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Sheaf representations of BL-algebras

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Abstract

In this paper we present a survey on sheaf representations of BL-algebras, based on our PhD thesis [36]. We define sheaf spaces and sheaf representations of BL-algebras, we study completely regular and compact sheaf spaces of BL-algebras, compact representations of BL-algebras and we develop a Gelfand theory for BL-algebras. Thus, we prove that the category of nontrivial BL-algebras is equivalent to the category of compact local sheaf spaces of BL-algebras. The last section of our paper is a contribution to the representation theory of BL-algebras by (weak) Boolean products. We characterize the (weak) Boolean products of BL-chains, the weak Boolean products of local BL-algebras and the weak Boolean products of perfect BL-algebras.

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References

  1. Belluce LP (1995) α-complete MV-algebras. In: Höhle U et al (ed) Non-classical logics and their applications to fuzzy subsets. A handbook of the mathematical foundations of fuzzy set theory. Kluwer Dordrecht, Theory Decis Libr Ser B 32:7–21

  2. Belluce LP (1986) Semisimple algebras of infinite valued logic and bold fuzzy set theory. Can J Math 38(6):1356–1379

    Google Scholar 

  3. Belluce LP (1991) Spectral spaces and non-commutative rings. Commun Algebra 19:1855–1865

    Google Scholar 

  4. Belluce LP, Di Nola A, Lettieri A (1993) Local MV-algebras. Rend Circolo Mat Palermo Serie II 42:347–361

    Google Scholar 

  5. Bigard A, Keimel K, Wolfenstein S (1977) Groupes et anneaux réticulés. Lecture Notes in Mathematics 608. Springer, Berlin Heidelberg New York

  6. Birkenmeier GF, Park JK (2000) Self-adjoint ideals in Baer *-rings. Commun Algebra 28(9):4259–4268

    Google Scholar 

  7. Brezuleanu A, Diaconescu R (1969) Sur la duale de la catégorie des treillis. Rev Roum Math Pures Appl 14(3):311–329

    Google Scholar 

  8. Burris S, Werner H (1979) Sheaf constructions and their elementary properties. Trans Am Math Soc 248:269–309

    Google Scholar 

  9. Chang CC Algebraic analysis of many-valued logic. Trans Am Math Soc 88:467–490

  10. Cignoli R (1971) Stone filters and ideals in distributive lattices. Bull Math de la Soc Sci Math de Roumanie 15(2):131–137

    Google Scholar 

  11. R. Cignoli, D'Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Trends in Logic–Studia Logica Library 7. Kluwer, Dordrecht

  12. Cignoli R, Torrens A (1996) Boolean products of MV-algebras: hypernormal MV-algebras. J Math Anal Appl 199(3):637–653

    Google Scholar 

  13. Comer SD (1971) Representations by algebras of sections over Boolean spaces. Pacific J Math 38:29–38

    Google Scholar 

  14. Cornish W (1977) O-ideals, congruences, and sheaf representations of distributive lattices. Rev Roum Math Pures et Appl 22(8):1059–1067

    Google Scholar 

  15. Cornish W (1972) Normal lattices. J Aust Math Soc 14:200–215

    Google Scholar 

  16. Dauns J, Hofmann KH (1966) The representation of biregular rings by sheaves. Math Z 91:103–123

    Google Scholar 

  17. Davey BA (1973) Sheaf spaces and sheaves of universal algebras. Math Z 134:275–290

    Google Scholar 

  18. Di Nola A, Georgescu G, Leuştean L (2000) Boolean products of BL-algebras. J Math Anal Appl 251(1):106–131

    Google Scholar 

  19. Di Nola A, Leuştean L (2003) Compact representations of BL-algebras. Arch Math Logic 42(8):737–761

    Google Scholar 

  20. Filipoiu A (1994) About Baer extensions of MV-algebras. Math JPN 40(2)235–241

    Google Scholar 

  21. Filipoiu A, Georgescu G (1995) Compact and pierce representations of MV-algebras. Rev Roum Math Pures Appl 40(7–8):599–618

