On connections between hypergraphs and algebras

Konrad Pióro

Archivum Mathematicum (2000)

  • Volume: 036, Issue: 1, page 45-60
  • ISSN: 0044-8753

Abstract

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The aim of the present paper is to translate some algebraic concepts to hypergraphs. Thus we obtain a new language, very useful in the investigation of subalgebra lattices of partial, and also total, algebras. In this paper we solve three such problems on subalgebra lattices, other will be solved in [[Pio4]]. First, we show that for two arbitrary partial algebras, if their directed hypergraphs are isomorphic, then their weak, relative and strong subalgebra lattices are isomorphic. Secondly, we prove that two partial algebras have isomorphic weak subalgebra lattices iff their hypergraphs are isomorphic. Thirdly, for an arbitrary lattice 𝐋 and a partial algebra 𝐀 we describe (necessary and sufficient conditions) when the weak subalgebra lattice of 𝐀 is isomorphic to 𝐋 .

How to cite

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Pióro, Konrad. "On connections between hypergraphs and algebras." Archivum Mathematicum 036.1 (2000): 45-60. <http://eudml.org/doc/248536>.

@article{Pióro2000,
abstract = {The aim of the present paper is to translate some algebraic concepts to hypergraphs. Thus we obtain a new language, very useful in the investigation of subalgebra lattices of partial, and also total, algebras. In this paper we solve three such problems on subalgebra lattices, other will be solved in [[Pio4]]. First, we show that for two arbitrary partial algebras, if their directed hypergraphs are isomorphic, then their weak, relative and strong subalgebra lattices are isomorphic. Secondly, we prove that two partial algebras have isomorphic weak subalgebra lattices iff their hypergraphs are isomorphic. Thirdly, for an arbitrary lattice $\mathbf \{L\}$ and a partial algebra $\mathbf \{A\}$ we describe (necessary and sufficient conditions) when the weak subalgebra lattice of $\mathbf \{A\}$ is isomorphic to $\mathbf \{L\}$.},
author = {Pióro, Konrad},
journal = {Archivum Mathematicum},
keywords = {hypergraph; subalgebras (relative; strong; weak); subalgebra lattices; partial algebra; hypergraph; strong subalgebras; subalgebra lattices; partial algebra},
language = {eng},
number = {1},
pages = {45-60},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On connections between hypergraphs and algebras},
url = {http://eudml.org/doc/248536},
volume = {036},
year = {2000},
}

TY - JOUR
AU - Pióro, Konrad
TI - On connections between hypergraphs and algebras
JO - Archivum Mathematicum
PY - 2000
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 036
IS - 1
SP - 45
EP - 60
AB - The aim of the present paper is to translate some algebraic concepts to hypergraphs. Thus we obtain a new language, very useful in the investigation of subalgebra lattices of partial, and also total, algebras. In this paper we solve three such problems on subalgebra lattices, other will be solved in [[Pio4]]. First, we show that for two arbitrary partial algebras, if their directed hypergraphs are isomorphic, then their weak, relative and strong subalgebra lattices are isomorphic. Secondly, we prove that two partial algebras have isomorphic weak subalgebra lattices iff their hypergraphs are isomorphic. Thirdly, for an arbitrary lattice $\mathbf {L}$ and a partial algebra $\mathbf {A}$ we describe (necessary and sufficient conditions) when the weak subalgebra lattice of $\mathbf {A}$ is isomorphic to $\mathbf {L}$.
LA - eng
KW - hypergraph; subalgebras (relative; strong; weak); subalgebra lattices; partial algebra; hypergraph; strong subalgebras; subalgebra lattices; partial algebra
UR - http://eudml.org/doc/248536
ER -

References

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  1. Bartol W., Weak subalgebra lattices, Comment. Math. Univ. Carolinae 31 (1990), 405–410. (1990) Zbl0711.08007MR1078473
  2. Bartol W., Weak subalgebra lattices of monounary partial algebras, Comment. Math. Univ. Carolinae 31 (1990), 411–414. (1990) Zbl0711.08007MR1078474
  3. Bartol W., Rosselló F., Rudak L., Lectures on Algebras, Equations and Partiality, Technical report B–006, Univ. Illes Balears, Dept. Ciencies Mat. Inf, ed. Rosselló F., 1992. (1992) 
  4. Berge C., Graphs and Hypergraphs, North-Holland, Amsterdam 1973. (1973) Zbl0254.05101MR0357172
  5. Birkhoff G., Frink O., Representation of lattices by sets, Trans. AMS 64 (1948), 299–316. (1948) MR0027263
  6. Burmeister P., A Model Theoretic Oriented Approach to Partial Algebras, Math. Research Band 32, Akademie Verlag, Berlin, 1986. (1986) Zbl0598.08004MR0854861
  7. Evans T., Ganter B., Varieties with modular subalgebra lattices, Bull. Austr. Math. Soc. 28 (1983), 247–254. (1983) Zbl0545.08010MR0729011
  8. Grätzer G., Universal Algebra, second edition, Springer-Verlag, New York 1979. (1979) MR0538623
  9. Grätzer G., General Lattice Theory, Akademie-Verlag, Berlin 1978. (1978) MR0504338
  10. Grzeszczuk P., Puczyłowski E. R., On Goldie and dual Goldie dimensions, J. Pure Appl. Algebra 31(1984) 47–54. (1984) Zbl0528.16010MR0738204
  11. Grzeszczuk P., Puczyłowski E. R., On infinite Goldie dimension of modular lattices and modules, J. Pure Appl. Algebra 35(1985) 151–155. (1985) Zbl0562.16014MR0775467
  12. Jónsson B., Topics in Universal Algebra, Lecture Notes in Mathemathics 250, Springer-Verlag, 1972. (1972) MR0345895
  13. Kiss E. W., Valeriote M. A., Abelian algebras and the Hamiltonian property, J. Pure Appl. Algebra 87 (1993), 37–49. (1993) Zbl0779.08004MR1222175
  14. Lukács E., Pálfy P. P., Modularity of the subgroup lattice of a direct square, Arch. Math. 46 (1986), 18–19. (1986) Zbl0998.20500MR0829806
  15. Pálfy P. P., Modular subalgebra lattices, Alg. Univ. 27 (1990), 220–229. (1990) MR1037863
  16. Pióro K., On some non–obvious connections between graphs and unary partial algebras, - to appear in Czechoslovak Math. J. Zbl1046.08002MR1761388
  17. Pióro K., On the subalgebra lattice of unary algebras, Acta Math. Hungar. 84(1–2) (1999), 27–45. (1999) Zbl0988.08004MR1696550
  18. Pióro K., On a strong property of the weak subalgebra lattice, Alg Univ. 40(4) (1998), 477–495. (1998) MR1681837
  19. Pióro K., On some properties of the weak subalgebra lattice of a partial algebra of a fixed type, - in preparation. 
  20. Sachs D., The lattice of subalgebras of a Boolean algebra, Canad. J. Math. 14 (1962), 451–460. (1962) MR0137666
  21. Shapiro J., Finite equational bases for subalgebra distributive varieties, Alg. Univ. 24 (1987), 36–40. (1987) MR0921528
  22. Shapiro J., Finite algebras with abelian properties, Alg. Univ. 25 (1988), 334–364. (1988) MR0969156

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