Almost body

An almost solid is an algebraic structure that is used in synthetic geometry as a coordinate range for certain affine and projective translation planes . He generalizes the term inclined body insofar as only one of the distributive laws is required: for a left fast body the left law, for a right fast body the right distributive law. A fast body that only fulfills one of the distributive laws is also called a real fast body . To distinguish between different meanings of the term, the structures described here are sometimes (for example by Zassenhaus ) called complete fast bodies .

The projective planes of classes IVa.2 in the Lenz-Barlotti classification of projective planes can always be coordinated by a real left fast body, as well as the only plane of class IVa.3 (except for isomorphism). The dual classes IVb.2 and IVb.3 can be coordinated by real legal fast bodies. In addition, by changing the multiplication, which is related to the Moulton plane construction, one can construct models for arranged ternary bodies from arranged almost solids, which coordinate planes of Lenz class I.

One can always build a weakly affine space on a finite almost body as a coordinate area .

Every oblique body is an almost body. Every almost body is a quasi-body and therefore a Cartesian group and a ternary body.

definition

A left fast body or short fast body is an algebraic structure so that two two- digit links addition and multiplication are defined on the set , for which applies: ${\displaystyle (F,+,\cdot ),}$ ${\displaystyle F}$ ${\displaystyle +}$ ${\displaystyle \cdot }$

1. ${\displaystyle (F,+)}$is a group with the neutral element ${\displaystyle 0.}$
2. ${\displaystyle (F\setminus \{0\},\cdot )}$ is a group with the neutral element ${\displaystyle 1.}$
3. The zero is absorbent : applies to everyone . ${\displaystyle a\cdot 0=0\cdot a=0}$ ${\displaystyle a\in F}$
4. It applies the Linksdistributivgesetz : for all ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$ ${\displaystyle a,b,c\in F.}$

If the right distribution law applies instead of the left distribution law, then one speaks of a right almost body.

Equivalently, a left fast body can be defined as a left quasi-body with associative multiplication. The same naturally also applies to the respective “legal” structures.

Core of the fast body

As with a quasi-body , for an almost-body too, the amount becomes

${\displaystyle \operatorname {Kern} (F):=\{x\in F\mid \forall a,b\in F\colon (a+b)x=ax+bx\}}$ for left chamfer or

${\displaystyle \operatorname {Kern} (F):=\{x\in F\mid \forall a,b\in F\colon x(a+b)=xa+xb\}}$ for legal caste

defined as the core of the fast body.

Features and remarks

• A fast body is a fast ring with a group. ${\displaystyle (F,+,\cdot ),}$ ${\displaystyle (F\setminus \{0\},\cdot )}$
• The addition of an almost body is always commutative , in other words: it is an Abelian group . ${\displaystyle (F,+)}$
• The term "(complete) almost body" of geometry stands between the terms "quasi-body" and "oblique body":

• Every almost body is a quasi body and a quasi body is an almost body if and only if the associative law of multiplication applies in it.
• Every oblique body is an almost body and an almost body is an oblique body if and only if both distributive laws apply in it.

• Also between "quasi-body" and "oblique body" are the terms " half-body " (in the sense of geometry) and the sharper term " alternative body ", both terms also describe structures that are not almost bodies and an almost body does not need a half body and thus certainly not one To be alternate body.

• A half body is an almost body if and only if the associative law of multiplication applies in it, it is then even a skew body. In other words: an algebraic structure that is half-body and almost-body at the same time is necessarily a skew.
• The two previous statements apply word for word to “alternative body” instead of “half body”.

• The core of a fast body is an oblique body and the fast body is a module above its core. (For geometrical conclusions from this fact see affine translation plane !)
• According to Wedderburn's theorem, the kernel of a finite almost-field is a finite field . So every finite near-body is a finite-dimensional vector space over such a finite field and therefore has elements ( prime number, ). ${\displaystyle p^{n}}$ ${\displaystyle p}$ ${\displaystyle n\in \mathbb {N} }$
• An almost body is a half body - and thus also a sloping body - if and only if it agrees with its core.

