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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