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Implicative subreducts of MV-algebras: free and weakly projective objects

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In this article, we explore in some detail the free and weakly projective objects of the variety of Łukasiewicz implication algebras (the implicative subreducts of MV-algebras). We review the two already known descriptions of finitely generated free algebras, giving new insights into their structure and their connection, as well as providing new proofs of the characterizations. We give a representation theorem for weakly projective algebras as algebras of certain McNaughton functions restricted to rational polyhedra and prove that finitely generated weakly projective algebras coincide with finitely presented ones. We also prove that finite chains are the only totally ordered weakly projective examples in this variety.

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References

  1. Berman J., Blok W.J.: Free Łukasiewicz and hoop residuation algebras. Studia Logica 77, 153–180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cabrer L., Mundici D.: Projective MV-algebras and rational polyhedra. Algebra Universalis 62, 63–74 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Campercholi M., Castaño D., Díaz Varela J.P.: Quasivarieties and congruence permutability of Łukasiewicz implication algebras. Studia Logica 98, 267–283 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cignoli, R.L., D’Ottaviano, I.M.L., Mundici, D.: Algebraic foundations of many-valued reasoning. Kluwer Academic Publishers, Dordrecht (2000)

  5. Cintula P., Metcalfe G.: Structural completeness in fuzzy logics. Notre Dame J. Form. Log. 50, 153–182 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Díaz Varela J.P.: Free Łukasiewicz implication algebras. Arch. Math. Logic 47, 25–33 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Díaz Varela J.P., Torrens A.: Decomposability of free Łukasiewicz implication algebras. Arch. Math. Logic 45, 1011–1020 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Komori Y.: Super-Łukasiewicz implicational logics. Nagoya Math. J. 72, 127–133 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  9. Panti G.: Varieties of MV-algebras. J. Appl. Non-Classical Logics 9, 141–157 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rose, A.: Extensions de quelques théorèmes de McNaughton. C. R. Acad. Sci. Paris Sér. I Math. 292, 979–981 (1981) (French)

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  1. Departamento de Matemática, (Universidad Nacional del Sur), INMABB (UNS-CONICET), Bahía Blanca, 8000, Argentina

    Diego N. Castaño

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  1. Diego N. Castaño
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Correspondence to Diego N. Castaño.

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Presented by C. Tsinakis.

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Castaño, D.N. Implicative subreducts of MV-algebras: free and weakly projective objects. Algebra Univers. 78, 579–600 (2017). https://doi.org/10.1007/s00012-017-0474-8

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  • DOI: https://doi.org/10.1007/s00012-017-0474-8

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