Deterministic coloring of a family of complexes

TL;DR

This paper presents a method for deterministic coloring of complexes, ensuring that the color of each square's corner is uniquely determined by the colors of three corners, contributing to the construction of a finitely presented infinite nil semigroup.

Contribution

It introduces a finite, deterministic coloring scheme for complexes that is crucial for constructing a specific algebraic structure, advancing prior work on nil semigroups.

Findings

Complexes are uniform elliptic with shortest paths forming disks of bounded width.

A finite color system with deterministic rules is established for complexes.

The coloring scheme ensures corner colors are uniquely determined by adjacent corners.

Abstract

This is the second paper devoted to construction of finitely presented infinite nil semigroup with identity $x^9=0$. This construction answers to the problem of Lev Shevrin and Mark Sapir. In the first part we constructed the sequence of complexes with some set of properties. Namely, all these complexes are uniform elliptic: any two points $A$ and $B$ with distance $d$ can be connected with a system of shortest paths forming a disk of width $ \lambda \cdot D $ for some global constant $ \lambda> 0 $. In the second part of the proof, a finite system of colors with determinism is introduced: for each minimum square that the complex consists of, the color of the three angles determines the color of the fourth corner. The present paper is devoted to the second part of the proof.