Journal of Mahani Mathematical Research

Journal of Mahani Mathematical Research

Discrete dualities for Monteiro's tetravalent modal algebras

Document Type : Research Paper

Author
Gustavo Pelaitay
Instituto de Ciencias Basicas, Universidad Nacional de San Juan, Avda. Ignacio de la Roza 230 (O), 5400 San Juan, Argentina
10.22103/jmmr.2025.24322.1717
Abstract
Discrete duality refers to a form of duality in which a class of abstract relational systems serves as the dual counterpart to a class of algebras. These relational systems are called \textit{frames}, following the terminology of non-classical logics. No topology is involved in the construction of these frames; hence, they can be considered as having a discrete topology. In 1978, A. Monteiro introduced a class of algebras known as tetravalent modal algebras, which constitute a generalization of the three-valued {\L}ukasiewicz algebras defined by Moisil. The theory of these tetravalent modal algebras was initially developed by I. Loureiro, followed by significant contributions from A. V. Figallo, later enriched by the work of J. Font and M. Rius, and more recently by the works of M. Coniglio and M. Figallo. In this paper, we present two discrete dualities for Monteiro's tetravalent modal algebras. Each of these dualities involves a distinct class of frames and a unique definition of a complex algebra.
Keywords
De Morgan algebras
frames
discrete duality
tetravalent modal algebras
Subjects
[1] Boicescu, V., Filipoiu, A., Georgescu, G., & Rudeanu, S. (1991).  Lukasiewicz - Moisil Algebras. Annals of Discrete Mathematics, 49. North-Holland.
[2] Celani, S. A. (1997). A note on homomorphisms of tetravalent modal algebras. (Spanish). Boletim da Sociedade Paranaense de Matematica, 17(2), 65{70. [In Spanish]
[3] Celani, S. A. (2011). Classical modal De Morgan algebras. Studia Logica, 98(2), 251{266. https://doi.org/10.1007/s11225-011-9328-0
[4] Coniglio, M. E., & Figallo, M. (2014). Hilbert-style presentations of two logics associated to tetravalent modal algebras. Studia Logica, 102(3), 525{539. https://doi.org/10.1007/s11225-013-9489-0
[5] Coniglio, M. E., & Figallo, M. (2024). Normal proofs and tableaux for the Font-Rius tetravalent modal logic. Logic and Logical Philosophy, 33(1), 171{203. https://doi.org/10.12775/LLP.2024.006
[6] Dzik, W., Or lowska, E., & van Alten, C. (2006). Relational representation theorems for general lattices with negations. In Relations and Kleene Algebra in Computer Science (pp. 162{176). Lecture Notes in Computer Science, 4136. Springer, Berlin. https://doi.org/10.1007/11828563_11
[7] Figallo, A. V. (1989). Notes on generalized N-lattices. Revista de la Union Matematica Argentina, 35, 61{65.
[8] Figallo, A., & Ziliani, A. (1991). Symmetric tetra-valued modal algebras. IX Latin American School of Mathematics: Algebra (Spanish). Notas de la Sociedad Matematica de Chile, 10, 133{141. [In Spanish]
[9] Figallo, A. V. (1992). On the congruences in four-valued modal algebras. Portugaliae Mathematica, 49, 249{261.
[10] Figallo, A. V., & Landini, P. (1995). On generalized I-algebras and 4-valued modal algebras. Reports on Mathematical Logic, 29, 3{18.
[11] Figallo, A. V., & Landini, P. (2014). Several characterizations of the 4-valued modal algebras. Annals of the University of Craiova, Mathematics and Computer Science Series. https://doi.org/10.52846/ami.v41i2.560
[12] Figallo, A. V., & Pelaitay, G. (2018). Localization of tetravalent modal algebras. Asian-European Journal of Mathematics, 11(4), 1850067. https://doi.org/10.1142/S1793557118500675
[13] Figallo{Orellano, A., & Pascual, I. (2019). On monadic operators over modal pseu-docomplemented De Morgan algebras and tetravalent modal algebras. Studia Logica, 107(3), 591{611. https://doi.org/10.1007/s11225-018-9802-z
[14] Font, J. M., & Rius, M. (2000). An abstract algebraic logic approach to tetravalent modal logics. Journal of Symbolic Logic, 65(2), 481{518. https://doi.org/10.2307/2586552
[15] Gomez, C., Marcos, M., & San Martn, H. J. (2021). On the relation of negations in Nelson algebras. Reports on Mathematical Logic, 56, 15{56. https://doi.org/10.4467/20842589RM.21.002.14374
[16] Jonsson, B., & Tarski, A. (1951). Boolean algebras with operators I. American Journal of Mathematics, 73(4), 891{939. https://doi.org/10.2307/2372074
[17] Loureiro, I. (1982). Axiomatisation et proprietes des algebres modales tetravalentes. Comptes Rendus de l'Academie des Sciences de Paris, Serie I, Mathematiques, 295, 555{557. [In French]
[18] Loureiro, I. (1983). Prime Spectrum of a Tetravalent Modal Algebra. Notre Dame Journal of Formal Logic, 24(3), 255{259. https://doi.org/10.1305/ndjfl/1093870382
[19] Loureiro, I. (1984). Finite Tetravalent Modal Algebras. Revista de la Union Matematica Argentina, 31(4), 187{191.
[20] Monteiro, L. (1963). Axiomes independants pour les algebres de  Lukasiewicz trivalentes. Bulletin de la Societe des Sciences Mathematiques et Physiques de la R. P. Roumanie, Nouvelle Serie, 7, 199{202.
[21] Or lowska, E., & Rewitzky, I. (2005). Duality via Truth: Semantic frameworks for lattice-based logics. Logic Journal of the IGPL, 13(4), 467{490. https://doi.org/10.1093/jigpal/jzi035
[22] Or lowska, E., & Rewitzky, I. (2007). Discrete duality and its applications to reasoning with incomplete information. In M. Kryszkiewicz, J. F. Peters, H. Rybinski, & A. Skowron (Eds.), Rough Sets and Intelligent Systems Paradigms (pp. 51{56). Lecture Notes in Arti cial Intelligence, 4585. Springer, Heidelberg. https://doi.org/10.1007/978-3-540-73451-2_7
[23] Pelaitay, G. (2017). Discrete duality for 3-valued  Lukasiewicz{Moisil algebras. Asian-European Journal of Mathematics, 10(1), 1750003. https://doi.org/10.1142/S1793557117500036
[24] Priestley, H. A. (1970). Representation of distributive lattices by means of ordered Stone spaces. Bulletin of the London Mathematical Society, 2(2), 186{190. https://doi.org/10.1112/blms/2.2.186

  • Receive Date 04 November 2024
  • Revise Date 21 December 2024
  • Accept Date 29 January 2025