# Upper bounds for the length of non-associative algebras

## TL;DR

This paper establishes a precise upper bound for the length of non-associative algebras using a novel linear algebra-based method, demonstrating the bound's sharpness with examples and applications to locally complex algebras.

## Contribution

Introduces a new characteristic sequence method based on linear algebra to determine the length bounds of non-associative algebras, including locally complex cases.

## Key findings

- Derived a sharp upper bound for the length of non-associative algebras.
- Provided an example demonstrating the bound's sharpness.
- Linked the length bound in locally complex algebras to Fibonacci numbers.

## Abstract

We obtain a sharp upper bound for the length of arbitrary non-associative algebra and present an example demonstrating the sharpness of our bound. To show this we introduce a new method of characteristic sequences based on linear algebra technique. This method provides an efficient tool for computing the length function in non-associative case. Then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra. We also show that the obtained bound is sharp. In the last case the length is bounded in terms of Fibonacci sequence.