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On Gelfand residuated lattices

Soft Computing volume 27, pages 2147–2158 (2023)Cite this article

Abstract

In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand’s residuated lattices strongly tied up with the hull–kernel topology. Particularly, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull–kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull–kernel and the \(\mathscr {D}\)-topology coincide on the set of maximal filters.

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References

  • Aghajani M, Tarizadeh A (2020) Characterizations of Gelfand rings specially clean rings and their dual rings. Results Math 75(3):1–24

    Article  MathSciNet  MATH  Google Scholar 

  • Al-Ezeh H (1989) Pure ideals in commutative reduced Gelfand rings with unity. Arch Math 53(3):266–269

    Article  MathSciNet  MATH  Google Scholar 

  • Al-Ezeh H (1990a) Exchange PF-rings and almost PP-rings. Matematički Vesnik 42(109):77–80

    MathSciNet  MATH  Google Scholar 

  • Al-Ezeh H (1990b) Topological characterization of certain classes of lattices. Rendiconti del Seminario Matematico della Università di Padova 83:13–18

    MathSciNet  MATH  Google Scholar 

  • Banaschewski B (2000) Gelfand and exchange rings: their spectra in pointfree topology. Arab J Sci Eng 25(2; PART C):3–22

    MathSciNet  MATH  Google Scholar 

  • Bigard A, Keimel K, Wolfenstein S (2006) Groupes et anneaux réticulés, vol 608. Springer

    MATH  Google Scholar 

  • Bkouche R (1970) Couples spectraux et faisceaux associés. applications aux anneaux de fonctions. Bull Soc Math France 98:253–295

    Article  MathSciNet  MATH  Google Scholar 

  • Borceux F, VanDenBossche G (1983) Algebra in a localic topos with applications to ring-theory. Lect Notes Math 1038:R3

    MathSciNet  Google Scholar 

  • Buşneag C, Piciu D (2012) The stable topology for residuated lattices. Soft Comput 16(10):1639–1655

    Article  MATH  Google Scholar 

  • Cheptea D, Georgescu G (2020) Boolean lifting property in quantales. Soft Comput 24(8):6169–6181

    Article  MATH  Google Scholar 

  • Cignoli RL, d’Ottaviano IM, Mundici D (2013) Algebraic foundations of many-valued reasoning, vol 7. Springe, Cham

    MATH  Google Scholar 

  • Ciungu LC (2006) Classes of residuated lattices. Ann Univ Craiova-Math Comput Sci Ser 33:189–207

    MathSciNet  MATH  Google Scholar 

  • Contessa M (1982) On PM-rings. Comm Algebra 10(1):93–108

    Article  MathSciNet  MATH  Google Scholar 

  • Contessa M (1984) On certain classes of PM-rings. Comm Algebra 12(12):1447–1469

    Article  MathSciNet  MATH  Google Scholar 

  • Cornish WH (1972) Normal lattices. J Aust Math Soc 14(2):200–215

    Article  MathSciNet  MATH  Google Scholar 

  • De Marco G (1983) Projectivity of pure ideals. Rendiconti del Seminario Matematico della Università di Padova 69:289–304

    MathSciNet  MATH  Google Scholar 

  • De Marco G, Orsatti A (1971) Commutative rings in which every prime ideal is contained in a unique maximal ideal. Proc Am Math Soc 30(3):459–466

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Engelking R (1989) General topology. Heldermann, Berlin

    MATH  Google Scholar 

  • Filipoiu A, Georgescu G (1995) Compact and Pierce representations of MV-algebras. Rev Roumaine Math Pures Appl 40(7):599–618

    MathSciNet  MATH  Google Scholar 

  • Freytes H (2004) Injectives in residuated algebras. Algebra Universalis 51:373–393

    Article  MathSciNet  MATH  Google Scholar 

  • Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics, vol 151. Elsevier, London

