Algebras of length one

TL;DR

This paper characterizes algebras over fields of any characteristic that have a generating set where every element can be expressed as a linear combination of just the generators themselves, indicating a length of one.

Contribution

It provides a classification of non-associative algebras over arbitrary fields with length equal to one, expanding understanding of algebraic generation properties.

Findings

Algebras with length one are fully characterized.

The classification applies to non-associative algebras over any field.

Results extend previous work on algebraic length and generation.

Abstract

If $\A$ is an algebra over a field $\F$ and $\s$ is a generating set of $\A$, the length of $\s$ indicates the maximal length needed to express an arbitrary element of $\A$ as a linear combination of words in the elements of $\s$. The length of an algebra $\A$ is defined as the maximum of lengths of its generating sets. In this paper (not necessarily associative) algebras, over fields of arbitrary characteristic, having length equal to 1 are determined.