Independence algebras, basis algebras and the distributivity condition

Abstract

Stable basis algebras were introduced by Fountain and Gould and developed in a series of articles. They form a class of universal algebras, extending that of independence algebras, and reflecting the way in which free modules over well-behaved domains generalise vector spaces. If a stable basis algebra $\mathbb{B}$ satisfies the distributivity condition (a condition satisfied by all the previously known examples), it is a reduct of an independence algebra $\mathbb{A}$ . Our first aim is to give an example of an independence algebra not satisfying the distributivity condition.

Gould showed that if a stable basis algebra $\mathbb{B}$ with the distributivity condition has finite rank, then so does the independence algebra $\mathbb{A}$ of which it is a reduct, and that in this case the endomorphism monoid ${\rm End}(\mathbb{B})$ of $\mathbb{B}$ is a left order in the endomorphism monoid ${\rm End}(\mathbb{A})$ of $\mathbb{A}$. We complete the picture by determining when ${\rm End}(\mathbb{B})$ is a right, and hence a two-sided, order in ${\rm End}(\mathbb{A})$. In fact (for rank at least 2), this happens precisely when every element of ${\rm End}(\mathbb{A})$ can be written as ${\alpha}^{\sharp} \beta$ where $\alpha,\beta\in{\rm End}(\mathbb{B})$, ${\alpha}^{\sharp}$ is the inverse of $\alpha$ in a subgroup of ${\rm End}(\mathbb{A})$ and $\alpha$ and $\beta$ have the same kernel. This is equivalent to ${\rm End}(\mathbb{B})$ being a special kind of left order in ${\rm End}(\mathbb{A})$ known as straight.