Paraconsistent and Paracomplete Logics Based on k-Cyclic Modal Pseudocomplemented De Morgan Algebras

Equivalence between varieties of square root rings and Boolean algebras with a distinguished automorphism

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 ?1 of the variety $${\mathcal{V}(L_{p,k})}$$ generated by L p,k into the variety $${\mathcal{V}(F(p^k))}$$ generated by F(p k ) and an interpretation ?2 of $${\mathcal{V}(F(p^k))}$$ into $${\mathcal{V}(L_{p,k})}$$ such that ?2?1(B) = B for every $${B \in \mathcal{V}(L_{p,k})}$$ and ?1?2(R) = R for every $${R \in \mathcal{V}(F(p^k))}$$ 

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, Grobner bases and algorithms programmed in Maple

https://www.jstor.org/stable/pdfplus/41475182.pdf

Resolution of Algebraic Systems of Equations in the Variety of 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 Grobner 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.

Gröbner Bases over Cyclic Post Algebras


 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 $p$.

Equivalence between Varieties of Łukasiewicz–Moisil Algebras and Rings

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L p,k, and the finite field F(p k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain 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 e V(L p,k) and Φ1Φ2(R) = R for every R e V(F(p k)).

M. del C. Vannicola

Papers

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