Ternary bodies

A ternary body is an algebraic structure that is used in synthetic geometry as the coordinate range of any affine plane . As a set, the ternary body consists of the points of a fixed straight line of the plane, namely the first coordinate axis of the coordinate system that is introduced on this plane. On this point set, the ternary construction defines a three-digit link with which the straight line receives the algebraic structure of a ternary body. Conversely, for every structure that fulfills the axioms of a ternary body, there is an affine plane whose points are the pairs and whose straight lines can be represented as solution sets of equations in with the help of the ternary connection . ${\displaystyle T}$ ${\displaystyle (K,T)}$ ${\displaystyle (x_{1},x_{2})\in K^{2}}$ ${\displaystyle K}$ ${\displaystyle T}$

To put it somewhat casually: every affine plane “is” a two-dimensional plane over a ternary body and for every affine plane there is exactly one ternary body as a set of coordinates, apart from isomorphism. The thickness of the ternary body corresponds to the order of the associated affine plane.

If the affine plane is an affine translation plane , then its coordinate ternary body can be made into a quasi-body , for desargue planes this is even a skewed body , for pappus planes a body .

A ternary body in which the ternary link can be represented by an addition and a multiplication is called linear . If the addition satisfies the associative law in a linear ternary field , then it is called a Cartesian group . Quasi-bodies are always Cartesian groups. A quasi-body whose multiplication is associative is called an almost-body . If both distributive laws hold, the quasi- body is called a half- body in geometry . Alternative bodies are always such half bodies, oblique bodies are always alternative bodies.

The coordinate areas described here, which are called coordinate bodies in synthetic geometry , even if they are not bodies in the algebraic sense, can also be used to introduce projective coordinates on a projective plane . The relationship between affine and projective inference clauses and the implications for the algebraic structure of the coordinate range of the planes that satisfy the inference clause is presented in this article and summarized below in the section #Closure clauses and coordinate areas . When classifying projective planes , it turns out that each class of projective planes (in the sense of the classification according to Hanfried Lenz ) can be assigned a class of coordinate areas with additional properties that are characteristic of this plane class.

This article describes algebraizations of affine planes based on a coordinate system and the links that result from the geometric structure on a coordinate axis. Another approach, which proves to be particularly fruitful for non-Desargue affine translation levels, is to describe certain endomorphisms of the translation group algebraically , namely the true- to-track endomorphisms . In the case of Desargue planes, this approach leads to an inclined body that is isomorphic to the coordinate inclined body described in this article. This other approach is described in the main article Affine Translation Plane. For a synonymous algebraization of affine planes, in particular the non-desargue planes, without a dedicated system of coordinates, reference is made to the main article Geometric Relation Algebra .

Geometric construction

Here, its coordinate ternary body and its linkage are geometrically constructed for an affine plane . For this purpose, three points in the affine plane that are not on a straight line must be selected as a coordinate reference system (point base). The point is the origin , the other points are the unit points of this coordinate system. The points of the connecting line , the first coordinate axis, form the set of coordinates . The ternary construction provides this set of coordinates with a three-digit link with which it becomes a ternary body. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle (O;E_{1},E_{2})}$ ${\displaystyle O}$ ${\displaystyle OE_{1}}$ ${\displaystyle K}$

Coordinate construction

Construction of the coordinate pair to a point and vice versa. ${\displaystyle (x_{1},x_{2})}$ ${\displaystyle P}$

At one point is ${\displaystyle P\in A}$

1. ${\displaystyle x_{1}}$: The parallel to through cuts the first coordinate axis in . ${\displaystyle OE_{2}}$ ${\displaystyle P}$ ${\displaystyle OE_{1}}$ ${\displaystyle x_{1}}$
2. ${\displaystyle x_{2}}$: The parallel to through intersects the second coordinate axis in . The parallel to through intersects the first coordinate axis in . ${\displaystyle OE_{1}}$ ${\displaystyle P}$ ${\displaystyle OE_{2}}$ ${\displaystyle P_{2}}$ ${\displaystyle E_{1}E_{2}}$ ${\displaystyle P_{2}}$ ${\displaystyle x_{2}}$

