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A topological approach to MTL-algebras

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Abstract

We give a dualized construction of Aguzzoli–Flaminio–Ugolini of a large class of MTL-algebras from quadruples \((\mathbf{B},\mathbf{A},\vee _e,\delta )\), consisting of a Boolean algebra \(\mathbf{B}\), a generalized MTL-algebra \(\mathbf{A}\), and maps \(\vee _e\) and \(\delta \) parameterizing the connection between these two constituent pieces. Our dualized construction gives a uniform way of building the extended Priestley spaces of MTL-algebras in this class from the Stone spaces of their Boolean skeletons, the extended Priestley spaces of their radicals, and a family of maps connecting the two. In order to make this dualized construction possible, we also present novel results regarding the extended Priestley duals of MTL-algebras and GMTL-algebras, in particular emphasizing their structure as Priestley spaces enriched by a partial binary operation.

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References

  1. Aguzzoli, S., Flaminio, T., Ugolini, S.: Equivalences between subcategories of MTL-algebras via Boolean algebras and prelinear semihoops. J. Log. Comput. 27(8), 2525–2549 (2017)

    Article  MathSciNet  Google Scholar 

  2. Bezhanishvili, G., Ghilardi, S.: An algebraic approach to subframe logics: intuitionistic case. Ann. Pure Appl. Log. 147, 84–100 (2007)

    Article  MathSciNet  Google Scholar 

  3. Blok, W., Pigozzi, D.: Algebraizable logics. Mem. Am. Math. Soc. 77, 287–345 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Busaniche, M., Cignoli, R., Marcos, M.: A categorical equivalence for Stonean residuated lattices. Stud. Log. 107(2), 399–421 (2019)

    Article  MathSciNet  Google Scholar 

  5. Busaniche, M., Marcos, M., Ugolini, S.: Representation by triples of algebras with an MV-retract. Fuzzy Sets Syst. 369, 82–102 (2019). https://doi.org/10.1016/j.fss.2018.10.024

    Article  MathSciNet  Google Scholar 

  6. Cabrer, L., Celani, S.: Priestley dualities for some lattice-ordered algebraic structures, including MTL, IMTL, and MV-algebras. Cent. Eur. J. Math. 4(4), 600–623 (2006)

    Article  MathSciNet  Google Scholar 

  7. Celani, S.: Distributive lattices with fusion and implication. Southeast Asian Bull. Math. 28, 999–1010 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Chen, C., Grätzer, G.: Stone lattices I: construction theorems. Can. J. Math. 21, 884–994 (1969)

    Article  MathSciNet  Google Scholar 

  9. Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124, 271–288 (2001)

    Article  MathSciNet  Google Scholar 

  10. Fussner, W., Galatos, N.: Categories of models of R-mingle. Ann. Pure Appl. Log. 170, 1188–1242 (2019)

    Article  MathSciNet  Google Scholar 

  11. Fussner, W., Palmigiano, A.: Residuation algebras with functional duals. In: Adaricheva, K., et al. (eds.) Algebras and Lattices in Hawaii: Honoring Ralph Freese, pp. 39–46. Bill Lampe, and J.B. Nation, Lulu (2018)

    Google Scholar 

  12. Galatos, N.: Varieties of residuated lattices. Ph.D. thesis, Vanderbilt University (2003)

  13. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices: an algebraic glimpse at substructural logics. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  14. Gehrke, M.: Stone duality, topological algebra, and recognition. J. Pure Appl. Algebra 220, 2711–2747 (2016)

    Article  MathSciNet  Google Scholar 

  15. Goldblatt, R.: Varieties of complex algebras. Ann. Pure Appl. Log. 44, 173–242 (1989)

    Article  MathSciNet  Google Scholar 

  16. Jansana, R., Rivieccio, U.: Dualities for modal N4-lattices. Log. J. IGPL 22(4), 608–637 (2014)

    Article  MathSciNet  Google Scholar 

  17. Montagna, F., Ugolini, S.: A categorical equivalence for product algebras. Stud. Log. 103(2), 345–373 (2015)

    Article  MathSciNet  Google Scholar 

  18. Noguera, C.: Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. thesis, University of Barcelona (2007)

  19. Pogel, A.: Stone triples and self-duality. Ph.D. thesis, New Mexico State University (1998)

  20. Priestley, H.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2(2), 186–190 (1970)

    Article  MathSciNet  Google Scholar 

  21. Priestley, H.: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. 24(3), 507–530 (1972)

    Article  MathSciNet  Google Scholar 

  22. Priestley, H.: Stone lattices: a topological approach. Fund. Math. 84, 127–143 (1974)

    Article  MathSciNet  Google Scholar 

  23. Routley, R., Meyer, R.: The semantics of entailment II. J. Phil. Log. 1, 53–73 (1972)

    Article  MathSciNet  Google Scholar 

  24. Routley, R., Meyer, R.: The semantics of entailment I. In: Leblanc, H. (ed.) Truth, syntax, modality. North-Holland Publ. Co., Amsterdam (1973)

    Google Scholar 

  25. Ugolini, S.: Varieties of residuated lattices with an MV-retract and an investigation into state theory. Ph.D. thesis, University of Pisa (2018)

  26. Urquhart, A.: Duality for algebras of relevant logics. Stud. Log. 56(1/2), 253–276 (1996)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Nick Galatos for a number of helpful suggestions regarding this work.

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, Denver, CO, 80231, USA

    Wesley Fussner

  2. Department of Computer Science, University of Pisa, Pisa, Italy

    Sara Ugolini

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  1. Wesley Fussner
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  2. Sara Ugolini
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Correspondence to Wesley Fussner.

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Fussner, W., Ugolini, S. A topological approach to MTL-algebras. Algebra Univers. 80, 38 (2019). https://doi.org/10.1007/s00012-019-0612-6

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