# Evaluation

An interpretation $v^\ast\colon T(\Sigma,\emptyset)\longrightarrow A$ only defined on the ground terms $t\in T(\Sigma)$ of a signature $\Sigma$ is called an evaluation. Since interpretations are $\Sigma$-algebra-morphisms, evaluations are $\Sigma$-algebra-morphisms as well. Furthermore, evaluations are uniquely determined, i.e. there exists exactly one mapping $e\colon T(\Sigma)\longrightarrow A$. This specific property has remarkable consequences. Consider for example a $\Sigma$-algebra-morphism $f\colon A\longrightarrow B$ between $\Sigma$-algebras $A$ and $B$. Then the equality $e_A=f\circ e_B$ holds for evaluations $e_A$ and $e_B$. In effect, each assignement can be extended to a functor between the term algebra $T(\Sigma)$ and $A$.
For reasons of simplicity, the application of the (uniquely determined) evaluation $e\colon T(\Sigma)\longrightarrow A$ to a term $t\in T(\Sigma)$ is often designated as $t^A := e(t)$.