Initial algebra

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

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

单词	Initial algebra
释义	Initial algebra 英语百科 Initial algebra In mathematics, an initial algebra is an initial object in the category of $F$ -algebras for a given endofunctor $F$ . The initiality provides a general framework for induction and recursion. For instance, consider the endofunctor $1+(-)$ on the category of sets, where $1$ is the one-point (singleton) set, the terminal object in the category. An algebra for this endofunctor is a set $X$ (called the carrier of the algebra) together with a point $x\in X$ and a function $X\to X$ . 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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