Galois connections and modal algebras

Hércules de Araújo Feitosa; Marcelo Reicher Soares; Romulo Albano de Freitas

doi:10.23925/2316-5278.2024v25i1:e67779

Authors

Hércules de Araújo Feitosa Universidade Estadual Paulista "Júlio de Mesquita Filho"
Marcelo Reicher Soares Universidade Estadual Paulista "Júlio de Mesquita Filho"
Romulo Albano de Freitas Universidade Estadual Paulista "Júlio de Mesquita Filho"

DOI:

https://doi.org/10.23925/2316-5278.2024v25i1:e67779

Keywords:

Algebraic logic, Modal algebra, Galois connections, Non-distributive lattices

Abstract

We investigate the properties of a modal algebra, more specifically, a non-distributive lattice with operators via Galois connections. Pairs of Galois are very common in mathematical environments, and, in this article, they appear as unary operators in lattices even without the distributivity. In a previous paper, Castiglioni and Ertola-Biraben studied the meet-complemented lattices with two modal operators for necessary □ and possible ◊. We observed that this pair of operators determines an adjunction. Then, we used Galois pairs on the meet-complemented lattices, showing some properties of this structure that were already been proved in their paper, and some new laws non presented. Lastly, we define a new pair of operators that also constitute another Galois pair.

Metrics

Metrics Loading ...

Author Biographies

Hércules de Araújo Feitosa, Universidade Estadual Paulista "Júlio de Mesquita Filho"

Graduated in Mathematics from Fundação Educacional de Bauru (1984), Master in
Fundamentals of Mathematics from Universidade Estadual Paulista - UNESP - IGCE (1992)
and PhD in Logic and Philosophy of Science from Universidade Estadual de Campinas -
UNICAMP - IFCH (1998). Since 1988 he has been a professor at UNESP, Faculty of Sciences,
Department of Mathematics, Bauru Campus. He is currently an associate professor (livre
docente) and is accredited in the Postgraduate Program in Philosophy at UNESP - FFC -
Marília. His academic experience has an emphasis on teaching Logic and Fundamentals of
Mathematics and his scientific investigations are focused on logic, translations between logics,
algebraic models, quantifiers and non-classical logics.

Marcelo Reicher Soares, Universidade Estadual Paulista "Júlio de Mesquita Filho"

Post-Doctorate at the Centro de Lógica, Epistemologia e História da Ciência CLE-UNICAMP (2015), PhD in Mathematics from the Universidade de São Paulo - USP (2000), Master in Mathematics from the Universidade de São Paulo - USP (1989) and holds a Degree in Mathematics from Universidade São Francisco (1983). He is currently an Assistant Professor at the Universidade Estadual Paulista Júlio de Mesquita Filho - UNESP and works as a professor and advisor in the Postgraduate Program in Mathematics in the PROFMAT National Network. He has experience in teaching and research in the area of Mathematical Analysis, with an emphasis on Generalized Colombeau Functions. He currently works in Fundamentals and Mathematical Logic with an emphasis on Non-Standard Analysis and Algebraic Logic. Participates in the Research Groups, certified by CNPQ, "Sistemas Adaptativos, Lógica e Computação Inteligente" and " Lógica e Epistemologia ".

Romulo Albano de Freitas, Universidade Estadual Paulista "Júlio de Mesquita Filho"

Graduated in Mathematics from Universidade Estadual Paulista "Júlio de Mesquita Filho" -
Unesp, Bauru campus. He is a member of the research group, certified by CNPQ, "Sistemas
Adaptativos, Lógica e Computação Inteligente" (SALCI). He has experience in teaching and
research in Logic. Currently studying for a master's degree in the Postgraduate Program in
Philosophy at FFC - Unesp Marília, with an emphasis on Logic. Has interest in algebraic
developments for logics/algebraic logic, non-classical logics and proof theory.

References

BIRKHOFF, G. Lattice theory. 2. ed. Providence: American Mathematical Society, 1948.

CASTIGLIONI, J. L.; ERTOLA-BIRABEN, R. Modal operators for meet-complemented lattices. Logic Journal of the IGPL, v. 25, n. 4, p. 465-495, 2017. https://doi.org/10.1093/jigpal/jzx011.

DUNN, J. M.; HARDEGREE, G. M. Algebraic methods in philosophical logic. Oxford: Oxford University Press, 2001.

HERRLICH, H.; HUSEK, M. Galois connections categorically. Journal of Pure and Applied Algebra, v. 68, n. 1-2, p. 165-180, 1990. https://doi.org/10.1016/0022-4049(90)90141-4.

MIRAGLIA, F. Cálculo proposicional: uma interação da álgebra e da lógica. Campinas: UNICAMP/CLE, 1987. (Coleção CLE, v. 1).

ORE, O. Galois connexions. Transactions of the American Mathematical Society, v. 55, n. 3, p. 493-513, 1944. https://doi.org/10.2307/1990305.

ORLOWSKA, E.; REWITZKY, I. Algebras for Galois-style connections and their discrete duality. Fuzzy Sets and Systems, v. 161, n. 9, p. 1325-1342, 2010. https://doi.org/10.1016/j.fss.2009.12.013.

RASIOWA, H. An algebraic approach to non-classical logics. Warszawa: PWN - Polish Scientific Publishers, 1974.

RASIOWA, H.; SIKORSKI, R. The mathematics of metamathematics. 2. ed. Warszawa: PWN – Polish Scientific Publishers, 1968.

SMITH, P. The Galois connection between syntax and semantics. Technical report. Cambridge: University of Cambridge, 2010. Disponível em: https://www.logicmatters.net/resources/pdfs/Galois.pdf. Acesso em: 19 aug. 2024.