Initial algebra

In mathematics, an initial algebra is an initial object in the category of -algebras for a given endofunctor . The initiality provides a general framework for induction and recursion.

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