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The Archimedean property: new horizons and perspectives

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Abstract

Although there have been repeated attempts to define the concept of an Archimedean algebra for individual classes of residuated lattices, there is no all-purpose definition that suits the general case. We suggest as a possible candidate the notion of a normal-valued and e-cyclic residuated lattice that has the zero radical compact property—namely, a normal-valued and e-cyclic residuated lattice in which every principal convex subuniverse has a trivial radical (understood as the intersection of all its maximal convex subuniverses). We characterize the Archimedean members in the variety of e-cyclic residuated lattices, as well as in various special cases of interest. A theorem to the effect that each Archimedean and prelinear GBL-algebra is commutative, subsuming as corollaries several analogous results from the recent literature, is grist to the mill of our proposal’s adequacy. Finally, we revisit the concept of a hyper-Archimedean residuated lattice, another notion with which researchers have engaged from disparate angles, and investigate some of its properties.

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References

  1. Aglianò, P., Montagna, F.: Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181(2–3), 105–129 (2003)

    Article  MathSciNet  Google Scholar 

  2. Anderson, M., Feil, T.: Lattice Ordered Groups: An Introduction. Reidel, Dordrecht (1988)

    Book  Google Scholar 

  3. Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices. Algebra Univers. 50(1), 83–106 (2003)

    Article  MathSciNet  Google Scholar 

  4. Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13(4), 437–461 (2003)

    Article  MathSciNet  Google Scholar 

  5. Botur, M., Kühr, J., Liu, L., Tsinakis, C.: Conrad’s program: from \(\ell \)-groups to algebras of logic. J. Algebra 450, 173–203 (2016)

    Article  MathSciNet  Google Scholar 

  6. Botur, M., Kühr, J., Tsinakis, C.: Strong simplicity and states in ordered algebras: pushing the limits (in preparation)

  7. Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht (1999)

    MATH  Google Scholar 

  8. Ciungu, L.C.: Classes of residuated lattices. Ann. Univ. Craiova Math. Comp. Sci. Ser. 33, 189–207 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Conrad, P.: The structure of a lattice-ordered group with a finite number of disjoint elements. Mich. Math. J. 7, 171–180 (1960)

    Article  MathSciNet  Google Scholar 

  10. Conrad, P.: Some structure theorems for lattice-ordered groups. Trans. Am. Math. Soc. 99, 212–240 (1961)

    Article  MathSciNet  Google Scholar 

  11. Conrad, P.: The lattice of all convex \(\ell \)-subgroups of a lattice-ordered group. Czechoslov. Math. J. 15, 101–123 (1965)

    MathSciNet  MATH  Google Scholar 

  12. Conrad, P.: Lex-subgroups of lattice-ordered groups. Czechoslov. Math. J. 18, 86–103 (1968)

    Article  MathSciNet  Google Scholar 

  13. Darnel, M.: The Theory of Lattice-Ordered Groups. Dekker, New York (1995)

    MATH  Google Scholar 

  14. Dilworth, R.P.: Non-commutative residuated lattices. Trans. Am. Math Soc. 46(3), 426–444 (1939)

    Article  MathSciNet  Google Scholar 

  15. Dvurečenskij, A.: On pseudo-MV-algebras. Soft Comput. 5, 347–354 (2001)

    Article  Google Scholar 

  16. Dvurečenskij, A.: Pseudo-MV-algebras are intervals in \(\ell \)-groups. J. Aust. Math. Soc. 72, 427–445 (2002)

    Article  MathSciNet  Google Scholar 

  17. Dvurečenskij, A.: Aglianò–Montagna type decomposition of linear pseudo hoops and its applications. J. Pure Appl. Algebra 221(3), 851–861 (2007)

    Article  Google Scholar 

  18. Dvurečenskij, A., Kowalski, T.: Multipotent GBL-algebras. Algebra Univers. 64(1), 25–38 (2010)

    Article  MathSciNet  Google Scholar 

  19. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht (2000)

    Book  Google Scholar 

  20. Font, J.M., Rodriguez, A.J., Torrens, A.: Wajsberg algebras. Stochastica 8, 5–31 (1984)

    MathSciNet  MATH  Google Scholar 

  21. Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford (1963)

    MATH  Google Scholar 

  22. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  23. Galatos, N., Tsinakis, C.: Generalized MV-algebras. J. Algebra 283, 254–291 (2005)

    Article  MathSciNet  Google Scholar 

  24. Georgescu, G., Leuştean, L., Preoteasa, V.: Pseudo-hoops. J. Mult. Valued Log. Soft Comput. 11, 153–184 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Gil Férez, J., Ledda, A., Paoli, F., Tsinakis, C.: Projectable \(\ell \)-groups and algebras of logic: categorical and algebraic connections. J. Pure Appl. Algebra 220(10), 3345–3572 (2016)

