LATTICE UNIVERSALITY OF LOCALLY FINITE \(p\)-GROUPS

Vladimir B. Repnitskiǐ     (Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation)

Abstract


For an arbitrary prime \(p\), we prove that every algebraic lattice is isomorphic to a complete sublattice in the subgroup lattice of a suitable locally finite \(p\)-group. In particular, every lattice is embeddable in the subgroup lattice of a locally finite \(p\)-group.


Keywords


Subgroup lattice, Algebraic lattice, Complete sublattice, Lattice-universal class of algebras, Locally finite \(p\)-group, Group valuation.

Full Text:

PDF

References


  1. Grätzer G. General Lattice Theory, 2nd edn. Basel: Birkhäuser–Verlag, 2003. XX+663 pp.
  2. Kargapolov M.I., Merzljakov Ju.I. Fundamentals of the Theory of Groups. Translated from the second Russian edition by Robert G. Burns. Grad. Texts in Math., vol. 62. New York-Berlin: Springer–Verlag, 1979. XVIII+203 pp.
  3. RepnitskiÇ V.B. On the representation of lattices by lattices of subsemigroups. Russian Math. (Iz. VUZ), 1996. Vol. 40, No. 1. P. 55–64.
  4. RepnitskiÇ V.B. Lattice universality of free Burnside groups. Algebra and Logic, 1996. Vol. 35. P. 330–343. DOI: 10.1007/BF02367358
  5. RepnitskiÇ V.B. On lattice-universal varieties of algebras. Russian Math. (Iz. VUZ), 1997. Vol. 41, No. 5. P. 50–56.
  6. RepnitskiÇ V.B. A new proof of Tůma’s theorem on intervals in subgroup lattices. In: Contrib. General Algebra 16. Proc. 68th Workshop on General Algebra “68. Arbeitstagung Allgemeine Algebraâ€, Dresden, Germany, June 10–13, 2004 and Summer School 2004 on General Algebra and Ordered Sets, Malá Morávka, Czech Republic, September 5–11, 2004. Chajda I. et al. (eds.). Klagenfurt: Verlag Johannes Heyn, 2005. P. 213–230.
  7. RepnitskiÇ V.B. Lattice universality of locally finite 2-groups. Algebra Univers., 2020. Vol. 81. Art. no. 44. DOI: 10.1007/s00012-020-00672-8
  8. RepnitskiÇ V., Tůma J. Intervals in subgroup lattices of countable locally finite groups. Algebra Univers., 2008. Vol. 59, P. 49–71. DOI: 10.1007/s00012-008-2045-5
  9. Schmidt R. Subgroup Lattices of Groups. De Gruyter Expositions in Mathematics, vol. 14. Berlin–New York: Walter de Gruyter, 1994. XVI+572 pp. DOI: 10.1515/9783110868647
  10. Shevrin L.N., Ovsyannikov A.J. Semigroups and Their Subsemigroup Lattices. Ser. Math. Appl. Dordrecht: Kluwer Academic Publishers, 1996. XI+380 pp.
  11. Tůma J. Semilattice-valued measures. In: Contrib. General Algebra 18, Proc. 73rd Workshop on General Algebra 2007 (AAA 73) and 22nd Conf. Young Algebraists. Klagenfurt: Verlag Johannes Heyn, 2008. P. 199–210.
  12.  Whitman Ph.M. Lattices, equivalence relations, and subgroups. Bull. Amer. Math. Soc., 1946. Vol. 52. P. 507–522. DOI: 10.1090/S0002-9904-1946-08602-4




DOI: http://dx.doi.org/10.15826/umj.2023.1.011

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.