To the theorem of A. P. Sprague on spatial webs (Q1056905)

The authors define spatial \((=3\)-dimensional) webs by incidence properties only. Let then F be a (not necessarily commutative) field, R a right vector space over F. Then a spatial web over R is defined and called a spatial vector web. If G is an additive group and E a set of endomorphisms of G satisfying \(\forall \epsilon_ 1,\epsilon_ 2\in E\); \(\epsilon_ 1\neq \epsilon_ 2\) \(\forall b\in G\) \(\exists ! a\in G:a\epsilon_ 1=a\epsilon_ 2+b,\) then (G,E) is said to be a ''group with endomorphisms''. (If \(\epsilon\in E\) is nonzero, \(\epsilon\) is an automorphism). If G is the additive group of R, and E the set of all maps \(R\to R\), \(x\mapsto xa\), \(a\in F\), then a planar vector web over R can be defined by means of (G,E). The authors prove that every spatial web is isomorphic with some spatial vector map. For planar webs an analogous result holds only under special configuration assumptions, which are automatically satisfied in spatial webs: the minor and major Desargues and Reidemeister conditions. These conditions and their influence on the structure of embedded planar webs are thoroughly investigated in the paper. Finally, a coordinatization of spatial webs by means of a right vector space over a field is accomplished.