A semigroup of paths on a sequence of uniformly elliptic complexes
Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 2, pp. 41-74
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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.
Keywords:
aperiodic tiling, determinacy, substitution complex,
finitely presented semigroup,
Burnside-type problem, nil-semigroup.
@article{FAA_2023_57_2_a3,
author = {I. A. Ivanov-Pogodaev},
title = {A semigroup of paths on a sequence of uniformly elliptic complexes},
journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
pages = {41--74},
publisher = {mathdoc},
volume = {57},
number = {2},
year = {2023},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FAA_2023_57_2_a3/}
}
I. A. Ivanov-Pogodaev. A semigroup of paths on a sequence of uniformly elliptic complexes. Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 2, pp. 41-74. http://geodesic.mathdoc.fr/item/FAA_2023_57_2_a3/