Term algebra

Remarks

The term algebra is absolutely free over $X$. For every algebra $A$ of signature $\sigma$ and every map $\alpha \colon X \to A$, there is a unique homomorphism $\bar\alpha \colon \mathbf{T}_\sigma(X) \to A$ extending $\alpha$, namely $\bar\alpha(t) = t^A[\alpha]$, the value of $t$. When $X$ is empty, it is the algebra of closed terms, also called the Herbrand universe.

Example: let $\sigma$ have one binary symbol $\cdot$ and let $X = \set{x}$. The elements $x$, $x \cdot x$, $(x \cdot x) \cdot x$, $x \cdot (x \cdot x)$, … are pairwise distinct, and the operation sends $(s, t)$ to the term $s \cdot t$. This is the free magma on one generator.

Non-example: the positive integers with $\cdot$ read as addition are not isomorphic to this term algebra, even though they are generated by $1$. In the term algebra, $s \cdot t = s' \cdot t'$ implies $s = s'$ and $t = t'$, because terms have a unique reading. In the positive integers $1 + 2 = 2 + 1$ while $1 \ne 2$, so this property fails.