    Article  MathSciNet  Google Scholar 

  26. Gil Férez, J., Ledda, A., Tsinakis, C.: Hulls of ordered algebras: projectability, strong projectability and lateral completeness. J. Algebra 483, 429–474 (2017)

    Article  MathSciNet  Google Scholar 

  27. Hölder, O.: Die Axiome der Quantität und die Lehre vom Mass. Leipziger Berichte 53(1), 1–64 (1901). (German)

    MATH  Google Scholar 

  28. Holland, W.C.: The largest proper variety of lattice ordered groups. Proc. Am. Math. Soc. 57(1), 25–28 (1976)

    Article  MathSciNet  Google Scholar 

  29. Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Martinez, J. (ed.) Ordered Algebraic Structures, pp. 19–56. Kluwer, Dordrecht (2002)

    Chapter  Google Scholar 

  30. Jónsson, B., Tsinakis, C.: Products of classes of residuated structures. Studia Logica 77, 267–292 (2004)

    Article  MathSciNet  Google Scholar 

  31. Kühr, J.: Generalizations of pseudo MV-algebras and generalized pseudo effect algebras. Czechoslov. Math. J. 58(2), 395–415 (2008)

    Article  MathSciNet  Google Scholar 

  32. Ledda, A., Paoli, F., Tsinakis, C.: Lattice-theoretic properties of algebras of logic. J. Pure Appl. Algebra 218, 1932–1952 (2014)

    Article  MathSciNet  Google Scholar 

  33. Martinez, J.: Archimedean lattices. Algebra Univers. 3, 247–260 (1973)

    Article  MathSciNet  Google Scholar 

  34. Metcalfe, G., Paoli, F., Tsinakis, T.: Ordered algebras and logic. In: Hosni, H., Montagna, F. (eds.) Uncertainty and Rationality, vol. 10, pp. 1–85. Publications of the Scuola Normale Superiore, Pisa (2010)

    MATH  Google Scholar 

  35. Mostert, P.S., Shields, A.L.: On the structure of semigroups on a compact manifold with boundary. Ann. Math. 65, 117–143 (1957)

    Article  MathSciNet  Google Scholar 

  36. Mundici, D.: Interpretations of AF C* algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986)

    Article  MathSciNet  Google Scholar 

  37. Ono, H.: Logics without contraction rules and residuated lattices I, typescript (2001)

  38. Paoli, F.: Substructural Logics: A Primer. Kluwer, Dordrecht (2002)

    Book  Google Scholar 

  39. Rachůnek, J.: Radicals in non-commutative generalizations of MV-algebras. Math. Slovaca 52, 135–144 (2002)

    MathSciNet  MATH  Google Scholar 

  40. Turunen, E.: Hyper-Archimedean BL-algebras are MV-algebras. Math. Log. Q. 53(2), 170–175 (2007)

    Article  MathSciNet  Google Scholar 

  41. Wolfenstein, S.: Valeurs normales dans un groupe reticulé, Atti Accademia Nazionale dei Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 8(44), 337–342 (1968)

    MathSciNet  MATH  Google Scholar 

  42. Young, W.: Varieties generated by unital abelian \(\ell \)-groups. J. Pure Appl. Algebra 219(1), 161–169 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Substantive parts of the present paper were written when the first two authors were visiting the Department of Mathematics at Vanderbilt University (Nashville, TN) and when the third author was visiting the Department of Pedagogy, Psychology, Philosophy at the University of Cagliari, Italy. The assistance and facilities provided by both departments are gratefully acknowledged. We also acknowledge the following funding sources that made these visits possible: the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”); the Visiting Scientists Programme of the University of Cagliari, sponsored by Regione Autonoma della Sardegna; the project “Order properties in mathematics and physics”, CUP: F72F16002920002, sponsored by Regione Autonoma della Sardegna; and the Faculty Research Grants Program of the College of Arts and Science of Vanderbilt University. Finally, we thank the anonymous reviewer for his/her careful reading of the manuscript and helpful suggestions.

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Authors and Affiliations

  1. Università di Cagliari, Cagliari, Italy

    Antonio Ledda & Francesco Paoli

  2. Department of Mathematics, Vanderbilt University, Nashville, USA

    Constantine Tsinakis

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  1. Antonio Ledda
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  2. Francesco Paoli
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  3. Constantine Tsinakis
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Correspondence to Constantine Tsinakis.

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In memoriam: Bjarni Jónsson.

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This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J. B. Nation.

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Ledda, A., Paoli, F. & Tsinakis, C. The Archimedean property: new horizons and perspectives. Algebra Univers. 79, 91 (2018). https://doi.org/10.1007/s00012-018-0573-1

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