An algebra \(A= \langle A,\to,\neg\rangle\) is said to be a Wajsberg algebra if it satisfies the following equations: \[ \begin{aligned} (a\to b)\to b&= b \\ (a\to b)\to((b\to c)\to (a\to c))&= a\to a \\ (a\to b)\to b&= (b\to a)\to a \\ (\neg a\to \neg b)\to (b\to a)&= a\to a \end{aligned} \] Any finite subdirectly irreducible Wajsberg algebra is quasiprimal and semiprimal, but not each of them is primal. The author gives equational conditions under which the appropriate variety of Wajsberg algebras with additional constants is generated by a primal algebra. The same algebras (and only these algebras) without additional constants admit a lattice reduct which is a P-algebra (P-algebras are a generalization of Post algebras introduced by G. Epstein and A. Horn).