Term operation

Remarks

The term operation of the variable $x_i$ is the projection $(a_1, \ldots, a_n) \mapsto a_i$. The term operation of $f(t_1, \ldots, t_k)$ is the composition $f^A(t_1^A, \ldots, t_k^A)$. So the set $\operatorname{Clo}(A)$ of all term operations is the smallest set of operations on $A$ that contains the projections and the basic operations and is closed under composition. Every homomorphism $h \colon A \to B$ satisfies $h(t^A(a_1, \ldots, a_n)) = t^B(h(a_1), \ldots, h(a_n))$.

Example: in the group $(\Z, +, -, 0)$ the map $(a, b) \mapsto 2a - b$ is the term operation of $(x_1 + x_1) + (-x_2)$.

Non-example: the map $a \mapsto a + 1$ on $(\Z, +, -, 0)$ is not a term operation. Every term operation $t^A$ of this group satisfies $t^A(0, \ldots, 0) = 0$, since $0 + 0 = -0 = 0$, while $0 + 1 \ne 0$.