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 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 ε V(L p,k) and Φ1Φ2(R) = R for every R ε V(F(p k)).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abad: “Cyclic Post algebras of order n”, An. Acad. Brasil. Ciênc., Vol. 53(2), (1981), pp. 243–246.
R. Balbes and P. Dwinger: Distributive Lattices, University of Missouri Press, Columbia, MO., 1974.
V. Boicescu, A. Filipoiu, G. Georgescu and S. Rudeanu: Lukasiewicz-Moisil Algebras, Annals of Discrete Mathematics, Vol. 49, North-Holland, Amsterdam, 1991.
S. Burris and H. Sankappanavar: A Course in Universal Algebra, Graduate Texts in Mathematics, Vol 78, Springer, Berlin, 1981.
H. Cendra: “Cyclic Boolean algebras and Galois fields F(2k)”, Portugal. Math., Vol. 39(1–4), (1980), pp. 435–440.
G. Epstein: “The lattice theory of Post algebras”, Trans. Amer. Math. Soc., Vol. 95, (1960), pp. 300–317.
K. Kaarly and A.F. Pixley: Polynomial Completeness in Algebraic Systems, Chapman and Hall, Boca Raton, 2001.
S. Lang: Algebra, Addison-Wesley Publishing Company, CA., 1984.
R. Lewin: “Interpretability into Łukasiewicz algebras”, Rev. Un. Mat. Argentina, Vol. 41(3), (1999), pp. 81–98.
R. McKenzie, G. McNulty and W. Taylor: Algebras, Lattices, Varieties, Vol. I, Wadsworth and Brooks, Monterey, CA, 1987.
A. Monteiro: “Algèbres de Boole cycliques”, Rev. Roumaine de Mathématiques Pures Appl., Vol. 23(1), (1978), pp. 71–76.
G. Moisil: Algebra schemelor cu elemente ventil, Seria St. nat. 4–5, Revista Universitatii C.I. Parhon, Bucharest, 1954, pp. 9–15.
G. Moisil: “Algèbres universelles et automates”, In: Essais sur les Logiques non Chrysippiennes, Editions de L’Academie de la Republique Socialiste de Roumanie, Bucharest, 1972.
S. Rudeanu: Boolean functions and equations, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974.
M. Serfati: “Introduction aux Algèbres de Post et à leurs applications (logiques à r valeurs-équations postiennes-graphoïdes orientés)”, Cahiers du Bureau Universitaire de Recherche Opérationnelle Université Paris VI. Série Recherche, Vol. 21, (1973), pp. 35–42.
Author information
Authors and Affiliations
About this article
Cite this article
Abad, M., Díaz Varela, J.P., López Martinolich, B.F. et al. An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field. centr.eur.j.math. 4, 547–561 (2006). https://doi.org/10.2478/s11533-006-0023-8
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.2478/s11533-006-0023-8