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].
@article{bwmeta1.element.bwnjournal-article-fmv142i1p19bwm,
author = {K. Adaricheva and Wies\l aw Dziobiak and V. Gorbunov},
title = {Finite atomistic lattices that can be represented as lattices of quasivarieties},
journal = {Fundamenta Mathematicae},
volume = {142},
year = {1993},
pages = {19-43},
zbl = {0806.06005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv142i1p19bwm}
}
Adaricheva, K.; Dziobiak, Wiesław; Gorbunov, V. Finite atomistic lattices that can be represented as lattices of quasivarieties. Fundamenta Mathematicae, Tome 142 (1993) pp. 19-43. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv142i1p19bwm/
[00000] [1] K. V. Adaricheva, A characterization of finite lattices of subsemilattices, Algebra i Logika 30 (1991), 385-404 (in Russian). | Zbl 0773.06009
[00001] [2] K. V. Adaricheva and V. A. Gorbunov, Equaclosure operator and forbidden semidistributive lattices, Sibirsk. Mat. Zh. 30 (1989), 7-25 (in Russian). | Zbl 0711.08013
[00002] [3] K. V. Adaricheva, W. Dziobiak and V. A. Gorbunov, The lattices of quasivarieties of locally finite quasivarieties, preprint. | Zbl 0937.06002
[00003] [4] M. K. Bennett, Biatomic lattices, Algebra Universalis 24 (1987), 60-73. | Zbl 0643.06003
[00004] [5] G. Birkhoff and M. K. Bennett, The convexity lattice of a poset, Order 2 (1985), 223-242. | Zbl 0591.06009
[00005] [6] A. Day, Characterization of finite lattices that are bounded-homomorphic image of sublattices of free lattices, Canad. J. Math. 31 (1979), 69-78. | Zbl 0432.06007
[00006] [7] W. Dziobiak, On atoms in the lattice of quasivarieties, Algebra Universalis 24 (1987), 31-35. | Zbl 0642.08003
[00007] [8] R. Freese and J. B. Nation, Congruence lattices of semilattices, Pacific J. Math. 44 (1973), 51-58. | Zbl 0287.06002
[00008] [9] H. Gaskill, G. Grätzer and C. R. Platt, Sharply transferable lattices, Canad. J. Math. 27 (1975), 1246-1262. | Zbl 0284.06003
[00009] [10] V. A. Gorbunov, Lattices of quasivarieties, Algebra i Logika 15 (1976), 436-457 (in Russian). | Zbl 0359.06014
[00010] [11] V. A. Gorbunov and V. I. Tumanov, A class of lattices of quasivarieties, ibid. 19 (1980), 59-80 (in Russian).
[00011] [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). | Zbl 0523.08008
[00012] [13] G. Grätzer, General Lattice Theory, Birkhäuser, Basel 1979. | Zbl 0436.06001
[00013] [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. | Zbl 0299.08007
[00014] [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.
[00015] [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).
[00016] [17] A. I. Mal'cev, Algebraic Systems, Springer, 1973.
[00017] [18] R. McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1-43. | Zbl 0265.08006
[00018] [19] R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Wadsworth and Brooks/Cole, Monterey 1987.
[00019] [20] V. I. Tumanov, Finite distributive lattices of quasivarieties, Algebra i Logika 22 (1983), 168-181 (in Russian). | Zbl 0548.08006