The point pair is called the coordinate pair of the point , it is common to write for the point when the coordinate system is clear . This construction can be reversed: For a coordinate pair is ${\displaystyle (x_{1},x_{2})\in K^{2}=(OE_{1})^{2}}$ ${\displaystyle P}$ ${\displaystyle (O,E_{1},E_{2})}$ ${\displaystyle P(x_{1}|x_{2})}$ ${\displaystyle (x_{1},x_{2})\in K^{2}}$

1. ${\displaystyle P_{2}}$: The parallel to through intersects the second coordinate axis in . ${\displaystyle E_{1}E_{2}}$ ${\displaystyle x_{2}}$ ${\displaystyle OE_{2}}$ ${\displaystyle P_{2}}$
2. ${\displaystyle P}$: The parallel to through cuts the parallel to through in . ${\displaystyle OE_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle OE_{2}}$ ${\displaystyle x_{1}}$ ${\displaystyle P}$

As a result, a point of the affine plane is assigned to each coordinate pair in a reversible and unambiguous manner . ${\displaystyle A}$

The figure on the right shows the steps of both constructions, the coordinate construction and its inversion. Both constructions can be used in both directions for all coordinate areas from the ternary body to the body.

Ternary construction

Ternary construction.

For three coordinates , i.e. points on the axis , the point with the coordinates in the affine plane is first constructed by reverse coordinate construction. Then the parallel to through the first coordinate axis cuts into . ${\displaystyle a,x_{1},x_{2}\in K}$ ${\displaystyle K=OE_{1}}$ ${\displaystyle P}$ ${\displaystyle (x_{1}|x_{2})}$ ${\displaystyle aE_{2}}$ ${\displaystyle P}$ ${\displaystyle OE_{1}}$ ${\displaystyle t=T(a,x_{2},x_{1})}$

The figure on the right shows the ternary construction. The straight line shown then receives the coordinate equation, its family of parallels are the straight lines with the equations , where the respective “ axis section” is, ie intersects the first coordinate axis . The general definition of the straight line equations is given below. ${\displaystyle aE_{2}}$ ${\displaystyle g_{a,a}:T(a,x_{2},x_{1})=a}$ ${\displaystyle g_{a,d}:T(a,x_{2},x_{1})=d}$ ${\displaystyle d}$ ${\displaystyle x_{1}}$ ${\displaystyle g_{a,d}}$ ${\displaystyle (d|0)}$

The amount of points on the first coordinate axis complies with the link by the Ternärkonstruktion is given, the axiomatic requirements for a planar ternary ring. The structural constants, the existence of which is required in the axioms, are the points or , i.e. the origin or the first unit point of the point base. ${\displaystyle K}$ ${\displaystyle 0=O}$ ${\displaystyle 1=E_{1}}$

Algebraic definition

Here a ternary body is defined by its algebraic properties and an affine plane is built up on the structure defined in this way, in which the elements of serve as points. ${\displaystyle K}$ ${\displaystyle K^{2}}$

Axioms

A set together with a three-place link (the ternary link ) is called a ternary field if the following axioms hold: ${\displaystyle K}$ ${\displaystyle T:K^{3}\rightarrow K}$

1. There are two different elements and in , so that and and applies to everyone . ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle K}$ ${\displaystyle T(0,b,c)=T(a,0,c)=c}$ ${\displaystyle T(a,1,0)=a}$ ${\displaystyle T(1,b,0)=b}$ ${\displaystyle a,b,c\in K}$
2. For there is exactly one of the holds. ${\displaystyle a,x_{2},d\in K}$ ${\displaystyle x_{1}\in K}$ ${\displaystyle T(a,x_{2},x_{1})=d}$
3. For there is, if is exactly one pair for which applies. ${\displaystyle a,d,a',d'\in K}$ ${\displaystyle a\neq a'}$ ${\displaystyle (x_{1},x_{2})\in K^{2}}$ ${\displaystyle T(a,x_{2},x_{1})=d,\quad T(a',x_{2},x_{1})=d'}$
4. For two pairs there is, if is, exactly one for which applies. ${\displaystyle (x_{1},x_{2}),(y_{1},y_{2})\in K^{2}}$ ${\displaystyle x_{2}\neq y_{2}}$ ${\displaystyle a\in K}$ ${\displaystyle T(a,x_{2},x_{1})=T(a,y_{2},y_{1})}$

