Equivalence between varieties of Post algebras with a distinguished automorphism and pth root rings

In this paper we investigate the variety of Post algebras of order
$p$
with a distinguished homomorphism. We prove that this variety is generated by its finite members. Then we study the variety of
$p\text{th}$
root rings, that is commutative rings with identity, of characteristic
$p$
, with a
$p\text{th}$
root as an additional operation. We prove that this variety is generated by the finite Galois fields and we establish an equivalence between the variety of Post algebras of order
$p$
, with a distinguished homomorphism and the variety of pth root rings.