No abstract

In "An equivalence between Varieties of ciclic 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.

There is a constructive method to define a structure of simple k-cyclic Post algebra of order p, L p,k , on a given finite field F (p k ), and conversely. There exists an interpretation Φ1 of the variety V(L p,k ) generated by L p,k into the variety V(F (p k )) generated by F (p k ) and an interpretation Φ2 of V(F (p k )) into V(L p,k ) such that Φ2Φ1(B) = B for every B ∈ V(L p,k ) and Φ1Φ2(R) = R for every R ∈ V(F (p k )).In this paper we show how we can solve an algebraic system of equations over an arbitrary cyclic Post algebra of order p, p prime, using the above interpretation, Gröbner bases and algorithms programmed in Maple.

The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order $p$ and $p$-rings, $p$ prime and lift this result to the axled Łukasiewicz–Moisil algebra $L \cong B_s \times P$ and the ring $\prod ^s F_2 \times \prod ^l F_p$, where $B_s$ is a Boolean algebra of order $2^s$, $P$ a $p$-valued Post algebra of order $p^l$ and $F_p$ is the prime field of order $p$.