Absolutely free algebra

Remarks

This is a free algebra in the variety of all algebras of signature $\sigma$, the variety defined by the empty set of identities. The term algebra $T(X)$, whose elements are terms over $X$, is an absolutely free algebra on $X$. Every absolutely free algebra on $X$ is isomorphic to it. Let $F$ be generated by $\eta(X)$. Then $F$ is absolutely free on $X$ exactly when $\eta$ and each basic operation are injective, and the images of different operations are disjoint and do not meet $\eta(X)$. Example: for one unary symbol $s$ and $X = \set{x}$, the algebra $\N$ with $s(n) = n + 1$ and $\eta(x) = 0$ is absolutely free; the extension of $f$ sends $n$ to $s^n(f(x))$. Non-example: $\Z$ with $s(n) = n + 1$ and $\eta(x) = 0$ is not absolutely free on $\set{x}$, since it is not generated by $0$.