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More on the Strength of Engeler’s Lemma

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Abstract

We introduce semilattices equipped with a partial n-ary operation ρ. A useful separation lemma for such nontrivial complete partial ρ-semilattices is proved equivalent to PIT, the prime ideal theorem. The relation of various versions of the Lemma to each other and to PIT is also explored.

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References

  1. Banaschewski, B.: Prime elements from prime ideals. Order 2, 211–213 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banaschewski, B., Erné, M.: On Krull’s separation lemma. Order, 10, 253–260 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Casas, J.M., Loday, J.L., Pirashvili, T.: Leibniz n-algebras. Forum Math. 14, 189–207 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Engeler, E.: Eine Konstruktion von Modellerweiterungen. Z. Math. Log. Grundl. Math. 5, 126–131 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  5. Erné, M.: Prime ideal theorems and systems of finite character. Comment. Math. Univ. Carol. 38(3), 513–536 (1997)

    MATH  Google Scholar 

  6. Erné, M.: Prime ideal theory for general algebras. Appl. Categ. Struct. 8, 115–144 (2000)

    Article  MATH  Google Scholar 

  7. Erné, M.: Distributors and Wallman locales. Houst. J. Math. 34(1), 69–98 (2008)

    MATH  MathSciNet  Google Scholar 

  8. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A Compendium of Continuous Lattices. Springer-Verlag (1980)

  9. Jacobson, N.: Lie and jordan triple systems. Am. J. Math. 71(1), 149–170 (1949)

    Article  MATH  Google Scholar 

  10. Johnstone, P.T.: Almost maximal ideals. Fundam. Math. 123, 201–206 (1984)

    MathSciNet  Google Scholar 

  11. Keimel, K.: A unified theory of minimal prime ideals. Acta Math. Acad. Sci. Hung. 23, 51–69 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kruml, D.: Distributive quantales. Appl. Categ. Struct. 11, 561–566 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lister, W.G.: A structure theory of lie triple systems. Trans. Am. Math. Soc. 72, 217–242 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lister, W.G.: Ternary rings. Trans. Am. Math. Soc. 154, 37–55 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  15. Paseka, J.: Scott-open distributive filters and prime elements of quantales. Contributions to General Algebra 15, Proceedings of the Klagenfurt Conference 2003, (AAA 66), 87–98. Verlag Johannes Heyn, Klagenfurt (2004)

  16. Paseka, J.: The strength of Engeler’s lemma. Math. Struct. Comput. Sci. 16, 291–297 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rav, Y.: Variants of Rados selection lemma and their applications. Math. Nachr. 79, 145–165 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rav, Y.: Semiprime ideals in general lattices. J. Pure Appl. Sci. 56, 105–118 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rosický, J.: Multiplicative lattices and frames. Acta Math. Hung. 49, 391–395 (1987)

    Article  MATH  Google Scholar 

  20. Yesilot, G.: On the prime radical of a hypergroup. J. Math. Stat. 1(3), 234–238 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Jan Paseka.

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Supported by the Ministry of Education of the Czech Republic under the project MSM0021622409.

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Paseka, J. More on the Strength of Engeler’s Lemma. Order 25, 69–77 (2008). https://doi.org/10.1007/s11083-008-9078-4

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  • DOI: https://doi.org/10.1007/s11083-008-9078-4

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