Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2206.11031

Semigroup of paths on a family of complexes with uniform ellipticity

Ilya A. Ivanov-Pogodaev·Alexey Ya. Kanel-Belov
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.RA

Keep this discovery

BibTeXRIS

Ilya A. Ivanov-Pogodaev, Alexey Ya. Kanel-Belov. 2022-06-22. Semigroup of paths on a family of complexes with uniform ellipticity. https://arxiv.org/abs/2206.11031

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation

We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.

math.RA↗
arXiv2026

Quasilinear multiplication in the real Cayley--Dickson tower

Direct evaluation of the defining product in the real Cayley--Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge16$ and than the uniform Cariow--Cariowa method for $N\ge32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. In single-core benchmarks against direct multiplication and the uniform Cariow--Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.

math.RA↗
arXiv2026

Graded classification of Leavitt path algebras in terms of strong shift equivalence

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

math.RA↗