DOI: 10.2478/s11533-006-0023-8 Research article CEJM 4(4) 2006 547–561 An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field M. Abad 1∗ , J.P. D´ ıaz Varela 1 , B.F. L´opez Martinolich 2 , M. del C. Vannicola 2 , M. Zander 1 1 Departamento de Matem´atica, Universidad Nacional del Sur, 8000 Bah´ ıa Blanca, Argentina 2 Departamento de Matem´atica, Universidad Nacional del Comahue, 8300 Neuqu´ en, Argentina Received 18 April 2005; accepted 4 June 2006 Abstract: 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 Φ 1 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 Φ 2 of V(F (p k )) into 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 )). c  Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Varieties, Interpretations, equivalence, finite fields, post algebras MSC (2000): 06D25, 12E20, 03G25 1 Introduction The original motivation behind the research presented in this paper came from the works of Moisil [12, 13] and Cendra [5]. In those papers, Moisil established relationships between some finite fields F (2 k ) and some cyclic Boolean algebras, but it was Cendra in [5] who solved the problem of giving a constructive method to define a simple k-cyclic Boolean algebra 〈A; T 〉 on a given finite field F (2 k ), k ≥ 1, and conversely. In this paper we solve a similar problem for any finite field 〈F (p k ); +, ·,F (p)〉 and k-cyclic Post algebras of order p, p prime, k ≥ 1. In fact, we describe a constructive method to transform * E-mail: imabad@criba.edu.ar