Algebra and geometry of three-webs composed of foliations of different dimensions (Q941976)

Algebraic and geometric notions and some results of classical 3-web theory (three \(r\)-dimensional foliations on \(2r\)-dimensional manifold) are generalized for webs formed by foliations of different dimensions. In introduction the author gives a review of well-known results and indicates the main obstructions in developing the theory of 3-webs formed by foliations of different dimensions: the binary operation defined by the web equation \(z=f(x, y)\) does not admit the notions `unit', `commutativity', `associativity' since the variables \(x, y\) and \(z\) are of different dimensions. To overcome this obstruction the author introduces the notions of coordinate lattice, coordinate monoid, unit matrix element which are generalizations of the well-known notions `coordinate system', `loop', `unit'. In terms of these notions generalized Reidemeister configurations and generalized Reidemeister 3-webs are defined. The algebraic condition for a 3-web to be a generalized Reidemeister 3-web is given (generalized associativity identity). The 3-webs defined by transformation groups and Bol transformation quasigroups are introduced and some of their properties are considered. Examples of such webs are constructed.