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Strongly algebraically closed orthomodular near semirings

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Abstract

The notion of \(q^\prime \)-compactness on a modular near ring is introduced and considered. Our aim is to show that if an induced lattice with an antitone involution on an orthogonal near semiring is complete and \(q^\prime \)-compact, then the induced lattice is a strongly algebraically closed lattice. In particular, an open question proposed by A. Di-Nola, G. Georgescu and A. Iorgulescu about the connections of pseudo-BL algebras with other algebraic structures in Di Nola et al. (Mult Val Logic 8:717–750, 2002) is answered.

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References

  1. Beran, L.: Orthomodular Lattices: Algebraic Approach, Mathematics and Its Applications. Springer, Berlin (2011)

    Google Scholar 

  2. Bonzio, S., Chajda, I., Ledda, A.: Representing quantum structures as near semirings. Log. J. IGPL 24, 1–24 (2016)

    Article  MathSciNet  Google Scholar 

  3. Brandt, H.: Uber eine Verallgemeinerung der Gruppen-begriffes. Math. Ann. 96, 360–366 (1926)

    Article  Google Scholar 

  4. Chajda, I., Fazio, D., Ledda, A.: On the structure theory of Łukasiewicz near semirings. Log. J. IGPL (2018)

  5. Chajda, I., Halas, R., Kuhr, J.: Semilattice Structures. Research and Exposition in Mathematics. Heldermann, Berlin (2007)

    MATH  Google Scholar 

  6. Chang, C.C.: Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)

    Article  MathSciNet  Google Scholar 

  7. Daniyarova, E., Miasnikov, A., Remeslennikov, V.: Algebraic geometry over algebraic structures II: foundations. Fund. i prikl. mat. 17(1), 65–106 (2012)

    MathSciNet  Google Scholar 

  8. Daniyarova, E., Myasnikov, A., Remeslennikov, V.: Algebraic geometry over algebraic structures III: equationally noetherian property and compactness. Southeast Asian Bull. Math. 35, 35–68 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Daniyarova, E., Miasnikov, A., Remeslennikov, V.: Unification theorems in algebraic geometry. Algebra Discrete Math. 1, 80–112 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Di Nola, A., Gerla, B.: Algebras of Lstrok; Łukasiewicz Logic and their semiring reducts. In: Litvinov, G.L., Maslov, V.P. (eds) Proceedings of the Conference on Idempotent Mathematics and Mathematial Physics (2004)

  11. Di Nola, A., Georgescu, G., Iorgulescu, A.: Pseudo-BL algebras: Part II. Mult. Val. Logic 8, 717–750 (2002)

    MATH  Google Scholar 

  12. Georgescu, G., Iorgulescu, A.: Pseudo-BL algebras, a noncommutative extension of BL-algebras (Abstract). In: The Fifth International Conference FSTA 2000 on Fuzzy Sets Theory and Its Application, February, pp. 90–92 (2000)

  13. Hajek, P.: Fuzzy logics with noncommutative conjuctions. J. Log. Comput. 13, 469–479 (2003)

    Article  Google Scholar 

  14. Hajek, P.: Metamathematics of Fuzzy Logic (Trends in Logic, StudiaLogica Library). Kluwer, Dordrecht (1998)

    Book  Google Scholar 

  15. Kalmbach, G.: Orthomodular Lattices. Academic Press Inc (London) Ltd., London (1983)

    MATH  Google Scholar 

  16. Kim, K.H., Yon, Y.H.: Dual BCK-algebra and MV-algebra. Sci. Math. Jpn. e-2007, 393–399 (2007)

  17. Kuhr, J.: Pseudo BL-algebras and DRL-monoids. Math. Bohem. 128, 199–202 (2003)

    Article  MathSciNet  Google Scholar 

  18. Lacava, F.: Sulle classi delle \(\ell \)-algebre e degli \(\ell \)-gruppi abeliani algebricamente chiusi. Boll. Un. Mat. Ital. B(7)(1), 703–712 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Molkhasi, A.: On some strongly algebraically closed semirings. J. Intell. Fuzzy Syst. 36(6), 6393–6400 (2019)

    Article  Google Scholar 

  20. Molkhasi, A.: On strongly algebraically closed lattices. J. Sib. Fed. Univ. Math. Phys. 9(2), 202–208 (2016)

    Article  MathSciNet  Google Scholar 

  21. Motamed, S., Torkzadeh, L.: A new class of BL-algebras. Soft. Comput. 21, 686–693 (2017)

    Article  Google Scholar 

  22. Myasnikov, A., Remeslennikov, V.: Algebraic geometry over groups II: logical foundations. J. Algebra 234, 225–276 (2000)

    Article  MathSciNet  Google Scholar 

  23. Plotkin, B.: Algebras with the same (algebraic) geometry. Proc. Steklov Inst. Math. 242, 165–196 (2003)

    MathSciNet  MATH  Google Scholar 

  24. Ptak, P., Pulmannova, S.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht (1991)

    MATH  Google Scholar 

  25. Schmid, J.: Algebraically and existentially closed distributive lattices. Zeitschr Math. Logik U. G. M. 25, 525–530 (1979)

    Article  MathSciNet  Google Scholar 

  26. Shevlyakov, A.: Algebraic geometry over boolean algebras in the language with constants. J. Math. Sci. 206, 724–757 (2015)

    Article  Google Scholar 

  27. Turunen, E.: RS–BL-algebras are MV-algebras. Iran. J. Fuzzy Syst. 6, 153–154 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Walendziak, A.: On commutative BE-algebras. Sci. Math. Jpn. 69, 585–588 (2008)

    MathSciNet  MATH  Google Scholar 

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Molkhasi, A., Shum, K.P. Strongly algebraically closed orthomodular near semirings. Rend. Circ. Mat. Palermo, II. Ser 69, 803–812 (2020). https://doi.org/10.1007/s12215-019-00434-z

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