Half body (geometry)

This article describes the term “half-body” as it is used in synthetic geometry. For the structure of the same name in abstract algebra, see half-body .

A half body (English: Semi Field ) is in the synthetic geometry of a quasi body in which both distributive apply. Like the quasi-bodies, such half-bodies appear as coordinate areas of affine and projective translation planes . Half bodies are a generalization of the oblique bodies and the alternative bodies : The multiplicative link in the half body does not have to fulfill either the associative law or the (weaker) alternative .

A half body that is not an alternative body is called a real half body (English: proper semifield ). There are real finite half-bodies, which are not bodies , whereas the multiplicative connection in finite alternative bodies and finite skew bodies is always associative and commutative . Thus, among the ternary fields , i.e. among all finite algebraic structures that come into consideration as coordinate ranges for non-Desargue's affine or projective planes , real finite half-bodies (and almost-bodies ) are those that come closest to a body. In the same way as bodies, finite half-bodies can be assigned a prime number as a characteristic .

A projective plane that can be coordinated with a half-body K belongs in the classification of projective planes to one of Lenz classes V or VII , if K is not an alternative body, then it belongs to Lenz class V.

Definitions

Half body

A set with the two-digit connections and two different structural constants is called a half body if the following axioms hold: ${\displaystyle K}$ ${\displaystyle +,\;\cdot }$ ${\displaystyle 0,1\in K}$

1. ${\displaystyle (K,+)}$is an Abelian group with neutral element 0.
2. ${\displaystyle (K\setminus \lbrace 0\rbrace ,\cdot )}$is a loop with neutral element 1, i.e. a quasi-group with both left and right neutral element 1.
3. ${\displaystyle a\cdot 0=0\cdot a=0}$applies to everyone . ${\displaystyle a\in K}$
4. Both distributive laws apply : and for everyone . ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$ ${\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}$ ${\displaystyle a,b,c\in K}$

Equivalent to this axiom system is:

• ${\displaystyle (K,+,\cdot ,0,1)}$ is a left and at the same time a right quasibody.

Knuth has given the following equivalent axiom system:

1. ${\displaystyle (K,+)}$ is a group with neutral element 0.
2. Are , then there are unique elements with and ${\displaystyle a\in K\setminus \{0\};b\in K}$ ${\displaystyle x,y\in K}$ ${\displaystyle a\cdot x=b}$ ${\displaystyle y\cdot a=b.}$
3. Both distributive laws apply: and for everyone . ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$ ${\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}$ ${\displaystyle a,b,c\in K}$
4. It applies to everyone ${\displaystyle 1\cdot a=a\cdot 1=a}$ ${\displaystyle a\in K}$

Pre-semifield

Knuth describes an algebraic structure as a pre-semifield (no German name known) if the following axioms apply: ${\displaystyle (K,+,\cdot ,0)}$

1. ${\displaystyle (K,+)}$ is an Abelian group with neutral element 0.
2. ${\displaystyle (K\setminus \lbrace 0\rbrace ,\cdot )}$is a quasi-group .
3. ${\displaystyle a\cdot 0=0\cdot a=0}$applies to everyone . ${\displaystyle a\in K}$
4. Both distributive laws apply: and for everyone . ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$ ${\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}$ ${\displaystyle a,b,c\in K}$

The following axiom system is equivalent:

1. ${\displaystyle (K,+)}$ is a group with neutral element 0.
2. Are , then there are unique elements with and ${\displaystyle a\in K\setminus \{0\};b\in K}$ ${\displaystyle x,y\in K}$ ${\displaystyle a\cdot x=b}$ ${\displaystyle y\cdot a=b.}$
3. Both distributive laws apply: and for everyone . ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$ ${\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}$ ${\displaystyle a,b,c\in K}$

Nucleus, core and center

For a half body the quantities are called ${\displaystyle (K,+,\cdot )}$

1. ${\displaystyle {\mathcal {N}}_{l}(K)=\{a\in K|\;\forall x,y\in K:a\cdot (x\cdot y)=(a\cdot x)\cdot y\}}$ left nucleus ,
2. ${\displaystyle {\mathcal {N}}_{m}(K)=\{a\in K|\;\forall x,y\in K:x\cdot (a\cdot y)=(x\cdot a)\cdot y\}}$ middle nucleus ,
3. ${\displaystyle {\mathcal {N}}_{r}(K)=\{a\in K|\;\forall x,y\in K:x\cdot (y\cdot a)=(x\cdot y)\cdot a\}}$ right nucleus ,
4. ${\displaystyle {\mathcal {N}}(K)={\mathcal {N}}_{l}(K)\cap {\mathcal {N}}_{m}(K)\cap {\mathcal {N}}_{r}(K)}$the nucleus

of the half body. The left nucleus is at the same time the core of the quasi-body K and is always an oblique body. Like every quasi-body, K is always a left vector space over its kernel.

The set is called the center of the half-body. This center is always a commutative body and K is a vector space above this body. ${\displaystyle Z(K)=\{a\in {\mathcal {N}}(K)|\;\forall x\in K:x\cdot a=a\cdot x\}}$

Remarks

• In Knuth's system of axioms for half-bodies, the second axiom can be replaced by the formally weaker statement

“If holds for a pair , then follows or .” If K is finite. ${\displaystyle a\cdot b=0}$ ${\displaystyle (a,b)\in K^{2}}$ ${\displaystyle a=0}$ ${\displaystyle b=0}$

• A pre-semifield is a half-body if and only if it contains a universal unit on both sides . Obviously every half body is a pre-semifield.
• The additive group of every finite pre-semifield (in particular every finite half-field) K is an elementary Abelian p -group for a positive prime number p . This prime is as characteristic of K referred.
• If the multiplicative connection in a half-body is associative, then it is a skew.

