A Semigroup of Paths on a Sequence of Uniformly Elliptic Complexes

Abstract

The work is devoted to solving a problem of L. N. Shevrin and M. V. Sapir (Question 3.81b of the Sverdlovsk Notebook), namely, to constructing a finitely presented infinite nil-semigroup satisfying the identity \(x^9 = 0\). This problem is solved with the help of geometric methods of the theory of tilings and aperiodic tessellations. A semigroup of paths on a tiling, under certain conditions, inherits some properties of the tiling itself. Moreover, the defining relations in the semigroup correspond to a set of equivalent paths on the tiling.

The relationship between the geometric and the automaton approaches previously used in the construction of finitely presented objects is discussed. As noted by S. P. Novikov, the property of determinacy in the coloring of partition nodes and its extension inward is very similar to properties of a solution of a partial differential equation with a given boundary condition. The author believes that understanding this relationship between the theories of aperiodic mosaics and their arrangements and the theory of numerical methods and grids is very promising.