Determinicity properties, Mozes-Goodman-Strauss theorems and the construction of algebraic objects

The theorems of Moses and Goodman-Strauss are widely known in the theory of tilings. They link the language of local rules and the language of substitution systems. Let a substitution rule be given, according to which a tile of the next level can be assambled from polygon tiles of several types. Then the sides of the polygons can be decorated with a finite number of colors so that all tilings that follow the local rules (tiles are applied to each other only by sides of the same colors) will be generated by this substitution system. Thus, substitution systems can be specified using local rules. Determinicity property is also very interesting. For example, consider square Wang tiles, the sides of which are painted in a finite set of colors and only tiles with sides of the same color can be placed side by side. There are aperiodic sets of tiles that can be used to tile a plane only in an aperiodic manner. As Kari and Papasoglu showed, there are also aperiodic and deterministic sets, where the colors of two adjacent sides uniquely determine the colors of the other two sides (there is at most one tile with adjacent sides with a given pair of colors).
It is interesting that determinicity can also be achieved in more general situations. The talk deals with substitution systems of flat complexes, with a substitution rule, according to which a tile (4-cycle) of level k is subdivided into tiles of level k-1. The substitution system specifies a sequence of complexes of all possible levels.
Let the path consist of two adjacent edges of some minimal tile T and passes through three of its four vertices. Let X be the fourth vertex of T through which the path does not pass. If X lies on the boundary of a tile of strictly higher level than the other three vertices of T, we will call the path irregular. In other cases, we will call the path regular. Let the vertices and edges of the sequence of complexes obtained by some substitution be colored in a finite number of colors. We will assume that the coloring of a sequence of complexes has weak determinicity if known colors of the edges and vertices of the path P' are uniquely determine the colors of the edges and vertices of the path P', with the same beginning and end as P, but passing along the other two adjacent sides of the same tile.
The main result is that, for a given substitution system, a coloring in a limited number of colors with weak determinicity is always possible.
The specification of substitution tilings by local rules and determinicity (including weak determinicity) can be useful in constructing algebraic objects with a finite number of defining relations. Words from a semigroup or ring are considered as paths on a specially constructed sequence of complexes glued together from 4-cycles (tiles). Let a coloring with a globally bounded number of colors be introduced on the vertices and edges of a sequence of complexes.
The colors of the edges and vertices correspond to the letters of the finite alphabet, the vertices and edges traversed along the path correspond to a word. The defining relations correspond to pairs of equivalent paths of length 2, consisting of two adjacent sides of the square. For rings and semigroups, monomial relations are also introduced for some types of forbidden paths. In this case, the geometric properties of the complex correspond to some useful properties in the resulting object. This method can be useful for constructing finitely presented objects of Burnside type and has been applied in constructing a finitely presented nilsemigroup.