    Google Scholar 

  22. Gelfand I, Kolmogoroff A (1939) On rings of continuous functions on topological spaces. Dokl Akad Nauk SSSR 22:11–15

    Google Scholar 

  23. Georgescu G, Voiculescu I (1987) Isomorphic sheaf representations of normal lattices. J Pure Appl Algebra 45:213–233

    Google Scholar 

  24. Georgescu G, Voiculescu I (1989) Some abstract maximal ideal-like spaces. Algebra Univers 26:90–102

    Google Scholar 

  25. Gillman L, Jerison M (1960) Rings of continuous functions. Van Nostrand Princeton

  26. Grätzer G (1972) Lattice theory. First concepts and distributive lattices. A Series of Books in Mathematics. W. H. Freeman and Company San Francisco

  27. Grothendieck A, Dieudonné J (1960) Eléments de géométrie algébrique, I. Le langage des schémas. Inst Hautes Études Sci Publ Math 4

  28. Hájek P (1998) Metamathematics of fuzzy logic. trends in logic–studia logica library 4. Kluwer, Dordrecht

  29. Hofmann KH (1972) Representations of algebras by continuous sections. Bull Am Math Soc 91:291–373

    Google Scholar 

  30. Keimel K (1971) Baer extensions of rings and Stone extensions of semigroups. Semigroup Forum 2:55–63

    Google Scholar 

  31. Kist J (1963) Minimal prime ideals in commutative semigroups. Proc London Math Soc 13(3):31–50

    Google Scholar 

  32. Lambek J (1971) On the representation of modules by sheaves of factor modules. Can Math Bull 14:359–368

    Google Scholar 

  33. Leuştean L (2004) Representations of many-valued algebras. PHD Thesis, University of Bucharest

  34. Leuştean L (2003) The prime and maximal spectra and the reticulation of BL-algebras. Central Eur J Math 1(3):382–397

    Google Scholar 

  35. Leuştean L Sheaf representations of BL-algebras. Soft Comput (to appear)

  36. Leuştean L (2003) Co-annihilators, o-filters and sheaf representations of BL-algebras (submitted)

  37. Leuştean L (2003) Baer extensions of BL-algebras. (submitted)

  38. Mulvey CJ (1978) Compact ringed spaces. J Algebra 52:411–436

    Google Scholar 

  39. Mulvey CJ (1979) Representations of rings and modules. In: Applications of sheaves, Proc Res Symp, Durham 1977, Lecture Notes in Mathematics 753. Springer, Berlin Heidelberg New York pp 542–585

  40. Mulvey CJ (1979) A generalisation of Gelfand duality. J Algebra 56:499–505

    Google Scholar 

  41. Pierce RS (1967) Modules over commutative regular rings. Memoirs Am Math Soc 70

  42. Simmons H (1980) Reticulated rings. J Algebra 66:169–192

    Google Scholar 

  43. Simmons H (1989) Compact representations-the lattice theory of compact ringed spaces. J Algebra 126:493–531

    Google Scholar 

  44. Speed TP, Evans MW (1971) A note on commutative Baer rings. J Aust Math Soc 13:1–6

    Google Scholar 

  45. Sun S-H (1992) A remark on Gelfand duality. Bull Aust Math Soc 46:187–197

    Google Scholar 

  46. Swan RS (1964) The theory of sheaves, Chicago Lectures in Mathematics. The University of Chicago Press, Chicago

  47. Tennison BR (1975) Sheaf theory, London Mathematical Society Lecture Notes Series 20. Cambridge University Press, Cambridge

  48. Torrens A (1987) W-algebras which are Boolean products of members of SR[1] and CW-algebras. Studia Logica 46:265–274

    Google Scholar 

  49. Torrens A (1989) Boolean products of CW-algebras and pseudo-complementation. Rep Math Logic 23:31–38

    Google Scholar 

  50. Turunen E (2001) Boolean deductive systems of BL-algebras. Arch Math Logic 40(6):467–473

    Google Scholar 

  51. Turunen E, Sessa S (2001) Local BL-algebras. Mult Valued Log 6(1–2):229–249

    Google Scholar 

Download references

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  1. L. Leuştean

    Present address: Department of Mathematics, TU Darmstadt, Schlossgartenstrasse 7, Darmstadt

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  1. Institute of Mathematics, “Simion Stoilow” of the Romanian Academy, Calea Griviţei 21, P.O. Box 1-764, Bucharest

    L. Leuştean

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Leuştean, L. Sheaf representations of BL-algebras. Soft Comput 9, 897–909 (2005). https://doi.org/10.1007/s00500-004-0449-5

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