The geometry of double loops
By methods of classical geometry, the authors present an extensive study of not necessarily planar double loops \((Q,+,\cdot)\). A double loop is a set Q with two binary operations satisfying: \((Q,+)\) is a loop with neutral element 0, \((Q^*,\cdot)\) is a loop, \(Q^*:=Q\setminus \{0\}\), and \(0\cdot a=a\cdot 0=0\) for all \(a\in Q\). To every such double loop Q they assign a geometry \({\mathfrak N}\), called double net, which contains the 3-nets belonging to \((Q,+)\) and \((Q^*,\cdot)\) in a natural way [the authors, Adv. Math. 49, 1-105 (1983; Zbl 0518.20064)]. (In the classical case of a field Q, \({\mathfrak N}\) consists of the points in \(Q\times Q\), the horizontal and vertical lines (x,a) and (a,y), the transversal lines \(x+y=a,\) \(a\in Q\), and the hyperbolas \(xy=b\), \(b\in Q^*.)\) In general, \({\mathfrak N}\) is a rather weak geometry, but as in the paper cited above, the geometric approach turns out to be fruitful and leads to a wide variety of algebraic results on double loops. The main part of the paper is devoted to von Staudt's point of view, that is to characterizations of ``nice'' double loops Q via their group of projectivities \(\Theta\). Q is a commutative field if and only if \(\Theta\) is sharply 2-transitive on Q; a locally compact, not totally disconnected, topological double loop Q equals the field of real or complex numbers if and only if the pointwise stabilizer in \(\Theta\) of two different points is totally disconnected; a locally compact, 1- dimensional double loop Q is the field of the reals if and only if \(\Theta\) is locally compact; a finite double loop Q of order \(\neq 9,10,11,12,16\) is a field if and only if the pointwise stabilizer in \(\Theta\) of every five distinct points is trivial; in a finite double loop Q the loops \((Q,+)\) and \((Q^*,\cdot)\) are abelian groups if and only if \(\Theta\) has a regular abelian subgroup A, such that the stabilizer in \(\Theta\) of \(x\in Q\) has an abelian subgroup B regular on \(Q\setminus \{x\}\) and A and B generate \(\Theta\). The authors manage to determine all finite, 2-transitive groups with these properties, and thus to obtain a certain, complete overview of the finite double loops with commutative and associative addition and multiplication. Further, the authors give an account of free double loops and free double nets: as for free planes, they obtain that every double subnet of a free double net is free, that a free double net of rank m contains free double subnets of any rank \(n\geq 0\), and that every double net of rank m is the homomorphic image of a free double net of the same rank. Since they show that strongly isotopic free double loops are isomorphic, all these geometric results directly transfer to double loops. Using the theory developed by \textit{M. Funk}, \textit{O. H. Kegel} and the second author in Ann. Math. Pura Appl., IV. Ser. 141, 1-126 (1985; Zbl 0588.03021), they observe that for every given group G there exists a double loop Q (even with commutative addition and multiplication) such that G is the full automorphism group of Q. Besides this, they calculate the collineation groups of double loops over nearfields and fields, give an account of ratios and conics in double loops over di-associative division algebras (showing that a field Q is commutative if and only if the set of all lines belonging to the additive net together with all nondegenerate conics form an affine plane; and that in this case the projectivities are exactly those permutations of Q leaving the ratio invariant), and address the relations between the algebraic and the geometric notions of homomorphisms as well as between the algebraic and the geometric notions of orderings. In particular, using results from the first paper, cited above, (relying on \textit{J. Joussen}, Results Math. 4, 55-74 (1981; Zbl 0466.51003)), they prove that every free double loop of finite rank admits an archimedean ordering and is so embeddable into a topological double loop homeomorphic to the reals.
scientific article; zbMATH DE number 4086411
scientific article