Semigroup of paths on a family of complexes with uniform ellipticity

TL;DR

This paper constructs an infinite finitely presented nil-semigroup satisfying the identity x^9=0, using path encodings on specially designed complexes, and provides an algorithm for word reduction demonstrating the semigroup's properties.

Contribution

It introduces a new construction of an infinite nil-semigroup with uniform ellipticity, solving a problem posed by Shevrin and Sapir, with an explicit reduction algorithm.

Findings

A canonical form reduction algorithm for semigroup words.

Words with subwords of period 9 reduce to zero.

Long path encodings do not reduce to zero, ensuring infiniteness.

Abstract

This is the third part of a cycle of papers devoted to the construction of a finitely presented infinite nil-semigroup satisfying the identity $x^9 = 0$. This construction answers the problem of L. N. Shevrin and M. V. Sapir, posed, for example, in the Sverdlovsk notebook. A semigroup is realized as a set of path encodings on a family of special uniformly elliptic complexes. In the first paper of the cycle Finitely defined nil semigroup: complexes with uniform ellipticity, a sequence of complexes was constructed with a set of geometric properties. In the second work of the series Deterministic Coloring of a Family of Complexes a finite letter encoding was introduced on the vertices and edges of the constructed complexes. The deterministic property of such a coloring was proved, which makes it possible to introduce a finite set of defining relations on the set of words-codings of paths on complexes. In this paper, we describe an algorithm for reducing an arbitrary semigroup word to canonical form. It is also proved that a word containing a subword with period 9 can be reduced to zero using defining relations. Word encodings corresponding to sufficiently long paths are not reduced to zero and do not change their length, that is, the introduced semigroup is infinite.