An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field

Resolution of Algebraic Systems of Equations in the Variety of Cyclic Post Algebras

“…The varieties V(L p,k ) and V(F (p k )) are equivalent, that is, there exists an interpretation Φ 1 of V(L p,k ) in V(F (p k )) and an in- In the proof of the previous theorem [1] we give a general procedure, using Lagrange polynomials, to obtain explicitly the cyclic Post operations as terms in the language of fields, and conversely. In particular, the algebras L p,k and F (p k ) with constants F (p) are term equivalent.…”

“…In the following theorem [1] we obtain a structure of k-cyclic Post algebra of order p isomorphic to L p,k from the field F (p k ), and conversely.…”

“…In [1] we proved that there exists an interpretation Φ 1 of the variety V(L p,k ) generated by the k−cyclic simple Post algebra of order p, L p,k , in the variety V(F (p k )) generated by the field F (p k ); +, ·, F (p) and an interpretation Φ 2 of V(F (p k )) in V(L p,k ) such that Φ 2 Φ 1 (B) = B for every B ∈ V(L p,k ) and Φ 1 Φ 2 (R) = R for every R ∈ V(F (p k )).…”

“…We show how we can solve this equation using the interpretation described in [1] with the help of some algorithms programmed in Maple, and we also propose a solution when the polynomial does not satisfy the consistency condition. In that case we translate the equation f (x) = 0 and we obtain the postian polynomial f in the language of F (p k ); using a well known result of Galois theory, we consider it in an extension F (p t ) of F (p k ) where it is solvable.…”

“…In Section 3 we describe the constructive method to transform a field F (p k ); +, ·, F (p) into a k-cyclic Post algebra of order p, p prime, k ≥ 1, and conversely the field operations as terms in the language of cyclic Post algebras, and give the interpretations stated above and proved in [1]. Finally in Section 4 we explain how we can solve an algebraic system of equations over a cyclic Post algebra of order p, p prime and illustrate the whole process with some examples.…”