In the literature there are also axioms for ternary bodies in which the roles of the first and second positions in the ternary connection are reversed. This Ternärverknüpfung herein as "Rechtsternärverknüpfung" will be referred to, comes from the described "Linksternärverknüpfung" by interchanging the first two points out: . The third and fourth axioms must be reformulated accordingly for . ${\displaystyle T^{\text{op}}}$ ${\displaystyle T}$ ${\displaystyle T(a,x_{2},x_{1})=T^{\text{op}}(x_{2},a,x_{1})}$ ${\displaystyle T^{\text{op}}}$

Addition and multiplication, more specific ternary bodies

In general, one can define an addition and multiplication in ternary fields as follows:

${\displaystyle a+b=T(a,1,b)}$ and

${\displaystyle a\cdot b=T(a,b,0)}$.

The following applies:

• ${\displaystyle (K,+)}$is a quasi-group with the neutral element , i.e. a loop , ${\displaystyle 0}$
• ${\displaystyle (K\setminus \{0\},\cdot )}$ is also a loop with the neutral element 1 and
• it applies to every element . ${\displaystyle a\cdot 0=0\cdot a=0}$ ${\displaystyle a\in K}$

Also applies

${\displaystyle T(a,b,c)=T(T(a,b,0),1,c)}$, so for everyone , ${\displaystyle T(a,b,c)=a\cdot b+c}$ ${\displaystyle a,b,c\in K}$

so the ternary body is called linear . If the addition defined here is associative , then it even forms a group. In this case a linear ternary body is called a Cartesian group . ${\displaystyle K}$ ${\displaystyle (K,+)}$ ${\displaystyle (K,+,\cdot )}$

The ternary body of an affine translation plane is a quasi-body , a Cartesian group with commutative addition and other additional properties. See the main articles Affine Translation Plane and Quasi-Bodies .

Geometry of the plane

• The set of pairs forms the set of points , ${\displaystyle A=K^{2}}$
• Are straight

• are the solution sets of the equations , ( ) and ${\displaystyle x_{2}=c}$ ${\displaystyle g_{c}=\lbrace (x_{1},x_{2})\in A:x_{2}=c\rbrace }$
• the solution sets of the equations , ( or for linear ternary bodies, i.e. in particular quasi- and oblique bodies) ${\displaystyle T(a,x_{2},x_{1})=d}$ ${\displaystyle g_{a,d}=\lbrace (x_{1},x_{2})\in A:T(a,x_{2},x_{1})=d\rbrace }$ ${\displaystyle g_{a,d}=\lbrace (x_{1},x_{2})\in A:a\cdot x_{2}+x_{1}=d\rbrace }$

• The elements are called coefficients of the straight line or . They describe the straight lines one-to-one, i.e. two straight lines match exactly if they are of the same type and their coefficients match. ${\displaystyle c,a,d}$ ${\displaystyle g_{c}}$ ${\displaystyle g_{a,d}}$
• Two straight lines are exactly then parallel when they are both of the former type , or if they are both of the second-mentioned type, and correspond in their first coefficient: . ${\displaystyle g_{c_{1}}\parallel g_{c_{2}}}$ ${\displaystyle g_{a,d_{1}}\parallel g_{a,d_{2}}}$

variants

• The equations give explicit straight line equations that are preferred in recent literature. Then there is the " -axis intercept", the coefficients differ from those defined above. ${\displaystyle T(a,x_{1},d)=x_{2};x_{1}=c}$ ${\displaystyle d}$ ${\displaystyle x_{2}}$
• Corresponding straight line equations for the reversed “right” version can of course be obtained again simply by swapping the role of the first and second digit. ${\displaystyle T^{op}}$

