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.