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2014 | 12 | 7 | 1052-1108
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The subalgebra lattice of a finite algebra

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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.
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  • [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] 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] Berge C., Hypergraphs, North-Holland Math. Library, 45, North-Holland, Amsterdam, 1989
  • [4] Burmeister P., A Model Theoretic Oriented Approach to Partial Algebras I, Math. Res., 32, Akademie, Berlin, 1986
  • [5] Crawley P., Dilworth R.P., Algebraic Theory of Lattices, Prentice Hall, Englewood Cliffs, 1973
  • [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] Grätzer G., Universal Algebra, 2nd ed., Springer, New York, 1979
  • [8] Johnson J., Seifert R.L., A Survey of Multi-Unary Algebras, University of California, Berkeley, 1967
  • [9] Jónsson B., Topics in Universal Algebra, Lecture Notes in Math., 250, Springer, Berlin, 1972
  • [10] Ore O., Theory of Graphs, AMS Colloq. Publ., 38, American Mathematical Society, Providence, 1962
  • [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
  • [12] Pióro K., On connections between hypergraphs and algebras, Arch. Math. (Brno), 2000, 36(1), 45–60
  • [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
  • [14] Pióro K., On subalgebra lattices of a finite unary algebra I, Math. Bohem., 2001, 126(1), 161–170
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