Finite atomistic lattices that can be represented as lattices of quasivarieties

K. Adaricheva; Wiesław Dziobiak; V. Gorbunov

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 1, page 19-43
  • ISSN: 0016-2736

Abstract

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We prove that a finite atomistic lattice can be represented as a lattice of quasivarieties if and only if it is isomorphic to the lattice of all subsemilattices of a finite semilattice. This settles a conjecture that appeared in the context of [11].

How to cite

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Adaricheva, K., Dziobiak, Wiesław, and Gorbunov, V.. "Finite atomistic lattices that can be represented as lattices of quasivarieties." Fundamenta Mathematicae 142.1 (1993): 19-43. <http://eudml.org/doc/211969>.

@article{Adaricheva1993,
abstract = {We prove that a finite atomistic lattice can be represented as a lattice of quasivarieties if and only if it is isomorphic to the lattice of all subsemilattices of a finite semilattice. This settles a conjecture that appeared in the context of [11].},
author = {Adaricheva, K., Dziobiak, Wiesław, Gorbunov, V.},
journal = {Fundamenta Mathematicae},
keywords = {atomistic lattice; quasivariety; Mal'cev problem; equa-closure operator; semilattice; finite atomistic lattice; lattice of quasivarieties},
language = {eng},
number = {1},
pages = {19-43},
title = {Finite atomistic lattices that can be represented as lattices of quasivarieties},
url = {http://eudml.org/doc/211969},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Adaricheva, K.
AU - Dziobiak, Wiesław
AU - Gorbunov, V.
TI - Finite atomistic lattices that can be represented as lattices of quasivarieties
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 1
SP - 19
EP - 43
AB - We prove that a finite atomistic lattice can be represented as a lattice of quasivarieties if and only if it is isomorphic to the lattice of all subsemilattices of a finite semilattice. This settles a conjecture that appeared in the context of [11].
LA - eng
KW - atomistic lattice; quasivariety; Mal'cev problem; equa-closure operator; semilattice; finite atomistic lattice; lattice of quasivarieties
UR - http://eudml.org/doc/211969
ER -

References

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  1. [1] K. V. Adaricheva, A characterization of finite lattices of subsemilattices, Algebra i Logika 30 (1991), 385-404 (in Russian). Zbl0773.06009
  2. [2] K. V. Adaricheva and V. A. Gorbunov, Equaclosure operator and forbidden semidistributive lattices, Sibirsk. Mat. Zh. 30 (1989), 7-25 (in Russian). Zbl0711.08013
  3. [3] K. V. Adaricheva, W. Dziobiak and V. A. Gorbunov, The lattices of quasivarieties of locally finite quasivarieties, preprint. Zbl0937.06002
  4. [4] M. K. Bennett, Biatomic lattices, Algebra Universalis 24 (1987), 60-73. Zbl0643.06003
  5. [5] G. Birkhoff and M. K. Bennett, The convexity lattice of a poset, Order 2 (1985), 223-242. Zbl0591.06009
  6. [6] A. Day, Characterization of finite lattices that are bounded-homomorphic image of sublattices of free lattices, Canad. J. Math. 31 (1979), 69-78. Zbl0432.06007
  7. [7] W. Dziobiak, On atoms in the lattice of quasivarieties, Algebra Universalis 24 (1987), 31-35. Zbl0642.08003
  8. [8] R. Freese and J. B. Nation, Congruence lattices of semilattices, Pacific J. Math. 44 (1973), 51-58. Zbl0287.06002
  9. [9] H. Gaskill, G. Grätzer and C. R. Platt, Sharply transferable lattices, Canad. J. Math. 27 (1975), 1246-1262. Zbl0284.06003
  10. [10] V. A. Gorbunov, Lattices of quasivarieties, Algebra i Logika 15 (1976), 436-457 (in Russian). Zbl0359.06014
  11. [11] V. A. Gorbunov and V. I. Tumanov, A class of lattices of quasivarieties, ibid. 19 (1980), 59-80 (in Russian). 
  12. [12] V. A. Gorbunov and V. I. Tumanov, The structure of the lattices of quasivarieties, in: Trudy Inst. Mat. (Novosibirsk) 2, Nauka Sibirsk. Otdel., Novosibirsk 1982, 12-44 (in Russian). Zbl0523.08008
  13. [13] G. Grätzer, General Lattice Theory, Birkhäuser, Basel 1979. Zbl0436.06001
  14. [14] G. Grätzer and H. Lakser, A note on the implicational class generated by a class of structures, Canad. Math. Bull. 16 (1973), 603-605. Zbl0299.08007
  15. [15] B. Jónsson and J. B. Nation, A report on sublattices of a free lattice, in: Contributions to Universal Algebra, Szeged 1975, Colloq. Math. Soc. János Bolyai 17, 223-257. 
  16. [16] A. I. Mal'cev, On certain frontier questions in algebra and mathematical logic, in: Proc. Int. Congr. Mathematicians, Moscow 1966, Mir, 1968, 217-231 (in Russian). 
  17. [17] A. I. Mal'cev, Algebraic Systems, Springer, 1973. 
  18. [18] R. McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1-43. Zbl0265.08006
  19. [19] R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Wadsworth and Brooks/Cole, Monterey 1987. 
  20. [20] V. I. Tumanov, Finite distributive lattices of quasivarieties, Algebra i Logika 22 (1983), 168-181 (in Russian). Zbl0548.08006

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