Finite ternaries and Latin squares

→ Main article : Latin square

A list of pairwise orthogonal Latin squares of order n is used to determine a unique ternary field (except for isomorphism) with n elements. Each finite ternary field with a fixed arrangement of its elements determines a list of this type which is unique (except for the order of the squares and common renumbering of all their entries). ${\displaystyle n-1}$

Closure sets and coordinate areas

The following table summarizes the conclusions that result from geometric closures for the algebraic structure of a coordinate area that can be assigned to a plane. In addition, the table shows which closure theorems are fulfilled by a plane whose coordinate range fulfills the axioms for a certain “generalized body”.

Affine level			Projective plane
designation	Geometric characterization	Coordinate area	designation	Projective closure set
Affine incidence level	(Axioms of the affine plane )	${\displaystyle (K,T)}$ is a ternary body.	projective incidence level	(Axioms of the projective plane )
Translational plane	Desargues' little affine theorem holds.	${\displaystyle (K,+,\cdot )}$ is a quasi-body.	Moufang level	Small set of Desargues
Desargue plane	Desargues' large affine theorem holds.	${\displaystyle (K,+,\cdot )}$ is an oblique body.	Desargue's plain	Big set of Desargues
Pappus plane	Pappos's great affine theorem holds.	${\displaystyle (K,+,\cdot )}$ is a body.	Pappus plane	Big set of pappos

In the table, each line implies the one above, whereby the axioms of the affine or projective plane are required by each more specific plane and the connection in the ternary body is different from that in the more specific fields. Affine planes whose coordinate range is a sloping body, in which the large affine theorem of Desargues applies, are called affine Desargue planes , all other affine non- Desargues planes . The last two columns list the corresponding projective planes. By slotting a projective plane that fulfills the (projective) closure principle mentioned in the line, an affine plane of the type described in the same line always arises (→ see projective coordinate system ). Therefore, the coordinate areas of the affine planes can also be applied to the corresponding projective planes. Two affine planes obtained by cutting out different straight lines from a certain projective plane do not have to be isomorphic, nor do the coordinate ternary bodies determined by the slitting. This also means that the projective closures of two non-isomorphic affine planes can be isomorphic.

Projective expansion always creates a projective translation level from a translation level , but not always a Moufang level. The original Moufang plane is then created again through projective expansion from a translation plane that has emerged from a Moufang plane by slotting. This special case occurs precisely when every coordinate range of the affine and the projective plane is even an alternative body - and always the same apart from isomorphism . In all other lines, a projective extension of the type named in the same line arises quite generally from any affine plane of the type named in the line.

All coordinate ternary bodies that can be assigned to a projective plane as a coordinate range are ternary bodies isotopic to one another . → More detailed information about the relationship between the geometric properties of projective planes and the algebraic structure of their coordinate ternary bodies can be found in the article " Classification of projective planes ".

Examples and remarks

• Due to the structural constants required in the first axiom of the ternary body, each ternary body has at least two elements. The smallest ternary field is the remainder class field . The associated affine plane of order 2, i.e. the minimal model of an affine plane, is the affine Fano plane (the affine section is explained in Affine plane ). ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$
• Every oblique body fulfills the axioms of a quasi-body and those of a ternary body if the ternary connection is defined as described above for linear ternary bodies. ${\displaystyle (K,+,\cdot ,0,1)}$
• The ternary body clearly assigned to an affine plane except for isomorphism is a skew body if and only if the (large) affine theorem of Desargues applies in the affine plane, in this case one speaks of a Desargue plane and the addition and multiplication can be carried out as above for linear ones Express the ternary body described by the ternary link.

