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.

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独立代数、基代数和分配条件

稳定基代数由 Fountain 和 Gould 介绍并在一系列文章中得到发展。它们形成一类泛代数，扩展了独立代数的类。如果有限秩的稳定基代数 $\mathbb{B}$ 满足分配条件（所有先前已知示例都满足的条件），则它是独立代数 $\mathbb{A}$ 的约简。我们的第一个目标是给出一个不满足分配条件的独立代数的例子。Gould 证明，如果具有分配条件的稳定基代数 $\mathbb{B}$ 具有有限秩，那么它是归约的独立代数 $\mathbb{A}$ 也是如此，在这种情况下，自同态幺半群$\mathbb{B}$的End$(\mathbb{B})$是$\mathbb{A}$的自同态幺半群End$(\mathbb{A})$的左序。我们通过确定 End$(\mathbb{B})$ 何时是一个权利来完成图片，因此在 End$(\mathbb{A})$ 中是一个双边顺序。事实上（对于等级至少为 2），这恰好发生在 End$(\mathbb{A})$ 的每个元素都可以写成 $\alpha^\sharp\beta$ 其中 $\alpha,\beta\in$ End$(\mathbb{B})$, $\alpha^\sharp$ 是 End$(\mathbb{A})$ 和 $\alpha$ 和 $\beta$ 的子群中 $\alpha$ 的逆有相同的内核。这相当于 End$(\mathbb{B})$ 是 End$(\mathbb{A})$ 中一种特殊的左序，称为直。$\alpha^\sharp$ 是 End$(\mathbb{A})$ 的子群中 $\alpha$ 的逆元，并且 $\alpha$ 和 $\beta$ 具有相同的核。这相当于 End$(\mathbb{B})$ 是 End$(\mathbb{A})$ 中一种特殊的左序，称为直。$\alpha^\sharp$ 是 End$(\mathbb{A})$ 的子群中 $\alpha$ 的逆元，并且 $\alpha$ 和 $\beta$ 具有相同的核。这相当于 End$(\mathbb{B})$ 是 End$(\mathbb{A})$ 中一种特殊的左序，称为直。