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 $\mathbf {L}$.
@article{107717,
author = {Konrad Pi\'oro},
title = {On connections between hypergraphs and algebras},
journal = {Archivum Mathematicum},
volume = {036},
year = {2000},
pages = {45-60},
zbl = {1045.05070},
mrnumber = {1751613},
language = {en},
url = {http://dml.mathdoc.fr/item/107717}
}
Pióro, Konrad. On connections between hypergraphs and algebras. Archivum Mathematicum, Tome 036 (2000) pp. 45-60. http://gdmltest.u-ga.fr/item/107717/
Weak subalgebra lattices, Comment. Math. Univ. Carolinae 31 (1990), 405–410. (1990) | MR 1078473 | Zbl 0711.08007
Weak subalgebra lattices of monounary partial algebras, Comment. Math. Univ. Carolinae 31 (1990), 411–414. (1990) | MR 1078474 | Zbl 0711.08007
Lectures on Algebras, Equations and Partiality, Technical report B–006, Univ. Illes Balears, Dept. Ciencies Mat. Inf, ed. Rosselló F., 1992. (1992)
Graphs and Hypergraphs, North-Holland, Amsterdam 1973. (1973) | MR 0357172 | Zbl 0254.05101
Representation of lattices by sets, Trans. AMS 64 (1948), 299–316. (1948) | MR 0027263
A Model Theoretic Oriented Approach to Partial Algebras, Math. Research Band 32, Akademie Verlag, Berlin, 1986. (1986) | MR 0854861 | Zbl 0598.08004
Varieties with modular subalgebra lattices, Bull. Austr. Math. Soc. 28 (1983), 247–254. (1983) | MR 0729011 | Zbl 0545.08010
Universal Algebra, second edition, Springer-Verlag, New York 1979. (1979) | MR 0538623
General Lattice Theory, Akademie-Verlag, Berlin 1978. (1978) | MR 0504338
On Goldie and dual Goldie dimensions, J. Pure Appl. Algebra 31(1984) 47–54. (1984) | MR 0738204 | Zbl 0528.16010
On infinite Goldie dimension of modular lattices and modules, J. Pure Appl. Algebra 35(1985) 151–155. (1985) | MR 0775467 | Zbl 0562.16014
Topics in Universal Algebra, Lecture Notes in Mathemathics 250, Springer-Verlag, 1972. (1972) | MR 0345895
Abelian algebras and the Hamiltonian property, J. Pure Appl. Algebra 87 (1993), 37–49. (1993) | MR 1222175 | Zbl 0779.08004
Modularity of the subgroup lattice of a direct square, Arch. Math. 46 (1986), 18–19. (1986) | MR 0829806 | Zbl 0998.20500
Modular subalgebra lattices, Alg. Univ. 27 (1990), 220–229. (1990) | MR 1037863
On some non–obvious connections between graphs and unary partial algebras, - to appear in Czechoslovak Math. J. | MR 1761388 | Zbl 1046.08002
On the subalgebra lattice of unary algebras, Acta Math. Hungar. 84(1–2) (1999), 27–45. (1999) | MR 1696550 | Zbl 0988.08004
On a strong property of the weak subalgebra lattice, Alg Univ. 40(4) (1998), 477–495. (1998) | MR 1681837
On some properties of the weak subalgebra lattice of a partial algebra of a fixed type, - in preparation.
The lattice of subalgebras of a Boolean algebra, Canad. J. Math. 14 (1962), 451–460. (1962) | MR 0137666
Finite equational bases for subalgebra distributive varieties, Alg. Univ. 24 (1987), 36–40. (1987) | MR 0921528
Finite algebras with abelian properties, Alg. Univ. 25 (1988), 334–364. (1988) | MR 0969156