Examples

• Every oblique body and especially every body naturally provides an example of an almost body.
• The nine-element quasi-body , which is described in more detail in the article Ternary bodies (in the section Examples of order 9 ), is an example of a finite right caste body that is not a half-body. ${\displaystyle J_{9}}$
• For every odd prime number in the finite field with elements, the multiplication can be modified in such a way that a “real” complete left fast body of the order is created, which is a two-dimensional vector space above its core . A possible construction is described in detail in the article Quasi-Bodies in the section Quasi-Bodies of Finite Moulton Planes . In order to fulfill the associative law of multiplication, one can base the modified multiplication on the involutorial body automorphism and thus obtain a left fast body of the type described. ${\displaystyle p}$ ${\displaystyle p^{2n}}$ ${\displaystyle \mathbb {F} _{p^{2n}},\;(n\in \mathbb {N} \setminus \lbrace 0\rbrace )}$ ${\displaystyle F}$ ${\displaystyle p^{2n}}$ ${\displaystyle \mathrm {Kern} (F)=\mathbb {F} _{p^{n}}}$ ${\displaystyle \varphi :\mathbb {F} _{p^{2n}}\rightarrow \mathbb {F} _{p^{2n}};x\mapsto x^{(p^{n})}}$

literature

• Daniel R. Hughes: A class of non-Desarguesian projective planes . In: Canadian Journal of Mathematics . tape 9 . London Mathematical Society, 1957, ISSN  0008-414X , pp. 378-388 , doi : 10.4153 / CJM-1957-045-0 ( math.ca ).
• BH Neumann : On the commutativity of addition . In: Journal of the London Mathematical Society . s1-15, no. 3 . London Mathematical Society, 1940, pp. 203-208 , doi : 10.1112 / jlms / s1-15.3.203 .
• Günter Pilz: Near-Rings . North-Holland, Amsterdam / New York / Oxford 1977, ISBN 0-7204-0566-1 .
• Sibylla Prieß-Crampe : Arranged structures . Groups, bodies, projective levels (=  results of mathematics and its border areas . Volume 98 ). Springer, Berlin / Heidelberg / New York 1983, ISBN 3-540-11646-X .
• Emanuel Sperner : Affine Spaces with Weak Incidence and Associated Algebraic Structures . In: Journal for Pure and Applied Mathematics (Crelles Journal) . tape 204 . University of Berlin, 1960, ISSN  1435-5345 , p. 205-215 , doi : 10.1515 / crll.1960.204.205 .
• Charles Weibel : Survey of Non-Desarguesian Planes . In: Notices of the American Mathematical Society . tape 54 . American Mathematical Society, November 2007, pp. 1294–1303 ( ams.org [PDF; 702 kB ]).
• Hans Zassenhaus : About finite almost bodies . In: Treatises from the Mathematical Seminar of the University of Hamburg . No. 11 . Springer, 1935, p. 187-220 .
• JL Zemmer: The additive group of an infinite near-field is abelian . In: Journal of the London Math. Soc. s1-44, no. 1 . London Mathematical Society, 1969, pp. 65–67 , doi : 10.1112 / jlms / s1-44.1.65 (English, beginning of the article ).

Individual evidence

1. ↑ ^{a b c} Zassenhaus (1935)
2. ↑ Prieß-Crampe (1983) V. §5 Lenz-Barlotti classification of arranged projective planes
3. ↑ Sperner (1960)
4. ↑ ^{a b} Zemmer (1969)
5. ↑ In the English-language specialist literature, preference is more often given to the "right" versions, in the German-language literature rather the "left" versions. In all cases, the qualifying information (“ links quasi-bodies” etc.) is used at the beginning when defining the structures. Compare Weibel (2007) p. 1300.
6. ↑ Hauke Klein: Near Fields. In: Geometry. University of Kiel, November 29, 2002, accessed on December 15, 2011 (English). 
7. ^ Neumann (1940)
8. ↑ Hughes (1957)