By Gustave Choquet

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Greatly considered as a vintage of recent arithmetic, this accelerated model of Felix Klein's celebrated 1894 lectures makes use of modern strategies to envision 3 well-known difficulties of antiquity: doubling the quantity of a dice, trisecting an perspective, and squaring a circle. contemporary scholars will locate this quantity of specific curiosity in its solutions to such questions as: less than what situations is a geometrical development attainable?

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217] posed the following problem: Construct examples o f irreducible n-quasigroups f o r n > 3. Do there exist irreducible n-quasigroupsfor any n > 3? 14) for components of the torsion tensor of a web. 14), then the (n + 1)-web and its coordinate n-quasigroups are irreducible. This is the case, for example, for a general (n + 1)-web, a hexagonal (n + 1)-web, a transversally geodesic (n + 1)-web, an isoclinic (n + 1)-web, and a (2n + 2)-hedral (n + 1)-web. We shall call an (n + 1)-web a group (n + 1)-web if at least one of its coordinate n-quasigroups is an n-group.

U)) . w = u . (v . (u . w)) u . ((v . w) . v) = ((u . v) . w) . v u2. u = u. A. Akivis and V V. Goldberg and if one of the coordinate loops L p of a complete web W is a loop of any of these types, then all others are of the same type. 11) Here (Bin) denotes the middle Bol condition which is connected with properties of left and fight inverse loops for the loops L p (not with properties of the loops L p themselves). This was demonstrated in many papers (see, for example, [ 1,84], and [44]). When we will turn from complete three-webs to geometric webs W(3, 2, r) defined on a differentiable manifold M, we will come to the following situation.

12) For webs W(3, 2, r) in the cases r = 2, 3 the conditions (R) and (M) are equivalent. 1. This was noted in [136] and studied in detail in works of Akivis and his followers (see [86,23], and [44]). 2). 1, for the proof of their sufficiency one needs to use the apparatus developed in Sections 1 and 2 and perform rather complicated analytic and geometric constructions (see [7,38,44,161]). 2 Closure condition Conditions Akivis algebra References Trivial algebra [7] [7] [38,419] on tensors a and b (T) a)k = O, b}k I = 0 (R) b}kl = 0 9 i ] bjk I = b[jkl b i(jk)l =0 Lie algebra Mal'cev's algebra i b(jlkll ) --0 bj(kl ) = 0 b i(jkl) = 0 Triple Bol's system [161,413,222] Triple Bol's system [161,413,222] Not investigated [7] (M) (BI) (Br) (Bin) (H) Triple Bol's system [161,413,222] Nagy [368] investigated the reductive homogeneous spaces associated to Moufang loops.