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This is the third part of a cycle of papers devoted to the construction of afinitely 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 ofpath encodings on a family of special uniformly elliptic complexes. In thefirst paper of the cycle Finitely defined nil semigroup: complexes with uniformellipticity, a sequence of complexes was constructed with a set of geometricproperties. In the second work of the series Deterministic Coloring of a Familyof Complexes a finite letter encoding was introduced on the vertices and edgesof the constructed complexes. The deterministic property of such a coloring wasproved, which makes it possible to introduce a finite set of defining relationson the set of words-codings of paths on complexes. In this paper, we describean algorithm for reducing an arbitrary semigroup word to canonical form. It isalso proved that a word containing a subword with period 9 can be reduced tozero using defining relations. Word encodings corresponding to sufficientlylong paths are not reduced to zero and do not change their length, that is, theintroduced semigroup is infinite.

Semigroup of paths on a family of complexes with uniform ellipticity