On the determinicity of paths on substitution complexes

TL;DR

This paper investigates the combinatorial determinacy of paths in substitution complexes of quadrilaterals, with implications for constructing algebraic structures and understanding coloring properties.

Contribution

It introduces conditions for coloring substitution complexes to satisfy weak determinism, advancing methods for algebraic structure construction.

Findings

Established conditions for coloring complexes with weak determinism

Connected determinacy properties to algebraic structure construction

Provided insights into coloring constraints in substitution complexes

Abstract

The paper is devoted to the study of combinatorial determinacy properties of a family of substitution complexes consisting of quadrilaterals glued side-to-side with each other. These properties are useful in constructing algebraic structures with a finite number of defining relations. In particular, this method was used when constructing a finitely presented infinite nisemigroup satisfying the identity x^9 =0. This construction responds to the problem of L. N. Shevrin and M. V. Sapir. In this paper, we investigate the possibility of coloring the entire sequence of complexes into a finite number of colors, in which the property of weak determinism is fulfilled: if the colors of the three vertices of a certain quadrilateral are known, then the color of the fourth side is uniquely determined, except in some cases of a special arrangement of the quadrilateral.