    MATH  Google Scholar 

  • García-Pardo F, Cabrera IP, Cordero P, Ojeda-Aciego M (2013) On Galois connections and soft computing. In: International work-conference on artificial neural networks, Springer, pp 224–235

  • Gelfand I (1941) Normierte ringe. Mat Sbornik 9(1):3–24

  • Georgescu G (2020) Reticulation functor and the transfer properties. manuscript

  • Georgescu G, Cheptea D, Mureşan C (2015) Algebraic and topological results on lifting properties in residuated lattices. Fuzzy Sets Syst 271:102–132

    Article  MathSciNet  MATH  Google Scholar 

  • Georgescu G, Mureşan C (2014) Boolean lifting property for residuated lattices. Soft Comput 18(11):2075–2089

  • Georgescu G, Voiculescu I (1989) Some abstract maximal ideal-like spaces. Algebra Universalis 26(1):90–102

    Article  MathSciNet  MATH  Google Scholar 

  • Grätzer G (2011) Lattice theory: foundation. Springer, Cham

    Book  MATH  Google Scholar 

  • Johnstone PT (1982) Stone spaces, vol 3. Cambridge university press, Cambridge

  • Kowalski T, Ono H (2000) Residuated lattices: an algebaric glimpse at logics without contraction (preliminary report). The Japan Advanced Institute of Science and Technology (JAIST)

  • Levaro RA (1975) Projective quasi-coherent sheaves of modules. Pac J Math 57(2):457–461

    Article  MathSciNet  MATH  Google Scholar 

  • Mulvey CJ (1979) A generalisation of Gelfand duality. J Algebra 56(2):499–505

    Article  MathSciNet  MATH  Google Scholar 

  • Paseka J (1986) Regular and normal quantales. Archivum Mathematicum 22(4):203–210

    MathSciNet  MATH  Google Scholar 

  • Pawar Y (1994) Characterizations of normal lattices. Indian J Pure Appl Math 24:651–651

    MathSciNet  MATH  Google Scholar 

  • Pawar Y, Thakare N (1977) PM-Lattices. Algebra Universalis 7(1):259–263

    Article  MathSciNet  MATH  Google Scholar 

  • Rasouli S (2018) Generalized co-annihilators in residuated lattices. Ann Univ Craiova-Math Comput Sci Ser 45(2):190–207

    MathSciNet  MATH  Google Scholar 

  • Rasouli S (2019) The going-up and going-down theorems in residuated lattices. Soft Comput 23(17):7621–7635

    Article  MATH  Google Scholar 

  • Rasouli S (2020) Quasicomplemented residuated lattices. Soft Comput 24(9):6591–6602

    Article  MATH  Google Scholar 

  • Rasouli S (2021) Rickart residuated lattices. Soft Comput 25(22):13823–13840

    Article  MATH  Google Scholar 

  • Rasouli S, Dehghani A (2021) The hull-kernel topology on prime filters in residuated lattices. Soft Comput 25(16):10519–10541

    Article  MATH  Google Scholar 

  • Rasouli S, Kondo M (2020) \(n\)-Normal residuated lattices. Soft Comput 24(1):247–258

    Article  MATH  Google Scholar 

  • Simmons H (1980) Reticulated rings. J Algebra 66(1):169–192

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to give heartfelt thanks to the referees for their very careful reading of the paper and for their very valuable comments and suggestions which improved the paper.

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  1. Department of Mathematics, Persian Gulf University, Bushehr, Iran

    Saeed Rasouli & Amin Dehghani

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  1. Saeed Rasouli
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  2. Amin Dehghani
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Correspondence to Saeed Rasouli.

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Rasouli, S., Dehghani, A. On Gelfand residuated lattices. Soft Comput 27, 2147–2158 (2023). https://doi.org/10.1007/s00500-022-07640-5

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Keywords

  • Gelfand residuated lattice
  • Pure filters
  • Soft residuated lattice
  • Hull–kernel topology
  • D-topology