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 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* ∈ *X* and a function *X*→*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.

For a second example, consider the endofunctor 1+**N**×(-) on the category of sets, where **N** is the set of natural numbers. An algebra for this endofunctor is a set *X* together with a point *x* ∈ *X* and a function **N**×*X* → *X*. The set of finite lists of natural numbers is the initial such algebra. The point is the empty list, and the function is cons, taking a number and a finite list, and returning a new finite list with the number at the head.

Read more about Initial Algebra: Final Coalgebra, Theorems, Example, Use in Computer Science

### Other articles related to "algebras, initial, initial algebra, algebra, initial algebras":

*F*-algebra

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**algebras**for a given endofunctor F has an

**initial**object, it is called an

**initial algebra**... The

**algebra**in the above example is an

**initial algebra**... such as lists and trees, can be obtained as

**initial algebras**of specific endofunctors ...

**Initial Algebra**- Use in Computer Science

... used in programming, such as lists and trees, can be obtained as

**initial algebras**of specific endofunctors ... While there may be several

**initial algebras**for a given endofunctor, they are unique up to isomorphism, which informally means that the "observable" properties of a data structure can be adequately ... into one function, they give , which makes this an F-

**algebra**for the endofunctor F sending to ...

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