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Priestley duality for MV-algebras and beyond

Wesley Fussner, Mai Gehrke, Samuel J. van Gool ORCID logo and Vincenzo Marra
From the journal Forum Mathematicum


We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.

MSC 2010: 06D50; 06D35; 03G10

  1. Communicated by: Manfred Droste


[1] L. P. Belluce, A. Di Nola and A. Lettieri, Local MV-algebras, Rend. Circ. Mat. Palermo (2) 42 (1993), no. 3, 347–361. 10.1007/BF02844626Search in Google Scholar

[2] G. Birkhoff, Rings of sets, Duke Math. J. 3 (1937), no. 3, 443–454. 10.1007/978-1-4612-5373-0_23Search in Google Scholar

[3] W. J. Blok and D. Pigozzi, Algebraizable logics, Mem. Amer. Math. Soc. 77 (1989), 1–78. 10.1090/memo/0396Search in Google Scholar

[4] L. M. Cabrer and S. A. Celani, Priestley dualities for some lattice-ordered algebraic structures, including MTL, IMTL and MV-algebras, Cent. Eur. J. Math. 4 (2006), no. 4, 600–623. 10.2478/s11533-006-0025-6Search in Google Scholar

[5] R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Trends Log. Stud. Log. Libr. 7, Kluwer Academic, Dordrecht, 2000. 10.1007/978-94-015-9480-6Search in Google Scholar

[6] A. Di Nola and A. Lettieri, Perfect MV-algebras are categorically equivalent to abelian 𝑙-groups, Studia Logica 53 (1994), no. 3, 417–432. 10.1007/BF01057937Search in Google Scholar

[7] J. Fodor, Nilpotent minimum and related connectives for fuzzy logic, Proceedings of 1995 IEEE International Conference on Fuzzy Systems, IEEE Press, Piscataway (1995), 2077–2082. 10.1109/FUZZY.1995.409964Search in Google Scholar

[8] W. Fussner and P. Jipsen, Distributive laws in residuated binars, Algebra Universalis 80 (2019), no. 4, Paper No. 54. 10.1007/s00012-019-0625-1Search in Google Scholar

[9] W. Fussner and S. Ugolini, A topological approach to MTL-algebras, Algebra Universalis 80 (2019), no. 3, Paper No. 38. 10.1007/s00012-019-0612-6Search in Google Scholar

[10] N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Stud. Logic Found. Math. 151, Elsevier, Amsterdam, 2007. Search in Google Scholar

[11] M. Gehrke, Canonical extensions, Esakia spaces, and universal models, Leo Esakia on Duality in Modal and Intuitionistic Logics, Outst. Contrib. Log. 4, Springer, Dordrecht (2014), 9–41. 10.1007/978-94-017-8860-1_2Search in Google Scholar

[12] M. Gehrke, T. Jakl and L. Reggio, A duality theoretic view on limits of finite structures, Foundations of Software Science and Computation Structures, Lecture Notes in Comput. Sci. 12077, Springer, Cham (2020), 299–318. 10.1007/978-3-030-45231-5_16Search in Google Scholar

[13] M. Gehrke and B. Jónsson, Bounded distributive lattices with operators, Math. Japon. 40 (1994), no. 2, 207–215. Search in Google Scholar

[14] M. Gehrke and B. Jónsson, Bounded distributive lattice expansions, Math. Scand. 94 (2004), no. 1, 13–45. 10.7146/math.scand.a-14428Search in Google Scholar

[15] M. Gehrke and H. A. Priestley, Non-canonicity of MV-algebras, Houston J. Math. 28 (2002), no. 3, 449–455. Search in Google Scholar

[16] M. Gehrke and H. A. Priestley, Canonical extensions of double quasioperator algebras: An algebraic perspective on duality for certain algebras with binary operations, J. Pure Appl. Algebra 209 (2007), no. 1, 269–290. 10.1016/j.jpaa.2006.06.001Search in Google Scholar

[17] M. Gehrke and H. A. Priestley, Duality for double quasioperator algebras via their canonical extensions, Studia Logica 86 (2007), no. 1, 31–68. 10.1007/s11225-007-9045-xSearch in Google Scholar

[18] M. Gehrke, S. J. van Gool and V. Marra, Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality, J. Algebra 417 (2014), 290–332. 10.1016/j.jalgebra.2014.06.031Search in Google Scholar

[19] R. Goldblatt, Varieties of complex algebras, Ann. Pure Appl. Logic 44 (1989), no. 3, 173–242. 10.1016/0168-0072(89)90032-8Search in Google Scholar

[20] S. Jenei, On the structure of rotation-invariant semigroups, Arch. Math. Logic 42 (2003), no. 5, 489–514. 10.1007/s00153-002-0165-8Search in Google Scholar

[21] M. Kolařík, Independence of the axiomatic system for MV-algebras, Math. Slovaca 63 (2013), 1–4. 10.2478/s12175-012-0076-zSearch in Google Scholar

[22] R. McNaughton, A theorem about infinite-valued sentential logic, J. Symbolic Logic 16 (1951), 1–13. 10.2307/2268660Search in Google Scholar

[23] D. Mundici, Interpretation of AF C-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), no. 1, 15–63. 10.1016/0022-1236(86)90015-7Search in Google Scholar

[24] D. Mundici, Advanced Łukasiewicz Calculus and MV-Algebras, Trends Log. Stud. Log. Libr. 35, Springer, Dordrecht, 2011. 10.1007/978-94-007-0840-2Search in Google Scholar

[25] C. Noguera, F. Esteva and J. Gispert, Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops, Arch. Math. Logic 44 (2005), no. 7, 869–886. 10.1007/s00153-005-0276-0Search in Google Scholar

[26] H. A. Priestley, Representation of distributive lattices by means of ordered stone spaces, Bull. Lond. Math. Soc. 2 (1970), 186–190. 10.1112/blms/2.2.186Search in Google Scholar

[27] H. A. Priestley, Ordered topological spaces and the representation of distributive lattices, Proc. Lond. Math. Soc. (3) 24 (1972), 507–530. 10.1112/plms/s3-24.3.507Search in Google Scholar

[28] H. A. Priestley, Intrinsic spectral topologies, Papers on General Topology and Applications (Flushing 1992), Ann. New York Acad. Sci. 728, New York Academy of Science, New York (1994), 78–95. 10.1111/j.1749-6632.1994.tb44135.xSearch in Google Scholar

[29] S. Ugolini, Varieties of residuated lattices with an MV-retract and an investigation into state theory, Ph.D. Thesis, University of Pisa, 2018. Search in Google Scholar

Received: 2020-05-08
Revised: 2020-10-20
Published Online: 2021-05-12
Published in Print: 2021-07-01

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