Straight lines in the Moulton Plain. Explicit straight line equations were used as a basis for the illustration . ${\displaystyle y=T(m,x,d);\;x=c}$

• The Moulton plane is an example of a non-Desargue affine plane, compare the figure on the right. It can be described as a level above the ternary body , with the ternary link through ${\displaystyle (\mathbb {R} ,T)}$

${\displaystyle T(a,b,d)={\begin{cases}{\frac {1}{2}}\cdot a\cdot b+d,\quad a<0\land b<0\\a\cdot b+d\qquad {\mbox{sonst.}}\end{cases}}\quad {}}$

is defined. Since it is true with ordinary addition, this ternary field is linear, even a Cartesian group with commutative addition. However, it fulfills neither the Left nor the Right Distribution Act. So it is not a quasi-body and the moulton plane is therefore also not a translation plane. Analog can be constructed from any ordered body and with arbitrary "kink constants" instead of Cartesian groups. ${\displaystyle T(a,1,b)=T(1,a,b)=a+b}$ ${\displaystyle K}$ ${\displaystyle C>0,\,C\neq 1}$ ${\displaystyle 1/2}$

• Every division algebra over a field is a quasi-field and is only a skewed field if the multiplication is associative.
• An example of such a "real" quasi-body are the octonions , an 8-dimensional non-associative division algebra over . ${\displaystyle \mathbb {R} }$
• Every finite skew field is a finite field according to Wedderburn's theorem . Therefore, all Desarguean planes of finite order are pappusian, that is, every finite Desarguean affine plane has the order with a prime number and is a two-dimensional affine space (in the sense of linear algebra ) over the finite field ${\displaystyle p^{n},n\geq 1}$ ${\displaystyle p}$ ${\displaystyle \mathbb {F} _{p^{n}}.}$

Examples of order 9

All finite projective and affine planes, the order of which is less than 9, are desarguessian - so there is exactly one with the order and none with the order 6 except for isomorphism . The examples of the order 9, which are shown here, and all used Statements and terms can be found in Weibel (2007). There are exactly 4 different (non-isomorphic) projective levels of order 9. One of them is the projective level above the finite field , the other three are non-desargic. Non-Desargue projective planes of finite order are never Moufan planes , therefore the algebraic structure of the ternary field depends on the complete quadrilateral that is introduced as the projective point base on the plane. ${\displaystyle n\in \lbrace 2,3,4,5,7,8\rbrace }$ ${\displaystyle \mathbb {F} _{9}}$

A projective plane is called a translation plane with respect to one of its straight lines if it satisfies Desargues' little projective theorem with respect to this straight line as an axis . Only such projective planes can have a quasi-body as a coordinate area. An equivalent description of such a projective translation level : It belongs to one of the classes IVa, V or VII in the classification of projective levels according to Hanfried Lenz .

Two of the projective planes of order 9 are not translation planes in this sense; by slitting these planes one always arrives at an example of a non-Desarguean affine plane that is not a translation plane. The resulting coordinate ternary bodies are therefore not quasi-bodies.

The third non-Desarguean level, which was introduced by Veblen and Wedderburn in 1907, is a projective translational level; apart from isomorphism, it is the only level in the Lenz-Barlotti class IVa.3. It can be slit in such a way that a non-Desargue affine translation plane is created, which is a coordinate plane above the left quasi-body of order 9. ${\displaystyle J_{9}^{op}}$

The left quasi-body looks like this:, that means the multiplicative structure of the quasi-body is given by the quaternion group and therefore a group, products that contain 0 should be 0. The addition is obtained by identifying with the vector space for so : ${\displaystyle (J_{9}^{op\,*},\cdot )=(Q_{8},\cdot )}$ ${\displaystyle Q_{8}=\lbrace \pm 1,\pm i,\pm j,\pm k\rbrace }$ ${\displaystyle J_{9}^{op}=Q_{8}\cup \lbrace 0\rbrace }$ ${\displaystyle \mathbb {F} _{3}^{2}}$

1. ${\displaystyle (1,i)}$ be a base
2. ${\displaystyle j=1-i}$and . ${\displaystyle k=1+i}$

The other additions result from the vector space structure if the formal minus sign of the quaternion group is treated as an additive inverse formation. If you swap the identifications in 2., i.e. define and , then the right quasi-body arises . ${\displaystyle j=1+i}$ ${\displaystyle k=1-i}$ ${\displaystyle J_{9}}$