• In other words: a half-body, which is also an almost-body, is a sloping body.

Finite half-bodies, twisted bodies and their projective planes, half-body models

Knuth was able to show in his dissertation:

1. Every finite half-field K is a d -dimensional vector space over the remainder class field of its characteristic p . Only if and is can K be a true half-body. ${\displaystyle \mathbb {Z} /pZ}$ ${\displaystyle d\geq 3}$ ${\displaystyle p^{d}\geq 16}$
2. In the cases mentioned under 1., so prime powers for a true half-body exists with elements, which in most cases from the finite field by "twisting" ( twisting of the body can be constructed multiplication). ${\displaystyle q=p^{d}\geq 16,d\geq 3}$ ${\displaystyle p^{d}}$ ${\displaystyle \mathbb {F} _{q}}$
3. For a finite projective translation plane of Lenz class V, all coordinate ternary bodies are real, mutually isotopic half bodies.
4. Projective planes, whose coordinate ternary bodies are finite half bodies, always belong to Lenz class V or VII and are geometrically isomorphic to one another precisely when their coordinate half bodies are isotopic to one another.

The set of natural numbers n for which a real half body with n elements exists is sequence A088247 in OEIS .

Examples

All examples mentioned here can be found in Knuth's dissertation, unless another source is explicitly stated.

Finally real half-bodies with 16 elements

One explains a component-wise multiplication on the two-dimensional vector space with the help of the usual multiplication of the finite field on the pairs. The addition is in each case the vector space addition, one element of the multiplication is . The body K can be embedded through . The following multiplications lead to two half-bodies that are not isotopic to one another: ${\displaystyle V={\mathbb {F} _{4}}^{2}}$ ${\displaystyle \circ }$ ${\displaystyle \cdot }$ ${\displaystyle K=\mathbb {F} _{4}=\{0,1,\omega ,\omega ^{2}=\omega +1\}}$ ${\displaystyle (u,v)+(x,y)=(u+x,v+y)}$ ${\displaystyle (1,0)}$ ${\displaystyle \iota :K\hookrightarrow K^{2};x\mapsto (x,0)}$

1. ${\displaystyle (u,v)\circ (x,y)=\left(u\cdot x+v^{2}\cdot y,v\cdot x+u^{2}\cdot y+v^{2}\cdot y^{2}\right)}$,
2. ${\displaystyle (u,v)\circ (x,y)=\left(u\cdot v+\omega \cdot v^{2}\cdot y,v\cdot x+u^{2}\cdot y\right)}$.

Pre-semifields

Every field K that allows a non- identical automorphism becomes a commutative pre-semifield with its field addition and the new multiplication . This structure ${\displaystyle \varphi }$ ${\displaystyle a\circ b=\varphi (a\cdot b)}$ ${\displaystyle (K,+,\circ ,0)}$

• has no one element, because only the one of K would come into consideration, this is not neutral, ${\displaystyle 1\circ 1=1}$ ${\displaystyle a\neq \varphi (a)}$
• is associative for elements of the part of the body fixed by the used automorphism ${\displaystyle K_{\varphi }=\{x\in K:\varphi (x)=x\}.}$

literature

• Walter Benz: A Century of Mathematics, 1890–1990 . Festschrift for the anniversary of the DMV . Vieweg, Braunschweig 1990, ISBN 3-528-06326-2 .
• Donald Ervin Knuth : Finite Semifields and Projective Planes . In: Marshall Hall [Research Adviser] (Ed.): Selected Papers on Discrete Mathematics . Dissertation. California Institute of Technology, Stanford January 1, 1963 ( full text [PDF; accessed April 5, 2012]).
• Hanfried Lenz: Small Desarguessian sentence and duality in projective levels . In: Annual report of the German Mathematicians Association . tape 57 . Teubner, 1955, p. 20–31 ( Permalink to the digitized full text [accessed December 25, 2011]).
• 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 ]).

References and comments

1. ↑ ^{a b} Benz (1990)
2. ↑ ^{a b c d e f} Knuth (1963)
3. ↑ See for individual proofs: Moufang level for the statement "Every finite alternative body is a body!" And Wedderburn's proposition for the weaker statement "Every finite oblique body is a body!"
4. ↑ Note that the algebraic structure of the coordinate area in projective planes can depend on the choice of the point base! Weibel (2007)
5. ↑ Lenz (1955)
6. ↑ ^{a b} The commutativity of addition would not have to be required here, since it results from the other axioms for the group together with the other axioms 2 to 5. Knuth (1963), Theorem 2.4 ${\displaystyle (K,+)}$
7. ↑ Knuth (1963), II. Semifields and Pre-Semifields
8. ↑ ^{a b} Hauke ​​Klein: Semifields. Geometry. University of Kiel, accessed on April 9, 2012 (English). 
9. ↑ Knuth (1963) Abstract and Chapter VI-VII, he did not show all of the results mentioned below first, but he gives his own proofs for all of them.
10. ↑ Knuth (1963), sentence 6.4
11. ↑ Isotopy is a weakening of isomorphism: Isomorphic half-bodies are always isotopic, but isotopic half-bodies do not necessarily have to be isomorphic. The coordinate half-bodies of a class V plane are always not isomorphic to one another. Knuth (1963), Chapter VII
12. ↑ The powers in these definitions also relate to this ordinary body multiplication.