Tetration

Tetration, also known as hyper4, superpower, superdegree or powerlog, can refer to one of two different dyadic operators. The first and most common, written $$^yx$$ or x④y, is defined as $$^yx = \underbrace{x^{x^{x^{.^{.^.}}}}}_y$$, in analogy to $$x \cdot y = \underbrace{x + x + \ldots + x + x}_y$$ and $$x^y = \underbrace{x \cdot x \cdot \ldots \cdot x \cdot x}_y$$. It can also be written $$x\uparrow\uparrow y$$ (using Chained Arrow Notation) or $$\{x,y,2\}$$ (using BEAF.)

The second, less mathematically interesting version (also known as the hyper4 operator) is written x④y and is defined as x④y $$= \underbrace{\left(\left(\left(x^x\right)^x\right)^x\right)\ldots}_y$$, which simplifies to $$x^{x^{y - 1}}$$ as $$\left(a^b\right)^c = a^{b \cdot c}$$.

At the current time, mathematicians have not agreed on the function's behavior on $$^yx$$ where y is not an integer.