties is taken from {{harvtxt|Dembowski|1968|loc=p. 129}}.
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| A linear PTR whose additive loop is [[Associative property|associative]] (and thus a [[Group (mathematics)|group ]]), is called a '''cartesian group'''. In a cartesian group, the mappings
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| <math>x \longrightarrow -x \otimes a + x \otimes b </math>, and
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| <math>x \longrightarrow a \otimes x - b \otimes x </math>
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| must be permutations whenever <math>a \neq b</math>. Since cartesian groups are groups under addition, we revert to using a simple "+" for the additive operation.
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| A [[quasifield]] is a cartesian group satisfying the right distributive law:
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| <math> (x+y) \otimes z = x \otimes z + y \otimes z </math>.
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| Addition in any quasifield is [[Commutative property|commutative]].
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| A [[semifield]] is a quasifield which also satisfies the left distributive law:
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| <math> x \otimes (y + z) = x \otimes y + x \otimes z.</math>
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| A '''planar [[Near-field (mathematics)|nearfield]]''' is a quasifield whose multiplicative loop is associative (and hence a group). Not all nearfields are planar nearfields.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book|last1=Albert|first=A. Adrian|last2=Sandler|first2=Reuben|title=An Introduction to Finite Projective Planes|year=1968|publisher=Holt, Rinehart and Winston|location=New York}}
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| * [[Rafael Artzy]] (1965) ''Linear Geometry'', Chapter 4 Axiomatic Plane Geometry, [[Addison-Wesley]].
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| * {{Citation | last1=Dembowski | first1=Peter | title=Finite geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 | mr=0233275 | year=1968 | isbn=3-540-61786-8}}
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| * {{Citation | last1=Hall, Jr. | first1=Marshall | title=Projective planes | jstor=1990331 | mr=0008892 | year=1943 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=54 | pages=229–277 | issue=2 | publisher=American Mathematical Society}}
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| * {{citation|last= Hall, Jr.|first= Marshall|title=The Theory of Groups|year=1959|publisher=The MacMillan Company|place=New York}}
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| * {{citation|last1=Hughes|first1=Daniel R.|last2=Piper|first2=Fred C.|title=Projective Planes|year=1973|publisher=Springer-Verlag|place=New York|isbn=0387900446}}
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| * {{citation|last=Pickert|first=Günter|title=Projektive Ebenen|year=1975|publisher=Springer-Verlag|place=Berlin|isbn=3540072802}}
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| * {{citation|last= Stevenson|first=Frederick|title=Projective Planes|year=1972|publisher=W.H. Freeman and Company|place=San Francisco|isbn=071670443-9}}
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| [[Category:Algebraic structures]]
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| [[Category:Projective geometry]]
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