Gröbner Bases over Cyclic Post Algebras

In "An equivalence between Varieties of cyclic Post Algebras and Varieties generated by a finite field" we proved that the variety generated by the k-cyclic Post algebra of order p, and the variety generated by the finite field GF(p, k) are equivalents. In this paper we introduce the notion of Gröbner bases of an ideal in the first variety and show how to calculate it using the equivalence mentioned above. We give a division algorithm for p=2 and a theorem for calculating S-polynomials when k=1. We also show two different ways for solving algebraic systems of equations over the variety generated by the k-cyclic Post algebra of order p.