Abstract
It is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables. In this paper, we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem suggested by B. I. Plotkin.
References
[1] O. Arieli, A. Avron, The value of four values, Artificial Intelligence 102 (1998), 97-141.10.1016/S0004-3702(98)00032-0Search in Google Scholar
[2] R. Balbes, P. Dwinger, Distributive lattices, Univ. of Missouri Press, 1974.Search in Google Scholar
[3] N. D. Belnap, A useful for valued logic, in: G. Epstein, J. M. Dunn (Eds.), Modern Uses of Multiple-Valued Logic, Reidel Publishing Company, Boston, 1977, 7-73.10.1007/978-94-010-1161-7_2Search in Google Scholar
[4] J. Berman, W. Blok, Stipulations, multi-valued logic and De Morgan algebras, Multi- Valued Logic 7(5-6) (2001), 391-416.Search in Google Scholar
[5] A. Białynicki-Birula, H. Rasiowa, On the representation of quasi-Boolean algebras, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys. 5 (1957), 259-261.Search in Google Scholar
[6] G. Birkhoff, Lattice Theory, 3rd Ed., American Mathematical Society, Providence, Rhode Island, 1967. [7] F. Bou, U. Rivieccio, The logic of distributive bilattices, Log. J. IGPL 19 (2011), 183-216.10.1093/jigpal/jzq041Search in Google Scholar
[8] J. A. Brzozowski, A characterization of De Morgan algebras, Internat. J. Algebra Comput. 11 (2001), 525-527.10.1142/S0218196701000681Search in Google Scholar
[9] J. A. Brzozowski, De Morgan bisemilattices, Proceedings of the 30th IEEE International Symposium on Multiple-Valued Logic, (ISMVL 2000), May 23-25, (2000), p. 173.Search in Google Scholar
[10] J. A. Brzozowski, Partially ordered structures for hazard detection, Special Session: The Many Lives of Lattice Theory, Joint Mathematics Meetings, San Diego, CA, January 6-9, (2002).Search in Google Scholar
[11] Y. Crama, P. L. Hammer, Boolean Functions: Theory, Algorithms, and Applications, Cambridge University Press, New York, 2011.10.1017/CBO9780511852008Search in Google Scholar
[12] R. Dedekind, Uber Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler, Festschrift der Techn. Hochsch. Braunschwig bei Gelegenheit der 69, Versammlung deutscher Naturforscher und Arzte, (1897), 1-40.10.1007/978-3-663-07224-9_1Search in Google Scholar
[13] Z. Ésik, Free De Morgan Bisemigroups and Bisemilattices, Algebra Colloquium, Volume 10, Issue 1, June (2003), 23-32.10.1007/s100110300004Search in Google Scholar
[14] M. Gehrke, C. Walker, E. Walker, A mathematical setting for fuzzy logics, Internat. J.Uncertain. Fuzziness Knowledge-Based Systems 5(3) (1997), 223-238.10.1142/S021848859700021XSearch in Google Scholar
[15] M. Gehrke, C. Walker, E. Walker, Some comments on fuzzy normal forms, Proceedings of the ninth IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2000, vol. 2, 593-598.Search in Google Scholar
[16] G. Gratzer, Lattice Theory: Foundation, Springer, Basel AG, 2011.10.1007/978-3-0348-0018-1Search in Google Scholar
[17] G. Gratzer, Universal Algebra, Springer-Verlag, 2010.Search in Google Scholar
[18] J. A. Kalman, Lattices with involution, Trans. Amer. Math. Soc. 87 (1958), 485-491.10.1090/S0002-9947-1958-0095135-XSearch in Google Scholar
[19] L. H. Kauffman, De Morgan Algebras - completeness and recursion, Proceedings of the eighth International Symposium on Multiple-Valued Logic, IEEE Computer Society Press Los Alamitos, CA, USA, (1978), 82-86.Search in Google Scholar
[20] M. Kondo, Characterization theorem of 4-valued De Morgan logic, Mem. Fac. Sci. Eng.Shimane Univ. Ser. B Math. Sci. 31 (1998), 73-80.Search in Google Scholar
[21] A. D. Korshunov, Monotone Boolean functions, Uspekhi Mat. Nauk 58:5(353) (2003), 89-162. English translation in: Russian Math. Surveys 58(5) (2003), 929-1001.Search in Google Scholar
[22] A. A. Markov, Constructive logic, Uspekhi Mat. Nauk 5 (1950), 187-188, (in Russian).Search in Google Scholar
[23] B. Mobasher, D. Pigozzi, G. Slutzki, Multi-valued logic programming semantics, An algebraic approach, Theoret. Comput. Sci. 171 (1997), 77-109.10.1016/S0304-3975(96)00126-0Search in Google Scholar
[24] G. C. Moisil, Recherches sur l’algebre de la logique, Annales scientifiques de l’Universite de Jassy 22 (1935), 1-117.Search in Google Scholar
[25] Yu. M. Movsisyan, Introduction to the theory of algebras with hyperidentities, Yerevan State University Press, Yerevan, 1986, (in Russian).Search in Google Scholar
[26] Yu. M. Movsisyan, Hyperidentities and hypervarieties in algebras, Yerevan State University Press, Yerevan, 1990, (in Russian).Search in Google Scholar
[27] Yu. M. Movsisyan, Hyperidentities of Boolean algebras, Izv. Ross. Akad. Nauk Ser.Mat. 56 (1992), 654-672. English translation in: Russian Acad. Sci. Izv. Math. 40 (1993), 607-622.Search in Google Scholar
[28] Yu. M. Movsisyan, Algebras with hyperidentities of the variety of Boolean algebras, Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996), 127-168. English translation in: Russian Acad. Sci. Izv. Math. 60 (1996), 1219-1260. Search in Google Scholar
[29] Yu. M. Movsisyan, Hyperidentities in algebras and varieties, Uspekhi Mat. Nauk 53:1(319) (1998), 61-114. English translation in: Russian Math. Surveys 53(1) (1998), 57-108.Search in Google Scholar
[30] Yu. M. Movsisyan, Hyperidentities and hypervarieties, Sci. Math. Jpn. 54 (2001), 595-640.Search in Google Scholar
[31] Yu. M. Movsisyan, Binary representations of algebras with at most two binary operations.A Cayley theorem for distributive lattices, Internat. J. Algebra Comput. 19(1) (2009), 97-106.10.1142/S0218196709004993Search in Google Scholar
[32] Yu. M. Movsisyan, V. A. Aslanyan, Hyperidentities of De Morgan algebras, Log. J.IGPL 20 (2012), 1153-1174. doi:10.1093/jigpal/jzr05310.1093/jigpal/jzr053Search in Google Scholar
[33] B. I. Plotkin, Universal Algebra, Algebraic Logic, and Databases, Kluwer Academic Publisher, 1994.10.1007/978-94-011-0820-1Search in Google Scholar
[34] H. P. Sankappanavar, A characterization of principal congruences of De Morgan algebras and its applications, Math. Logic in Latin America, Proc. IV Latin Amer.Symp. Math. Logic, Santiago, (1978), 341-349. North-Holland Pub. Co., Amsterdam, 1980.Search in Google Scholar
[35] J. D. H. Smith, A. B. Romanowska, Post-Modern Algebra, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1999.Search in Google Scholar
[36] E. Sperner, Ein Satz uber Untermengen einer endlichen Menge, Math. Z. 27 (1928), 544-548. 10.1007/BF01171114Search in Google Scholar
© by Yu. M. Movsisyan
This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.