Abad, M.; Díaz Varela, J. P.; López Martinolich, B. F.; del C. Vannicola, M.; Zander, M. An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field. (English) Zbl 1117.06007 Cent. Eur. J. Math. 4, No. 4, 547-561 (2006). The authors give a term equivalence between the simply \(k\)-cyclic Post algebra of order \(p\), \( L_{p, k}\), and the finite field \(F(p^k)\) with constants \(F(p)\). By using Lagrange polynomials they give an explicit procedure to obtain an interpretation \(\Phi_1\) of the variety \(\nu (L_{p, k})\) generated by \(L_{p, k}\) into the variety \(\nu(F(p^k))\) generated by \(F(p^k)\) and an interpretation \(\Phi_2\) of \(\nu(F(p^k))\) into \(\nu (L_{p, k})\) such that \(\Phi_2 \Phi_1 (B) = B\) for every \(B \in \nu(L_{p, k})\) and \(\Phi_1 \Phi_2 (R) = R\) for every \(R \in \nu(F(p^k))\). Reviewer: Renata Majovská (Horní Suchá) Cited in 3 Documents MSC: 06D25 Post algebras (lattice-theoretic aspects) 12E20 Finite fields (field-theoretic aspects) Keywords:Varieties; Interpretations; Equivalence; Finite fields; Post algebra PDFBibTeX XMLCite \textit{M. Abad} et al., Cent. Eur. J. Math. 4, No. 4, 547--561 (2006; Zbl 1117.06007) Full Text: DOI References: [1] M. Abad: “Cyclic Post algebras of order n”, An. Acad. Brasil. Ciênc., Vol. 53(2), (1981), pp. 243-246.; · Zbl 0473.06009 [2] R. Balbes and P. Dwinger: Distributive Lattices, University of Missouri Press, Columbia, MO., 1974.; · Zbl 0321.06012 [3] V. Boicescu, A. Filipoiu, G. Georgescu and S. Rudeanu: Lukasiewicz-Moisil Algebras, Annals of Discrete Mathematics, Vol. 49, North-Holland, Amsterdam, 1991.; · Zbl 0726.06007 [4] S. Burris and H. Sankappanavar: A Course in Universal Algebra, Graduate Texts in Mathematics, Vol 78, Springer, Berlin, 1981.; · Zbl 0478.08001 [5] H. Cendra: “Cyclic Boolean algebras and Galois fields F(2k)”, Portugal. Math., Vol. 39(1-4), (1980), pp. 435-440.; · Zbl 0534.06005 [6] G. Epstein: “The lattice theory of Post algebras”, Trans. Amer. Math. Soc., Vol. 95, (1960), pp. 300-317. http://dx.doi.org/10.2307/1993293; · Zbl 0207.29403 [7] K. Kaarly and A.F. Pixley: Polynomial Completeness in Algebraic Systems, Chapman and Hall, Boca Raton, 2001.; [8] S. Lang: Algebra, Addison-Wesley Publishing Company, CA., 1984.; · Zbl 0712.00001 [9] R. Lewin: “Interpretability into Łukasiewicz algebras”, Rev. Un. Mat. Argentina, Vol. 41(3), (1999), pp. 81-98.; · Zbl 0957.06012 [10] R. McKenzie, G. McNulty and W. Taylor: Algebras, Lattices, Varieties, Vol. I, Wadsworth and Brooks, Monterey, CA, 1987.; · Zbl 0611.08001 [11] A. Monteiro: “Algèbres de Boole cycliques”, Rev. Roumaine de Mathématiques Pures Appl., Vol. 23(1), (1978), pp. 71-76.; · Zbl 0393.06007 [12] G. Moisil: Algebra schemelor cu elemente ventil, Seria St. nat. 4-5, Revista Universitatii C.I. Parhon, Bucharest, 1954, pp. 9-15.; [13] 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.; [14] S. Rudeanu: Boolean functions and equations, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974.; · Zbl 0321.06013 [15] 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.; This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.