Free Modal Pseudocomplemented De Morgan Algebras

Aldo V. Figallo; Nora Oliva; Alicia Ziliani

doi:10.18778/0138-0680.47.2.02

Authors

Aldo V. Figallo Aldo V. Figallo, Nora Oliva, Instituto de Ciencias Básicas, Universidad Nacional de San Juan, (5400) San Juan, Argentina
Nora Oliva
Alicia Ziliani Alicia Ziliani, Instituto de Matemática, Universidad Nacional del Sur, (8000) Bahía Blanca, Argentina

DOI:

https://doi.org/10.18778/0138-0680.47.2.02

Keywords:

Pseudocomplemented De Morgan algebras, congruences, free algebras

Abstract

Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ(∼x)* = (∼(xΛ(∼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18].

References

[1] R. Balbes, Ph. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, 1974. Zbl0321.06012, MR0373985.
Google Scholar

[2] G. Birkhoff, Lattice Theory, American Mathematical Society, Col Pub., 25 3rd ed., Providence, 1967. Zbl0153.02501, MR0227053.
Google Scholar

[3] V. Boicescu, A. Filipoiu, G. Georgescu, S. Rudeanu, Łukasiewicz-Moisil Algebras, North-Holland Publishing Co., Amsterdam, 1991. Zbl0726.06007, MR1112790.
Google Scholar

[4] S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics, Vol. 78, Springer-Verlag, Berlin, 1981. Zbl0478.08001, MR0648287.
Google Scholar

[5] A. V. Figallo, Tópicos sobre álgebras modales 4-valuadas, Proceeding of the IX Simposio Latino-Americano de Lógica Matemática, (Bahía Blanca, Argentina, 1992). Notas de Lógica Matemática 39(1992), pp. 145–157. MR1332541.
Google Scholar

[6] A. V. Figallo, P. Landini, Notes on 4-valued modal algebras, Preprints del Instituto de Ciencias Básicas, Universidad Nacional de San Juan 1(1990), pp. 28–37. Zbl0858.03062.
Google Scholar

[7] A. V. Figallo, N, Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 53, 1(2014), pp. 65–79. MR3331071.
Google Scholar

[8] J. Font, M. Rius, An abstract algebraic logic approach to tetravalent modal logics, Journal of Symbolic Logic 65(2000), pp. 481–518. Zbl1013.03075, MR1771068.
Google Scholar

[9] V. Glivenko, Sur quelques points de la logique de M. Brouwer, Academie Royale de Belgique. Bulletin de la Classe des Sciences 15 (1929), pp. 183–188. Zbl55.0030.05.
Google Scholar

[10] G. Grätzer, H. Lakser, The structure of pseudocomplemented distributive lattices II. Congruence extension and amalgamation, Transactions of the American Mathematical Society 156 (1971), pp. 343–358. Zbl0244.06011, MR0274359.
Google Scholar

[11] T. Hecht, T. Katriňák, Principal congruences of p-algebras and double p-algebras, Proceedings of the American Mathematical Society 58 (1976), pp. 25–31. Zbl0352.06006, MR0409293.
Google Scholar

[12] J. Kalman, Lattices with involution, Transactions of the American Mathematical Society 87 (1958), pp. 485–491. Zbl0228.06003, MR0095135.
Google Scholar

[13] I. Loureiro, Axiomatisation et propriétés des algèbres modales tétravalentes, Comptes Rendus Mathematique de l’Académie des Sciences Paris 295 (1982), Série I, pp. 555–557. Zbl0516.03010, MR0685023.
Google Scholar

[14] I. Loureiro, Algebras Modais Tetravalentes, PhD thesis, Faculdade de Ciências de Lisboa, Lisboa, Portugal, 1983.
Google Scholar

[15] A. Monteiro, La sémisimplicité des algèbres de Boole topologiques et les systémes déductifs, Revista de la Unión Matemática Argentina 25 (1975), pp. 417–448.
Google Scholar

[16] Yu. M. Movsisyan, V. A. Aslanyan, Boole–De Morgan Algebras and Quasi–De Morgan Functions, Communications in Algebra 42 (2014), pp. 4757-477, Zbl 1338.06010, MR3210411.
Google Scholar

[17] Yu. M. Movsisyan, V. A. Aslanyan, Super–De Morgan Functions and Free De Morgan Quasilattices, Central European Journal of Mathematics 12 (2014), pp. 1749–1761. Zbl 1346.08004, MR3232637.
Google Scholar

[18] Yu. M. Movsisyan, V. A. Aslanyan, De Morgan Functions and Free De Morgan Algebras, Demonstratio Mathematica 2 (2014), pp. 271–283.
Google Scholar

[19] P. Ribenboim, Characterization of the sup-complement in a distributive lattice with last element, Surma Brasil Mathematics 2 (1949), pp. 43–49. Zbl0040.01003, MR0030931.
Google Scholar

[20] A. Romanowska, Subdirectly irreducible pseudocomplemented De Morgan algebras, Algebra Universalis 12 (1981), pp. 70–75. Zbl0457.06009, MR0608649.
Google Scholar

[21] H. Sankappanavar, Pseudocomplemented Okham and Demorgan algebras, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 32 (1986), pp. 385–394. Zbl0612.06009, MR0860024.
Google Scholar

[22] H. Sankappanavar, Principal congruences of pseudocomplemented Demorgan algebras, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 33 (1987), pp. 3–11. Zbl0624.06016, MR0885477.
Google Scholar

[23] J. Varlet, Algèbres de Łukasiewicz trivalentes, Bulletin de la Société Royale des Sciences de Liège, (1968), pp. 9–10. Zbl0175.26604
Google Scholar