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