View source for 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 $A,B,C,D$ 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" />

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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c022610b.gif" />

Figure: c022610b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c022610c.gif" />

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,   "Einführung in die Geometrie der Waben" , Birkhäuser  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.D. Belousov,   "Algebraic nets and quasi-groups" , Stiintsa , Kishinev  (1971)  (In Russian)</TD></TR></table>



====Comments====
For the notion of isotopic groupoids cf. [[Isotopy|Isotopy]].