In this paper, we consider some three-dimensional noncommutative algebra ${\stackrel{\mathbb{~}}{\mathbb{A}}}_2$ over the field $\mathbb{C}$, which contains the algebra of bicomplex numbers $\mathbb{B}(\mathbb{C})$ as a subalgebra. Locally bounded and Gâteaux-differentiable mappings defined in the domains of the three-dimensional subspace of the algebra $\mathbb{B}(\mathbb{C})$ and taking values in the algebra ${\stackrel{\mathbb{~}}{\mathbb{A}}}_2$ are considered. Such mappings generalize holomorphic functions of bicomplex variable and mentioned mappings are described by means of three holomorphic functions of a complex variable.

中文翻译：

---

三维非交换代数中的 G-单基因映射

在本文中，我们考虑一些三维非交换代数${\stackrel{\mathbb{~}}{\mathbb{一个}}}_2$在场上$\mathbb{C}$，其中包含双复数的代数$\mathbb{乙}(\mathbb{C})$作为子代数。在代数的三维子空间域中定义的局部有界和 Gâteaux 可微映射$\mathbb{乙}(\mathbb{C})$并在代数中取值${\stackrel{\mathbb{~}}{\mathbb{一个}}}_2$被考虑。这种映射概括了双复变量的全纯函数，并且所提到的映射是通过一个复变量的三个全纯函数来描述的。