The subalgebra lattice of a finite algebra

Konrad Pióro

Open Mathematics (2014)

  • Volume: 12, Issue: 7, page 1052-1108
  • ISSN: 2391-5455

Abstract

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The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.

How to cite

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Konrad Pióro. "The subalgebra lattice of a finite algebra." Open Mathematics 12.7 (2014): 1052-1108. <http://eudml.org/doc/269100>.

@article{KonradPióro2014,
abstract = {The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.},
author = {Konrad Pióro},
journal = {Open Mathematics},
keywords = {Directed hypergraph; Finite algebra; Finite lattice; Subalgebra; Subalgebra lattice; Distributive lattice; Partial; directed hypergraph; finite algebra; finite lattice; subalgebra lattice; distributive lattice; partial},
language = {eng},
number = {7},
pages = {1052-1108},
title = {The subalgebra lattice of a finite algebra},
url = {http://eudml.org/doc/269100},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Konrad Pióro
TI - The subalgebra lattice of a finite algebra
JO - Open Mathematics
PY - 2014
VL - 12
IS - 7
SP - 1052
EP - 1108
AB - The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.
LA - eng
KW - Directed hypergraph; Finite algebra; Finite lattice; Subalgebra; Subalgebra lattice; Distributive lattice; Partial; directed hypergraph; finite algebra; finite lattice; subalgebra lattice; distributive lattice; partial
UR - http://eudml.org/doc/269100
ER -

References

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  1. [1] Bartol W., Introduction to the Theory of Partial Algebras, In: Lectures on Algebras, Equations and Partiality, Technical report B-006, Universitat de les Illes Balears, Palma, 1992, 113–137 
  2. [2] Berge C., Graphs and Hypergraphs, North-Holland Math. Library, 6, North-Holland, Amsterdam, 1973 http://dx.doi.org/10.1016/S0924-6509(09)70330-7 
  3. [3] Berge C., Hypergraphs, North-Holland Math. Library, 45, North-Holland, Amsterdam, 1989 
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  5. [5] Crawley P., Dilworth R.P., Algebraic Theory of Lattices, Prentice Hall, Englewood Cliffs, 1973 Zbl0494.06001
  6. [6] Grätzer G., General Lattice Theory, Pure and Applied Mathematics, 75, Academic Press, New York-London, 1978 http://dx.doi.org/10.1007/978-3-0348-7633-9 
  7. [7] Grätzer G., Universal Algebra, 2nd ed., Springer, New York, 1979 
  8. [8] Johnson J., Seifert R.L., A Survey of Multi-Unary Algebras, University of California, Berkeley, 1967 
  9. [9] Jónsson B., Topics in Universal Algebra, Lecture Notes in Math., 250, Springer, Berlin, 1972 Zbl0225.08001
  10. [10] Ore O., Theory of Graphs, AMS Colloq. Publ., 38, American Mathematical Society, Providence, 1962 Zbl0105.35401
  11. [11] Pióro K., On the subalgebra lattice of unary algebras, Acta Math. Hungar., 1999, 84(1–2), 27–45 http://dx.doi.org/10.1023/A:1006694618179 Zbl0988.08004
  12. [12] Pióro K., On connections between hypergraphs and algebras, Arch. Math. (Brno), 2000, 36(1), 45–60 Zbl1045.05070
  13. [13] Pióro K., On some non-obvious connections between graphs and unary partial algebras, Czechoslovak Math. J., 2000, 50(125), 295–320 http://dx.doi.org/10.1023/A:1022418818272 Zbl1046.08002
  14. [14] Pióro K., On subalgebra lattices of a finite unary algebra I, Math. Bohem., 2001, 126(1), 161–170 Zbl0978.08003

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