Initial algebra

In mathematics, an initial algebra is an initial object in the category of ${\displaystyle F}$-algebras for a given endofunctor ${\displaystyle F}$. This initiality provides a general framework for induction and recursion.

For instance, consider the endofunctor ${\displaystyle 1+(-)}$ on the category of sets, where ${\displaystyle 1}$ is the one-point (singleton) set, the terminal object in the category. An algebra for this endofunctor is a set ${\displaystyle X}$ (called the carrier of the algebra) together with a point ${\displaystyle x\in X}$ and a function ${\displaystyle X\to X}$. The set of natural numbers is the carrier of the initial such algebra: the point is zero and the function is the successor map.

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