# On connections between hypergraphs and algebras

Archivum Mathematicum (2000)

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

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topPióro, 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

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

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