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Varieties of Lattices with Geometric Descriptions

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Abstract

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann et al. proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem for free lattices. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite variety of join-semidistributive lattices.

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References

  1. Cozzens, J.H., Faith, C.: Simple Nœtherian Rings: Cambridge Tracts in Mathematics, no. 69. Cambridge University Press, Cambridge, xvii+135 pp. ISBN: 0 521 20734 7 (1975)

  2. Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice-Hall, New Jersey, vi+201 pp. ISBN: 0-13-022269-0 (1973)

  3. Freese, R.: The variety of modular lattices is not generated by its finite members. Trans. Am. Math. Soc. 255, 277–300 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Freese, R.: Free modular lattices. Trans. Am. Math. Soc. 261(1), 81–91 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  5. Frink, O. Jr.: Complemented modular lattices and projective spaces of infinite dimension. Trans. Am. Math. Soc. 60, 452–467 (1946)

    MathSciNet  MATH  Google Scholar 

  6. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous lattices and domains. In: Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press, Cambridge, xxxvi+591 pp. ISBN: 0-521-80338-1 (2003)

    Google Scholar 

  7. Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Robert E. Krieger, Malabar, xviii+412 pp. ISBN: 0-89464-632-X (1991)

  8. Goodearl, K.R., Menal, P., Moncasi, J.: Free and residually artinian regular rings. J. Algebra 156(2), 407–432 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gorbunov, V.A.: Algebraic Theory of Quasivarieties. (Algebraicheskaya Teoriya Kvazimnogoobrazij) (Russian) Sibirskaya Shkola Algebry i Logiki. 5. Novosibirsk: Nauchnaya Kniga, xii+368 pp. (1999). English translation by Plenum, New York, xii+298 pp. ISBN: 0-306-11063-6 (1998)

  10. Grätzer, G.: General Lattice Theory, 2nd edn. New appendices by the author with B.A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H.A. Priestley, H. Rose, E.T. Schmidt, S.E. Schmidt, F. Wehrung, and R. Wille. Birkhäuser, Basel, xx+663 pp. ISBN: 3-7643-5239-6; 3-7643-6996-5 (1998)

  11. Herrmann, C.: On the equational theory of submodule lattices. In: Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973), pp. 105–118. Dept. Math., Univ. Houston, Houston, Texas (1973)

  12. Herrmann, C., Huhn, A.P.: Zum Wortproblem für freie Untermodulverbände. Arch. Math. (Basel) 26(5), 449–453 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Herrmann, C.: On the word problem for the modular lattice with four free generators. Math. Ann. 265(4), 513–527 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Herrmann, C., Pickering, D., Roddy, M.: A geometric description of modular lattices. Algebra Univers. 31(3), 365–396 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Herrmann, C., Semenova, M.V.: Existence varieties of regular rings and complemented modular lattices. J. Algebra 314(1), 235–251 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huhn, A.P.: On nonmodular n-distributive lattices: the decision problem for identities in finite n-distributive lattices. Acta Sci. Math. (Szeged) 48(1–4), 215–219 (1985)

    MathSciNet  MATH  Google Scholar 

  17. Huhn, A.P.: On nonmodular n-distributive lattices. I. Lattices of convex sets. Acta Sci. Math. (Szeged) 52(1–2), 35–45 (1988)

    MathSciNet  MATH  Google Scholar 

  18. Hutchinson, G., Czédli, G.: A test for identities satisfied in lattices of submodules. Algebra Univers. 8(3), 269–309 (1978)

    Article  MATH  Google Scholar 

  19. Jipsen, P., Rose, H.: Varieties of lattices. In: Lecture Notes in Mathematics 1533. Springer, Berlin, x+162 pp. ISBN: 3-540-56314-8; 0-387-56314-8 (1992). Out of print, available online at http://www1.chapman.edu/˜jipsen/JipsenRoseVoL.html

  20. Jónsson, B.: Modular lattices and Desargues’ theorem. Math. Scand. 2, 295–314 (1954)

    MathSciNet  MATH  Google Scholar 

  21. Jónsson, B., Rival, I.: Lattice varieties covering the smallest nonmodular variety. Pac. J. Math. 82(2), 463–478 (1979)

    Article  MATH  Google Scholar 

  22. Maeda, F.: Kontinuierliche Geometrien. (German) Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. 95. Springer, Berlin, x+244 pp. ISBN: 978-1-114-52944-1 (1958)

  23. McKinsey, J.C.C.: The decision problem for some classes of sentences without quantifiers. J. Symb. Log. 8, 61–76 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nation, J.B.: An approach to lattice varieties of finite height. Algebra Univers. 27(4), 521–543 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Semenova, M.V., Wehrung, F.: Sublattices of lattices of order-convex sets. II. Posets of finite length. Int. J. Algebra Comput. 13(5), 543–564 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Semenova, M.V., Wehrung, F.: Sublattices of lattices of order-convex sets, I. The main representation theorem. J. Algebra 277, 825–860 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Semenova, M.V., Wehrung, F.: Sublattices of lattices of order-convex sets, III. The case of totally ordered sets. Int. J. Algebra Comput. 14(3), 357–387 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Semenova, M.V., Zamojska-Dzienio, A.: On lattices embeddable into lattices of order-convex sets. Case of trees. Int. J. Algebra Comput. 17(8), 1667–1712 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wehrung, F.: Sublattices of complete lattices with continuity conditions. Algebra Univers. 53(2–3), 149–173 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale de Marseille, Université de Provence, 39 rue F. Joliot Curie, 13453, Marseille Cedex 13, France

    Luigi Santocanale

  2. LMNO, CNRS UMR 6139 Département de Mathématiques, Université de Caen, 14032, Caen Cedex, France

    Friedrich Wehrung

Authors
  1. Luigi Santocanale
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  2. Friedrich Wehrung
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Correspondence to Friedrich Wehrung.

Additional information

Both authors were partially supported by the PEPS project TRECOLOCOCO.

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Santocanale, L., Wehrung, F. Varieties of Lattices with Geometric Descriptions. Order 30, 13–38 (2013). https://doi.org/10.1007/s11083-011-9225-1

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