Let $\Sigma$ be a signature and $V$ a set of variables. Consider the set of all terms of $T:=T(\Sigma)$ over $V$ . Define the following:

• For each constant symbol $c\in \Sigma$ , $c^T$ is the element $c$ in $T$ .
• For each $n$ and each $n$ -ary function symbol $f\in \Sigma$ , $f^T$ is an $n$ -ary operation on $T$ given by $$f^T(t_1,\ldots,t_n)=f(t_1,\ldots,t_n),$$ meaning that the evaluation of $f^T$ at $(t_1,\ldots,t_n)$ is the term $f(t_1,\ldots, t_n)\in T$ .
• For each relational symbol $R\in \Sigma$ , $R^T=\varnothing$ .

Then $T$ , together with the set of constants and $n$ -ary operations defined above is an $\Sigma$ -structure. Since there are no relations defined on it, $T$ is an algebraic system whose signature $\Sigma'$ is the subset of $\Sigma$ consisting of all but the relation symbols of $\Sigma$ . The algebra $T$ is aptly called the term algebra of the signature $\Sigma$ (over $V$ ).

The prototypical example of a term algebra is the set of all well-formed formulas over a set $V$ of propositional variables in classical propositional logic. The signature $\Sigma$ is just the set of logical connectives. For each $n$ -ary logical connective $\#$ , there is an associated $n$ -ary operation $[\#]$ on $V$ , given by $[\#](p_1,\ldots, p_n)=\# p_1 \cdots p_n$ .

Remark. The term algebra $T$ of a signature $\Sigma$ over a set $V$ of variables can be thought of as a free structure in the following sense: if $A$ is any $\Sigma$ -structure, then any function $\phi:V\to A$ can be extended to a unique structure homomorphism $\phi':T\to A$ . In this regard, $V$ can be viewed as a free basis for the algebra $T$ . As such, $T$ is also called the absolutely free $\Sigma$ -structure with basis $V$ .