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Elementary properties of free lattices

  • J. B. Nation and Gianluca Paolini EMAIL logo
Published/Copyright: May 15, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 2

Abstract

We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive ∃ ∀ -sentence true in 𝐅 3 and false in 𝐅 4 . Secondly, we show that every model of Th ⁢ ( 𝐅 n ) admits a canonical homomorphism into the profinite-bounded completion 𝐇 n of 𝐅 n . Thirdly, we show that 𝐇 n is isomorphic to the Dedekind–MacNeille completion of 𝐅 n , and that 𝐇 n is not positively elementarily equivalent to 𝐅 n , as there is a positive ∀ ∃ -sentence true in 𝐇 n and false in 𝐅 n . Finally, we show that DM ⁢ ( 𝐅 n ) is a retract of Id ⁢ ( 𝐅 n ) and that for any lattice 𝐊 which satisfies Whitman’s condition ( W ) and which is generated by join prime elements, the three lattices 𝐊 , DM ⁢ ( 𝐊 ) , and Id ⁢ ( 𝐊 ) all share the same positive universal first-order theory.

Keywords: Free lattices; profinite completions; model theory; positive first-order logic
MSC 2020: 03C05; 03C64; 06B05

Communicated by Manfred Droste

Funding statement: The second author was supported by project PRIN 2022 “Models, sets and classifications”, prot. 2022TECZJA. The second author also wishes to thank the group GNSAGA of the “Istituto Nazionale di Alta Matematica “Francesco Severi”” (INDAM) to which he belongs.

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Received: 2023-10-10
Revised: 2024-03-20
Published Online: 2024-05-15
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
  3. Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
  4. Estimates of Picard modular cusp forms
  5. Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
  6. Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
  7. Free groups generated by two unipotent maps
  8. Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
  9. Colored multizeta values in positive characteristic
  10. Simultaneous nonvanishing of central L-values with large level
  11. Laplace convolutions of weighted averages of arithmetical functions
  12. Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
  13. On the geometric trace of a generalized Selberg trace formula
  14. Elementary properties of free lattices
  15. Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
  16. Small generators of abelian number fields
  17. Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
  18. Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
https://doi.org/10.1515/forum-2023-0358
Keywords for this article
Free lattices; profinite completions; model theory; positive first-order logic

Articles in the same Issue

  1. Frontmatter
  2. Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
  3. Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
  4. Estimates of Picard modular cusp forms
  5. Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
  6. Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
  7. Free groups generated by two unipotent maps
  8. Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
  9. Colored multizeta values in positive characteristic
  10. Simultaneous nonvanishing of central L-values with large level
  11. Laplace convolutions of weighted averages of arithmetical functions
  12. Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
  13. On the geometric trace of a generalized Selberg trace formula
  14. Elementary properties of free lattices
  15. Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
  16. Small generators of abelian number fields
  17. Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
  18. Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
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