Abstract
The work is devoted to the study of the combinatorial properties of determinism in a family of substitution complexes consisting of quadrilaterals glued together side-to-side. These properties are useful in the construction of algebraic structures with a finite number of defining relations. In particular, this method was used in constructing an infinite, finitely presented nilsemigroup satisfying the identity \({{x}^{9}} = 0\) . This construction solves the problem posed by L.N. Shevrin and M.V. Sapir. This work investigates the possibility of coloring the entire family of complexes with a finite number of colors, for which the property of weak determinism holds: if the colors of three vertices of a given quadrilateral are known, then the color of the fourth vertex is uniquely determined, except in some cases of special arrangement of the quadrilateral. Even weak determinism is sufficient to construct a finitely presented nilsemigroup; when using this construction, the proof is shortened. Determinism properties help to correctly introduce defining relations in the semigroup of paths traversing the constructed complexes. The defining relations correspond to pairs of equivalent short paths. Properties of determinism have previously been studied in the context of tiling theory; in particular, Kari and Papasoglu constructed a set of square tiles that admits only aperiodic tilings of the plane and has the property of determinism: knowing the colors of two adjacent edges uniquely determines the colors of the remaining two edges.
