Primal algebra

Remarks

Example: the ring $(\Z_p, +, \cdot, -, 0, 1)$ for a prime $p$ is primal. Check: the polynomial $\delta(x) = 1 - x^{p - 1}$ equals $1$ at $0$ and $0$ elsewhere, by Fermat's little theorem. Then $f(a_1, \dots, a_n) = \sum_{b \in \Z_p^n} f(b) \prod_{i} \delta(a_i - b_i)$, and each $f(b)$ is a sum of copies of $1$. Non-example: the field $\F_4$ with its ring operations is not primal. Its Frobenius map $x \mapsto x^2$ is an automorphism, and term operations commute with automorphisms. A constant operation with value outside $\F_2$ does not commute with it, so it is not a term operation. Every primal algebra is quasi-primal.