Tue, 13 Oct 2026, 1:25 pm MDT
In order to study a certain 3-dimensional analogue of the Mandelbrot set, I needed a notion of holomorphy for the corresponding 3-dimensional real algebra. While this motivation (to generalize the basic elements of complex dynamics) is analytic, the focus of this talk will be the connection between holomorphic functions over a real algebra and first-order properties of the algebra. It turns out that, under the assumption of a single (left or right) divisible element, the holomorphic functions over an arbitrary real algebra form a vector space. Properties such as the Product Rule and Chain Rule are equivalent to purely algebraic properties of the underlying real algebra. When our real algebra has a multiplicative identity element, the Product Rule is precisely equivalent to being both commutative and associative. I will present proofs of some of these statements, which are quite succinct once the proper groundwork has been done.