In total there are 5 non-isomorphic left quasi-bodies with 9 elements, one of which is , the other four (including natural ) appear as the coordinate range of the projective translation plane if the projective point base is chosen appropriately. In addition, if the coordinate system is chosen differently as the coordinate area, a ternary body is created that is not a quasi-body. Some of the resulting ternary bodies are isomorphic to one another. ${\displaystyle \mathbb {F} _{9}}$ ${\displaystyle J_{9}^{op}}$ ${\displaystyle J_{9}^{op}}$

Additional properties of the coordinate areas

In the axiomatic structure of planar geometry, the structure of the coordinate area plays an important role, as many geometric properties are reflected in the algebraic properties of the coordinate area:

• The Fano axiom allows to define center points and point reflections in affine translation planes . Its validity for these levels is equivalent to the fact that no element of the coordinate quasi-body has the additive order 2. For (projective and affine) Desargue's planes, its validity is equivalent to the fact that the coordinate inclined body has one of two different characteristics .

In the axiomatic structure of Euclidean geometry in a Pappusian plane, the “coordinate body” is actually a body in the sense of algebra.

• If this body has at least two square classes, an orthogonality relation can be defined on its coordinate plane, if its characteristic is not 2, then perpendicular axis reflections and thus angle bisectors can be defined. The Pappus plane thus becomes a pre-Euclidean plane .
• A pre-Euclidean plane, in which bisectors always exist, is called a freely movable plane , its coordinate field is a Pythagorean field in which −1 is not a square number . These bodies always have the characteristic 0. Conversely, the affine plane over such a body can always be endowed with an orthogonality that makes it a freely movable plane. → See angle bisector # Synthetic Geometry .
• On the other hand, the coordinate bodies of a freely movable plane always allow an arrangement . Such an arrangement of the body then requires relationships in the plane .
• If the coordinate body is Euclidean , then the associated affine plane provides a model for a plane Euclidean geometry that cannot be distinguished from the conventional Euclidean plane over the real numbers using classical geometric methods . ${\displaystyle \mathbb {R} }$

literature

• Walter Benz: A Century of Mathematics, 1890-1990 . Festschrift for the anniversary of the DMV . Vieweg, Braunschweig 1990, ISBN 3-528-06326-2 .
• Marshall Hall : Projective planes . In: Transactions of the American Mathematical Society . tape 54 . American Mathematical Society , 1943, pp. 229-277 , JSTOR : 1990331 .
• Wendelin Degen and Lothar Profke: Fundamentals of affine and Euclidean geometry . In: Mathematics for teaching at high schools . 1st edition. Teubner, Stuttgart 1976, ISBN 3-519-02751-8 .
• Günter Pickert : Projective levels . 2nd Edition. Frankfurt am Main 1975.
• Günter Pickert: Level incidence geometry . 2nd Edition. Frankfurt am Main 1968.
• Oswald Veblen and Joseph Wedderburn : Non-Desarguesian and non-Pascalian geometries . In: Transactions of the American Mathematical Society . tape 8 . American Mathematical Society , 1907, pp. 379-388 .
• Charles Weibel: Survey of Non-Desarguesian Planes . In: Notices of the American Mathematical Society . tape 54 . American Mathematical Society, November 2007, pp. 1294–1303 ( full text [PDF; 702 kB ; accessed on April 25, 2012]).

Individual evidence

1. ↑ This is not a common term, it is used here because the "right" ternary body of an affine translation plane is a right quasi-body, while the ternary body defined in this article according to Degen (1976) is a left quasi-body in these cases.
2. ↑ Benz (1990), p. 244
3. ↑ Weibel (2007) p. 1296
4. ^ Peter Dembowski : Finite geometries . Springer, Berlin a. a. 1968, chapter 1
5. ^ Veblen-Wedderburn (1907)
6. ^ Hauke ​​Klein: Lenz Type IVa. In: Geometry. University of Kiel, November 29, 2002, accessed on December 13, 2010 (English).