# initial algebra

**initial algebra** An algebra *A*, from some class of algebras ** C**, such that for every algebra

*B*in

**there is a unique homomorphism from**

*C**A*to

*B*. Such an algebra is said to be initial in the class

**or, more precisely, initial in the category that has all the algebras in**

*C***as its objects and all the homomorphisms between them as its morphisms. Depending on the choice of**

*C***, there may or may not exist initial algebras; however if any do exist they will all be isomorphic to each other. If**

*C***is the class**

*C**Alg*(Σ,

*E*) of all Σ-algebras satisfying a set

*E*of equations or conditional equations, then

**has an initial algebra. If the set**

*C**E*is recursive enumerable then the initial algebra is

*semicomputable*.

Initial algebras have importance for the semantics of programming languages, abstract data types, and algebraic specifications. Of particular significance is the fact that, in the class of all Σ-algebras for a given signature Σ, an initial algebra is given by the terms or trees over Σ; this is often called the

*term algebra*for Σ.

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