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Difference between revisions of "Closure condition"

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The condition in the geometry of webs (cf. [[Webs, geometry of|Webs, geometry of]]) according to which certain incidences of points and lines of a web imply new incidences. For example, Thomsen's closure condition (see Fig.a) is as follows. The first and the second family of lines of a web are represented by parallel straight lines, the third by curves; the closure condition implies that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226101.png" /> lie on curves of the third family, then so do <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226103.png" />.
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The condition in the geometry of webs (cf. [[Webs, geometry of|Webs, geometry of]]) according to which certain incidences of points and lines of a web imply new incidences. For example, Thomsen's closure condition (see Fig.a) is as follows. The first and the second family of lines of a web are represented by parallel straight lines, the third by curves; the closure condition implies that if $A,B,C,D$ lie on curves of the third family, then so do $E$ and $F$.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c022610a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c022610a.gif" />
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Figure: c022610a
 
Figure: c022610a
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226105.png" /> are the parameters defining the lines of the three families, the closure condition can be presented in abstract form as a system of defining equations, regarding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226106.png" /> as the  "product"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226108.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c0226109.png" />,
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If $x,y$ and $z$ are the parameters defining the lines of the three families, the closure condition can be presented in abstract form as a system of defining equations, regarding $z$ as the  "product"  of $x$ and $y$: $z=xy$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c02261010.png" /></td> </tr></table>
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$$x_0y_1=x_1y_0,x_0y_2=x_2y_0\succeq x_1y_2=x_2y_1.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022610/c02261011.png" /> is regarded as a quasi-group operation, Thomsen's closure condition is equivalent to the condition that the quasi-group be isotopic to an Abelian group.
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If $z=xy$ is regarded as a quasi-group operation, Thomsen's closure condition is equivalent to the condition that the quasi-group be isotopic to an Abelian group.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c022610b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c022610b.gif" />

Latest revision as of 19:48, 7 July 2014

The condition in the geometry of webs (cf. Webs, geometry of) according to which certain incidences of points and lines of a web imply new incidences. For example, Thomsen's closure condition (see Fig.a) is as follows. The first and the second family of lines of a web are represented by parallel straight lines, the third by curves; the closure condition implies that if $A,B,C,D$ lie on curves of the third family, then so do $E$ and $F$.

Figure: c022610a

If $x,y$ and $z$ are the parameters defining the lines of the three families, the closure condition can be presented in abstract form as a system of defining equations, regarding $z$ as the "product" of $x$ and $y$: $z=xy$,

$$x_0y_1=x_1y_0,x_0y_2=x_2y_0\succeq x_1y_2=x_2y_1.$$

If $z=xy$ is regarded as a quasi-group operation, Thomsen's closure condition is equivalent to the condition that the quasi-group be isotopic to an Abelian group.

Figure: c022610b

Figure: c022610c

Fig.b and Fig.c illustrate the Reidemeister closure condition and the hexagonality condition (all three conditions are equivalent for a plane three-web, even without a differentiability assumption). In abstract form these conditions yield different classes of quasi-groups and loops; in the multi-dimensional geometry of webs, they yield different classes of webs.

Some theorems in projective geometry are essentially closure conditions (such as the theorems of Desargues and Pappus).

References

[1] W. Blaschke, "Einführung in die Geometrie der Waben" , Birkhäuser (1955)
[2] V.D. Belousov, V.V. Ryzhkov, "Geometry of webs" J. Soviet Math. , 2 (1974) pp. 331–348 Itogi Nauk. i Tekhn. Alg. Topol. Geom. , 10 (1972) pp. 159–188
[3] V.D. Belousov, "Algebraic nets and quasi-groups" , Stiintsa , Kishinev (1971) (In Russian)


Comments

For the notion of isotopic groupoids cf. Isotopy.

How to Cite This Entry:
Closure condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closure_condition&oldid=32395
This article was adapted from an original article by V.V. Ryzhkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article