Polynomial operation

Remarks

Equivalently, the polynomial operations of $A$ are the term operations of the algebra $A_A$ obtained from $A$ by adding a constant symbol for each element of $A$. Every term operation is a polynomial operation, with $m = 0$. An equivalence relation on $A$ is a congruence exactly when it is preserved by every unary polynomial operation of $A$. For a commutative ring $R$, the polynomial operations of $R$ are the polynomial functions $R^n \to R$ given by polynomials with coefficients in $R$.

Example: in the group $(\Z, +, -, 0)$, the map $p(a) = 2a + 5$ is a polynomial operation. Take $t = (x_1 + x_1) + x_2$ and $c_1 = 5$; then $t^A(a, 5) = 2a + 5$.

Non-example: on $(\Z, +, -, 0)$ the map $a \mapsto a^2$ is not a polynomial operation. By induction on terms, every unary polynomial operation of this group has the form $a \mapsto k a + c$ with $k, c \in \Z$. The map $a \mapsto a^2$ is not of this form, since its values $0, 1, 4$ at $0, 1, 2$ are not in arithmetic progression.