# 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

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topAdaricheva, 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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- [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] K. V. Adaricheva, W. Dziobiak and V. A. Gorbunov, The lattices of quasivarieties of locally finite quasivarieties, preprint. Zbl0937.06002
- [4] M. K. Bennett, Biatomic lattices, Algebra Universalis 24 (1987), 60-73. Zbl0643.06003
- [5] G. Birkhoff and M. K. Bennett, The convexity lattice of a poset, Order 2 (1985), 223-242. Zbl0591.06009
- [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] W. Dziobiak, On atoms in the lattice of quasivarieties, Algebra Universalis 24 (1987), 31-35. Zbl0642.08003
- [8] R. Freese and J. B. Nation, Congruence lattices of semilattices, Pacific J. Math. 44 (1973), 51-58. Zbl0287.06002
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- [10] V. A. Gorbunov, Lattices of quasivarieties, Algebra i Logika 15 (1976), 436-457 (in Russian). Zbl0359.06014
- [11] V. A. Gorbunov and V. I. Tumanov, A class of lattices of quasivarieties, ibid. 19 (1980), 59-80 (in Russian).
- [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] G. Grätzer, General Lattice Theory, Birkhäuser, Basel 1979. Zbl0436.06001
- [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] 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] 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] A. I. Mal'cev, Algebraic Systems, Springer, 1973.
- [18] R. McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1-43. Zbl0265.08006
- [19] R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Wadsworth and Brooks/Cole, Monterey 1987.
- [20] V. I. Tumanov, Finite distributive lattices of quasivarieties, Algebra i Logika 22 (1983), 168-181 (in Russian). Zbl0548.08006

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